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PHYSICS REPORTS (Review Section of Physics Letters) 52, No. 3 (1979) 133—201. North-Holland Publishing Company
EXPERIMENTS IN PHENOMENOLOGICAL ELECTRODYNAMICS AND
THE ELECTROMAGNETIC ENERGY-MOMENTUM TENSOR*
I. BREVIK
Institute of Theoret,cal Physics, University of Trondheim, N-7034 Trondheim-NTH, Norway
and Luftkrigsskolen, Trondheim-Mil, 7000 Trondheim, Norway
Received November 1978
Contents:
1. Introduction
1.1. Presentation ofthe problem. Survey
1.2. Definition of some energy-momentum tensors
2. Electrostriction and magnetostriction in static and
quasi-stationary fields
2.1. Introductory remarks
2.2. Condenser partially immersed in a dielectric
liquid
2.3. Hakim and Higham’s pressure experiment
2.4. Goetz and Zahn’s non-equilibrium experiment
3. Measurement of Abraham’s force in quasi-stationary
fields
3.1. The experiment of G.B. Walker, D.G. Lahoz and
G. Walker
3.2. James’s experiment
135
135
139
142
142
143
146
149
152
4. Optical phenomena
4.1. General remarks
4.2. Jones and Richards’s radiation pressure experiment
4.3. Ashkin and Dziedzic’s experiment on the radiation pressure on a liquid surface
4.4. Torque experiments. Remarks on the quantized
theory
4.5. Further experiments in optics. Measurement of
the integrated effect of the Abraham term?
4.6. Further remarks and conclusion
Appendix
References
152
155
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EXPERIMENTS IN PHENOMENOLOGICAL
ELECTRODYNAMICS AND THE
ELECTROMAGNETIC ENERGY-MOMENTUM
TENSOR
I. BREVIK
Institute of Theoretical Physics, University of Trondheim, N-7034 Trondheim-NTH, Norway
and Luftkrigsskolen, Trondheim-Mil, 7000 Trondheim, Norway
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
135
Abstract:
The microscopic and the macroscopic (phenomenological) approach to the electromagnetic theory in matter can be looked upon as
complementary to each other. In view of the complexity of the deductive microscopic approach it is desirable to use experimental
information to test the predictions of the more simple macroscopic theory. The purpose of the present paper is to review and give
the theory of experiments and Gedanken experiments in relation to various expressions for the electromagnetic energy-momentum
tensor. The paper is divided into three parts. The first part discusses the electrostriction effect, mainly in electrostatic experiments.
It turns out that both the Hakim—Higham experiment (1962) and the Goetz—Zahn experiment (1958) agree with the predictions of the
Helmholtz theory and disagree with the predictions of the Einstein—Laub theory. The second part of the paper is concerned with quasistationary crossed electric and magnetic fields. The electrostriction effect is here unimportant. Both the experiment of Walker et a!.
(1975) and that of James (1968) support the Abraham tensor in contradiction to the Minkowski tensor. The third and most extensive
part of the paper discusses optical phenomena, in which case a direct observation of the fluctuating Abraham force cannot be made.
Under usual circumstances, it turns out to be simplest to use the divergence-free Minkowski tensor. This tensor describes satisfactorily
all existing force or torque experiments in optics that we are aware of. Some attention is given also to the tensor alternative recently
introduced by Peierls (1976).
1. Introduction
1.1. Presentation of the problem. Survey
It is now nearly 70 years since Minkowski [1, 2] and Abraham [3] gave their famous expressions
for the electromagnetic four-tensor in a material medium and thereby started a debate that has
been going on more or less intensively up to this day. It is perhaps surprising that the electrodynamics of material media should be so delicate that it has hindered common agreement among
physicists on such a central entity as the electromagnetic energy-momentum tensor for several
decades. One reason for this circumstance is that the problem has no unique solution. Phenomenological electrodynamics is a simplified description of a complicated interaction situation
where both external fields, internal fields, and the constituent molecules in the medium are involved.
One can construct the microscopical energy-momentum tensor of the closed system (field plus
matter) and average it over suitable space-time regions in the usual way, thereby obtaining besides
the pure field or matter terms a number of complicated correlation terms. There exists however no
unique prescription for the separation of this total energy-momentum tensor into a field part and
a matter part. To obtain a definite energy-momentum tensor for the field one may impose extra
criteria built on physical arguments, for instance the criterion due to von Laue [4] and Møller [5]
that the propagation velocity of the energy of a light wave in a moving body shall transform like a
particle velocity under Lorentz transformations. Minkowski’s tensor satisfies this transformation
criterion, whereas Abraham’s tensor does not. A criterion of this type is however a test of a tensor’s
convenience rather than its correctness. The von Laue—Møller transformation criterion is an
illustrative example of the convenience of Minkowski’s tensor in optical situations, although it is
widely recognized now that Abraham’s tensor is also capable of describing optical experiments.
In general the problem on the phenomenological level is to choose the tensor form which, as simply
and directly as possible, predicts experimental effects in agreement with those one actually measures. A tensor alternative is rejectable only ifit definitely disagrees with observation. On the phenomenological level there is therefore an urgent need for critical experiments of various kinds.
Another main reason for the long-lasting discussion on the electromagnetic energy-momentum
tensor is the smallness of the differing terms. The energy-momentum tensor is intimately related to
the ponderomotive force density; in the Abraham—Minkowski case the differing force density term
136
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
is called the “Abraham term”. It has usually been concluded in the past that an experimental
detection of this term would be almost hopeless. The last years have however seen a radical change
of this situation. Using time-varying, crossed electric and magnetic fields, giving a time-dependent
electromagnetic momentum density, James [6], and independently, G.B. Walker, D.G. Lahoz and
G. Walker [7], were able to measure the Abraham term directly. We consider these experiments
to be perhaps the most important progress in phenomenological electrodynamics during the last
years. Moreover, beautiful experiments of importance for the electromagnetic energy-momentum
tensor have also been made with optical fields. For example, we may mention Ashkin and Dziedzic’s
observation [8] of the outward bulge on a water surface resulting when a strong laser beam
impinges upon it. The importance of this experiment lies primarily in the fact that it is a local
experiment, which shows also the direction of the force acting in the free surface. As far as we know,
this is the first time it has been shown explicitly that the surface force from an optical field is
directed from the medium towards the vacuum side. The description of the effect requires the
components of the electromagnetic energy-momentum tensor in the medium just inside the
boundary. Of course there are numerous experiments showing the over-all effect (force or torque)
from an optical field on a dielectric medium. The torque on a dielectric glass plate resulting when
an optical wave propagates through it obliquely had already been measured by Barlow [9] in 1912.
The plate was surrounded by practically a vacuum (actually hydrogen gas at low pressure). To
describe the Barlow kind of experiments it is however not necessary to know the electromagnetic
energy-momentum tensor in the medium; the overall effect can be computed simply from the energymomentum tensor on the vacuum side just outside the boundary. The differences between the
tensor alternatives disappear in vacuum, and so it is clear that these kind of experiments are not
critical. Observation of the total force or torque on a dielectric body does not give any information
as to whether the local surface force is directed normally to the surface, tangentially to it, or in some
other direction.
Some years ago the author wrote two review papers on the electromagnetic energy-momentum
tensor in material media [10, 11], emphasizing the experimental aspects of the problem. In view
of the recent progress in the field, especially on the experimental side, we have thought it would
be useful to give a new survey over important experimental achievements and their relation to the
energy-momentum tensor. We mainly consider recent experiments, but include also some important earlier material, especially in section 2 on electrostriction. Further, our aim has been to supply
with theoretical developments whenever appropriate throughout the text, mainly in section 4 on
optical phenomena. Some proposals of possible new experiments are also given, in the hope to
excite some experimentalists.
The theory is mainly classical and kept on the phenomenological level throughout. One might
perhaps wonder if it is really a favourable line of approach to pursue the phenomenological theory
and lean heavily on experimental information, such as we do. The basic equations are after all
Maxwell’s equations in intermolecular space, and it may seem advantageous to start from them
and derive the theory of all macroscopical phenomena by appropriate averaging procedures.
Important progress along the microscopical line of approach has really also been obtained in
recent years, notably by de Groot and Suttorp [12], and by Peierls [13]. We think however that
both the macroscopical and the microscopical methods are welljustified because each of them has
definite advantages. The microscopical method is advantageous from a fundamental viewpoint,
but its drawback is that it easily becomes formally complicated and tends to obscure a simple
physical interpretation. The advantage of the simple and less fundamental macroscopical method
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
137
is its close connection with observation. The two methods may be looked upon as complementary
to each other.
The next subsection summarizes some actual tensor expressions and their corresponding force
densities. Section 2 then concentrates on measurements of pressures and forces in static and quasistationary (slowly varying) fields. Of main interest in our context is actually the electrostriction
effect (or the analogous magnetostriction effect). The electrostriction term, as derived from the
Helmholtz variational principle, is not included in Abraham’s and Minkowski’s tensors. In fact
it is actually unnecessary to take it into account, because under conditions of mechanical equilibrium electrostriction gives no contribution to the observable over-all force or torque on a dielectric body. It is possible however to measure the electrostriction effect directly under special circumstances. The Hakim—Higham experiment [14], considered in section 2.3, is a local experiment
demonstrating the Helmholtz excess pressure produced in a non-polar isotropic liquid by an
electric field. Another important aspect of the electrostriction effect is that, under conditions of
mechanical non-equilibrium, it is possible to measure even its total effect on a solid body immersed
in a dielectric liquid. Non-equilibrium conditions can be produced by imposing an alternating
electric field of such a high frequency that the elastic pressure cannot equilibrate throughout the
liquid. The effect is demonstrated in the Goetz—Zahn experiment [15, 16]. We give some theory of
the non-equilibrium aspect of this experiment, since this aspect does not seem to have been considered before. The general conclusion to be drawn is that, in view of the importance of the electrostriction effect in special cases, one should always bear its existence in mind, and add the Helmholtz
electrostriction term to Abraham’s or Minkowski’s tensors when necessary.
Section 3 surveys and comments upon the Walker—Lahoz experiment and the James experiment
that were referred to above.
Section 4 on optical phenomena is the most extensive one in the paper. It may be worthwhile
already here to summarize the basic picture we advocate for the propagation of optical fields in an
infinite, non-dispersive, isotropic medium. In contradistinction to the quasi-stationary case one
cannot in the optical region make a direct measurement of the rapidly fluctuating Abraham term.
Let us assume that the field is a pulse of finite dimensions. At the leading edge of the pulse there
is generated a mechanical momentum, which will be assumed to be distributed in space in proportion to the field (Abraham) momentum density. The generated mechanical momentum dies away
at the rear end of the pulse. The reason for the production of this momentum is twofold: (1) The
Abraham term acting at the leading edge. Addition of the corresponding mechanical momentum
to the Abraham field momentum yields simply Minkowski’s momentum, which accordingly turns
out to be a composite quantity. (2) The electrostriction force at the leading edge, similarly producing
an accompanying momentum.
It should immediately be recognized that our assumption about the mechanical momentum
being distributed in space in proportion to the field momentum density cannot on physical grounds
be expected to hold in more complex situations. Notably, we have ignored the elastic coupling
between the constituent molecules. The picture should be expected to be most appropriate for
fluids. (Ifthe body is a solid, kept in fixed position, the elastic contribution from the clamping force
must also be considered.) Moreover, one should in principle consider also the transverse mechanical
momentum produced by the inward electrostriction forces at the lateral boundary of the pulse.
The above picture therefore seems to be most appropriate in the case of a wide pulse, for which
the transverse momentum is ignorable, cf. also Peierls [13]. We think, however, that the picture
should be appropriate also for a stationary narrow beam, since the transverse forces should
138
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
essentially act to produce a mechanical stress (rather than momentum) within the beam that
should not interfere seriously with the propagation of longitudinal mechanical momentum. In
any case, we shall assume throughout that the picture gives a reasonable description of the local
physical state of medium plus field.
Now it turns out to be possible also in optics, in most experimental situations, to omit the
electrostriction term from the formalism without any observational consequences. Under usual
circumstances, such as when mechanical equilibrium prevails in a dielectric liquid, the electrostriction effect has no effect upon the total force or torque on a body immersed in the liquid.
Accordingly, we shall in the following usually omit the electrostriction term, except in the cases
where electrostriction is directly subject to measurement. See also the later discussion in section 4.1.
To avoid misunderstanding, we stress that we do not claim that the electrostrictive contribution
to the force is non-existent in optics; the neglection is purely a matter of convenience. Further, this
formal simplification obviously has no bearing on the fact that the present theory is macroscopic
and not microscopic.
The omission of electrostriction means that the Minkowski momentum is imagined to be the
total momentum of the travelling disturbance in the medium. The effectiveness of the Minkowski
tensor in explaining optical phenomena is demonstrated, in particular, by the accurate Jones—
Richards radiation pressure experiment [17]. The practical effectiveness of Minkowski’s tensor,
together with its attractive formal simplicity, were actually the main reasons why the first part
of the author’s earlier investigation [10] was devoted to this tensor.
Section 4 starts with general remarks on travelling electromagnetic waves and the accompanying mechanical momenta, as interpreted by various energy-momentum tensors. Section 4.2
reviews and explains in various ways the Jones—Richards experiment. Application of the Peierls
tensor [13] is also examined. Section 4.3 presents an extensive theory of the Ashkin—Dziedzic
radiation pressure experiment [8]. This experiment demonstrates how a free liquid surface bulges
outwards when a laser beam impinges upon it. The first part of that section, dealing with incompressible theory, is in essence similar to the theory recently developed by Lai and Young [19],
but in contradistinction to these authors we assume a Gaussian intensity distribution of the laser
beam. The next part of section 4.3 analyses the transient pressure variation caused by the electrostriction effect. We are not aware that such an analysis has been given before. Section 4.4 discusses
the Roosen—Imbert torque experiment [48] and also some quantal aspects related to angular
momentum. A proposal is given of measuring the photon spin angular momentum in a dielectric
liquid, in analogy with the Beth experiment [55] which assumes vacuum surroundings. The
section concludes with some comments on the uncertainty relation in quantum mechanics.
Section 4.5 analyses further experiments in optics, among them the recent levitation experiment
of Ashkin and Dziedzic [20, 21], and also Roosen et al. [63, 64]. It is concluded that these experiments are not able to distinguish between the different electromagnetic energy-momentum tensors.
Section 4.5 further contains some remarks on relativity aspects, and finally it discusses some
experimental possibilities that might be able to demonstrate the existence of the integrated Abraham term (integrated over the volume of a test body, or over the whole story of an electromagnetic
pulse) also in optics. The final section 4.6 discusses, among other things, why theoretical arguments
based on the global conservation theorems for a closed system cannot be used to give an unambiguous derivation of the correct field momentum density in matter. This discussion is motivated
by the conflicting opinions that have been stated in the literature.
Readers interested in a general introduction to the subject are referred to Møller’s book [5].
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
139
Recent reviews have been given by Ginzburg [22, 23], Lahoz [24] (who also gives a thorough
survey of the earlier historical development), Robinson [25] and Skobel’tsyn [26]. Another recent
review, written from the standpoint of continuum mechanics, has been given by Pao [77].
1.2. Definition of some energy-momentum tensors
This subsection summarizes basic expressions for some energy-momentum tensors and force
densities that are in the focus of interest. The expressions, written in SI-units, refer to the inertial
frame where the medium is at rest.
The simplest form is that of Minkowski, henceforth designated by a superscript M:
S~= —E1D,~ HLBk + 2t5~~(ED + HB)
(1.la)
S~=~-Sj~, SM=EXH
(1.lb)
—
M
S~= icg~,
=
—
Dx B
=
(1.lc)
g
wM
WM,
=
~(E~D + H B).
.
(1.ld)
Here SM denotes Poynting’s vector, gM the momentum density, and WM the energy density. The
tensor is in general non-symmetric. If dispersion is negligible one can write, permitting the medium
to be spatially inhomogeneous and anisotropic, D. = ;kEk, B. = ptkHk (index k summed from 1 to
3). The components of the four-force density are given by
1M =
pE + j x B
—
~E~EkV~tk
—
(1.2a)
(1.2b)
f4M~E.j
where p and j are the external charge and current densities. In deriving these expressions the
symmetry property of e~and Pa was used. The two first terms in (1 .2a) give the density of force
acting on the external charges and currents, whereas the two last terms give the density of force
acting on the medium itself in regions of spatial inhomogeneity, for instance in the boundary
layer between two media. Of main interest in the following is the special case where the medium is
isotropic and non-conducting,
2Ve ~H2V p = 0, j = 0, ;~= ~ik’ Pik = Ith~k. Then (1.2) reduce to
= —~E
1z, f4M = 0.
(1.3)
—
In this special case the space components (1.1 a) form a symmetrical three-dimensional tensor,
while the non-symmetry still persists for the2,fourth
row and
column
in the of
energy-momentum
expressing
Planck’s
principle
inertia of energy,
tensor.
implies
the relation
= S/c
does
notThis
hold
in thethat
Minkowski
case.g One
finds instead ~M = CrILrSM/C2, where i~is the relative
permittivity and Pr the relative permeability.
It is important to notice the following point: Minkowski’s tensor does not describe electrostriction
or magnetostriction. This tensor, therefore, is unable to give a complete description of the local
electromagnetic state in the medium. Yet it turns out to be possible in most cases simply to omit
the electrostriction and magnetostriction terms from the formalism without any consequences
for the experimental predictions. For instance, in a measurement of the total electrostatic force on
140
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
a dielectric body immersed in a liquid, the total electrostriction force will be balanced by an increased hydrostatic pressure in the liquid. Just this vanishing of the net electrostriction or magnetostriction effect in most instances makes it possible to apply the simple Minkowski expression
directly in these cases. There are on the other hand special situations in which electrostriction or
magnetostriction terms are of direct experimental importance, so that their neglect is no longer
allowable. We shall discuss situations of this kind in the next section.
The next tensor that we shall consider is the symmetric one of Abraham, designated by a superscript A:
S~= —~{E~D~
+ EkDI)
SA=ExH,
WA
=
—
(1.4a)
~(H~Bk+ H~B~)
+ ~ö~k(E~D + H B)
YA2EXH
(1.4b)
~(E D + H B).
(1.4c)
Note that Planck’s principle is satisfied by (1.4b).
The components of the four-force density are
JA = 1M
—
~-
(ED,~ EkD)
~-
—
—
~-
~-
(HBk
f~A= £E.j,
—
HkB~+
~-
(D x B
—4 E x H)
(1.5a)
(1.5b)
where 1M is given by (1 .2a). In the special case of an isotropic, non-conducting body one has
1A = 1M
+~
1 ~ (E x H),
f~
=
0,
(1.6)
C
where 1M is given by (1.3). The second term in the expression for 1A is called the Abraham term.
Note that it is different from zero, although usually very small, even in the interior of an isotropic
and homogeneous body. Similarly as in the Minkowski case, electrostriction and magnetostriction
contributions have been omitted in the Abraham tensor (1.4).
Another energy-momentum tensor that has gained some attention is the one introduced by
Einstein and Laub [27], henceforth designated by a superscript E:
2)
(1.7a)
= —EIDk
H~Bk+ 2öIk(COE + p0H
—
SE=ExH,
9E4EXH
(1.7b)
2+ p 2).
(1.7c)
4(~0E
0H
The energy density component was actually not given by Einstein and Laub; we have added it
in the form (1.7c) so as to make the energy-momentum tensor, in the special case of non-magnetic
media, equal to one of the tensor forms recently derived by de Groot and Suttorp [12]. The force
density components obtained from (1.7) are in the general case
=
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
=
pE + Pci x H + (P~V)E + p0(M~V)H + Po~X H +
= 1M
+ ~V(E P + p0H~M) + ~ (D x B
f~=‘-(Ej
4E
141
(1.8a)
X
—4 E x H)
(1.8b)
+ E-P + p0H~I),
(1.8c)
where
D = s0E + P,
1M
B
=
p0(H + M),
(1.8d)
being given by (1.2a).
The force density in an isotropic non-conducting body can conveniently be written as
1E
fA + ~V(E- P + p0H M).
.
(1.9)
The Einstein—Laub tensor is meant to be sufficiently general to describe also electrostriction and
magnetostriction. These effects are taken into account by the divergence term in (1.8b) or (1.9).
One reason why this tensor has gained attention, is that the various terms in (1.8a) can be easily
interpreted in the case where the medium is non-magnetic. The two first terms to the right in (1.8a)
then describe the Lorentz force, the next term has the same form as the expression for the force
acting on an electric dipole in an external field E, and finally one recognizes the force acting on
the polarization current (the terms containing M are absent). Although arguments of this kind
are physically appealing one cannot, however, look upon them as forming a physical derivation.
For instance, it may seem more natural to expect the presence of the “effective” Lorentz field rather
than the mean field E in the term (P~V)E. There are also correlation effects between the microscopical fields than may change the above simple picture.
The final alternative for the electromagnetic force density to be considered in this paper, is the
one recently given by Peierls [13]. The force density expression, applicable to the case when a
light wave propagates in an isotropic non-magnetic medium, is
2
1)](n2
1)(E~V)E i~
2 1)2E x B + ~ 2 1)E x B,
1” = c~[1 + (a + x)(n
0a(n
0(n
(1.10)
—
—
—
—
—
where a, x are numerical constants for which Peierls arrived at values a = = ~ in case of media
satisfying the Clausius—Mossotti relation. The derivation is based on microscopical arguments
and will not be considered in this paper. What is of interest for us is to search for possibilities to
test (1.10) by experiment, assuming a and r to be fixed but unspecified numbers. For the electromagnetic field part of the momentum density, Peierls adopted the Abraham value:
= gA
=4E x H,
(1.11)
and a simple calculation shows that (1.10) and (1.11) correspond to the following space components
of the energy-momentum tensor:
S~= —e~[n2 + r(n2
—
1)2]E~E~ POHtHk +
—
oöIk[l
—
a(n2
—
1)2]E2 + ~po5tkH2. (1.12)
142
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
In deriving this expression, the absence of external charges and currents was explicitly taken into
account.
In general, for any electromagnetic energy-momentum tensor S~(p, v going from 1 to 4), the
three-dimensional force density is given by
=
~k~ik
—
ôg1/~t.
—
(1.13)
Integration over a volume with fixed boundaries yields the components of the total force acting
on the interior region:
F.
=
j’fi
dV
=
—
JSiknk dS
—
~-
Jgj dv,
(1.14)
where nk is the kth component of the outward normal to the surface element dS. An important
special case is when the electromagnetic field is stationary and oscillating so fast that only the
mean effect averaged over a period (cf. optical fields) is observable. Then the last term in (1.14)
can be set equal to zero, and we obtain
F~=
—
JSiknk dS.
(1.15)
2. Electrostriction and magnetostriction in static and quasi-stationary fields
2.1. Introductory remarks
When attempting to test the electromagnetic energy-momentum tensor experimentally, the
simplest situation one can imagine is a force measurement on a dielectric body in an electrostatic
field. The
relevant
of the energy-momentum
tensor
thistensor
case with
are the
space compo5a• The
stresscomponents
tensor is conventially
defined to be equal
to inthis
opposite
sign.
nents
We shall limit ourselves to fluid media. The free energy of a dielectric fluid in an electrostatic
field is
(2.1)
F=4JE.DdV.
Under reversible, isothermal conditions the variation of the free energy is equal to the mechanical
work done, and one derives the Helmholtz force density (see, for instance, [28])
f” = pE 4E2Ve + ~v[E2Pm(~)],
(2.2)
—
Pm
where Pm is the mass density of the fluid. The superscript H refers to the Helmholtz theory. The
magnetostatic case is analogous, and yields
fit
=j
xB
—
+H2Vp + ~V[H2Pm(~)
Pm
1•
We now combine these two cases, and write the composite force density as f.1’
(2.3)
=
—
ÔkS~ where
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
143
the three-dimensional tensor S~will henceforth be called the Helmholtz tensor. We find
S~= —E~D~ HLBk + ~t5ik[1
—
—
~~T]~’0
+ ~c5~k[1
—
(2.4)
In the previous subsection we mentioned that neither Minkowski’s nor Abraham’s tensor contains
any contribution from electrostriction or magnetostriction. If we want to include these effects,
we must replace the expression (1.la) by the expression (2.4). The Abraham expression (1.4a)
becomes simply equal to the Minkowski expression (1.la) when the medium is isotropic.
We noted earlier that the Einstein—Laub tensor is intended to be sufficiently general to include
electrostriction and magnetostriction effects. From the tensor family we have introduced there are
thus two members competing to describe phenomena where electrostriction or magnetostriction
are involved. The first alternative is the Helmholtz form (the natural extension of the Abraham—
Minkowski form), giving the following expression for the electrostatic force density in a fluid with
no external charges:
fH =
+~
—~E2V~
[E2P
(-~)]
(2.5)
T
(we omit henceforth the subscript m on Pm). The second alternative is the Einstein—Laub form,
giving the corresponding electrostatic force density
= (P~
V)E = —~E2V~
+ 4V(E~P).
(2.6)
The difference between (2.5) and (2.6) is contained in the electrostriction terms. The important
question is now: Is this difference in the electrostriction terms observable? As we shall see, the
answer depends on whether one measures the pressure distribution throughout the fluid or only
the total integrated force. Experimental evidence in the first case agrees with (2.5) and disagrees
with (2.6). Moreover, we shall investigate also a situation with quasi-stationary fields, where non-
equilibrium conditions allow an independent support for (2.5).
2.2. Condenser partially immersed in a dielectric liquid
We shall first give some supplementary comments on the well known situation shown in fig. 1,
where two parallel condenser plates are partially immersed in a dielectricliquid. When a horizontal
Fig. 1. Two condenser plates partially immersed in a dielectric liquid.
144
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
electric field E is applied between the plates, the liquid will rise within the condenser to some
equilibrium height h. This simple example is in fact sufficient to demonstrate the essential features
of the various force expressions in electrostatics. Even electrostriction has, as we shall see, an
important stabilizing effect.
What makes the liquid rise between the plates? Certainly not the electrostriction force: The
z-component of the total electrostriction force on the liquid between the plates can be written as
a surface integral which vanishes, at the upper surface (above the free surface) because the mass
density p = 0 there, at the lower surface (far beneath the plates) because E = 0. The single property
of the electrostriction force density expression needed to obtain this result, is that it is the gradient
of a scalar function which vanishes when p or E vanish. It follows immediately that a measurement
of the height h will not distinguish between the two force density expressions (2.5) and (2.6). The
elevation of the liquid can in both cases be calculated simply by considering the action of the force
density term —4E2VE in the boundary layer at the free surface.
Equating the corresponding vertical surface force density ~ — t~ 2to the gravity pressure
0)E
pgh, we obtain
h
2/(2pg).
(2.7)
0)E
As an example, consider benzene at 20°C, for which ~ = 2.28, p = 880 kg/m3. With
E = 3 x 106 V/m one finds the elevation h = 5.9 mm.
An additional effect ignored so far is the surface tension. This effect will play an important role
if the distance d between the plates is small. Let us digress to give a simple quantitative estimate,
assuming the dimensions of the plates in the y-direction to be large. The important parameter
here is a = (2x/pg)112, where ~ is the surface tension coefficient [29]. If d ~ a the influence from
surface tension will be negligible. The free surface between the plates will essentially remain as a
plane surface after the elevation in the condenser. Only in the vicinity of the walls may the free
surface be curved. At the walls themselves, the extra elevation of the free surface because of surface
tension will be equal to a(1 — sin 0)1/2, where 0 is the angle of contact (see fig. 1. We have assumed
for definiteness that the liquid wets the surface, so that 0 <90°.)In the case of an air—benzene
interface, ~x= 2.89 x 10_2 N/rn, corresponding to a = 2.6 mm. Accordingly, when d ~ 2.6 mm,
the position of the free surface is practically independent of the surface tension. On the other hand,
in the opposite extreme case d e~a, the surface tension will be very important and will cause the
liquid between the plates to rise approximatively to the height (a2/d) cos 0 in the absence of any
field.
Most experiments in this area are old. Lahoz and G. Walker [24] have recently undertaken to
repeat some experiments of this type, mainly in the analogous magnetostatic case with a paramagnetic saturated solution of Fe Cl
3 as liquid. Theory and experiment were found to agree to
within 2 %. Some electrostatic experiments were also carried out, with comparable results.
Now turn to a calculation of the hydrostatic pressure distribution in the liquid shown in fig. 1.
At this point the electrostriction effect becomes essential. Consider first the Helmholtz force
density expression (2.5). One then has for the pressure differences throughout the liquid, when
the notation refers to fig. 1 (the air pressures Pi and p5 can be assumed to be equal),
2[p(~/ôp)T (i~ c
P2
Pi = ~E
0)]
(2.8a)
2(~— ~)
(2.8b)
P3
P2 = pgh = ~-E
J~3 f~
5= ~EP(oC/aP)T.
(2.8c)
(i~—
=
c
—
—
—
—
—
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
145
The region between the lower edges of the plates (around point 4 in fig. 1) is a transition region
in which the electrostriction pressure gradually dies out with decreasing values of z. For the
numerical evaluation of (2.8) assume simply that the liquid is non-polar. Use of the Clausius—
Mossotti relation then gives
co)2/(6~o)
(2.9a)
P2
p1 = E2(fl
P3
P2 = ~-E2(E— ~)
(2.9b)
—
—
—
p
2(E —
3
p5
i~)(i~+
=
E
2c
(2.9c)
2.28, E = 3 x 106 V/rn, one finds
In
thePi example
with benzene mentioned
P2
= 21.8 N/rn2, p
2, pabove,
2. Figure 2 illustrates schema3
P2
=
51.0
N/rn
p5 1 =to72.8
tically the pressure variation along a path from 3points
5 in N/rn
fig 1. Note again that the difference
between the air pressures Pt and p
5 is negligible. (Actually, the ratio between the differences
(i’s Pi) and (p~ P2) is equal to the ratio between the densities of air and liquid.)*
—
0)/(6c0).
1~r=
—
—
—
—
—
80
Fig. 2. Sketch of the pressure variation along a path from points 1 to 5 in fig. 1, in the numerical example considered in the text. Note
the linear increase in pressure from 2 to 3.
Equation (2.9a) gives the answer to the following question: Why are the molecules in 2(~
the —
liquid
kept
together
when
the
free
surface
is
pulled
upwards
due
to
the
surface
force
density
~E
As a liquid cannot take up stresses, the column of liquid between the plates must be subjected to
compressive forces that produce an isotropic hydrostatic pressure greater than -~E2(E
If this
were not the case it would be impossible for the liquid to rise as a single column. This necessary
compressible force is provided just by the electrostriction force: Its action across the free surface
1—2 in fig. 1 is to give a vertical force which is directed downwards and which is stronger than the
surface force directed upwards at the same surface. This property is shown by the fact that (2.9a),
as a quadratic expression, is always non-negative. The electrostriction effect thus stabilizes the
system. Note again that the electrostriction effect does not contribute to the total elevating force
on the column of liquid. For the electrostrictive depressing surface force density ~E2p(ôc/ôp)~ at
the free surface is balanced by the equally large and opposite electrostriction force acting in the
inhomogeneous field region between the lower edges of the plates. The net force pulling up the
liquid acts at the free surface.
— ~.
*
Figure 2 appears to be different from the corresponding fig. 6-8 in [28].
146
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
Naturally, there are in addition horizontal electrostriction forces acting at the surfaces of the
plates. These forces are directed towards the interior region of the liquid. It must be borne in
mind that under conditions of mechanical equilibrium we cannot measure these forces by measuring the attractive force between the plates. The reason is that the electrostrictive contribution to
the attractive force is balanced by the increased elastic (i.e. hydrostatic) pressure in the liquid.
The measurable attractive force between the plates is simply 1~E2per unit surface area, i.e. the
same result as one would obtain using Abraham’s or Minkowski’s tensors (without electrostriction)
directly.
In magnetostatics, the paramagnetic case is just analogous. Some care is needed, however, in
the diamagnetic case where i~< Po~One will observe a depression of a diamagnetic liquid between
the plates, as 4H2(p
Po) <0. The magnetostriction force acts upwards at the free surface and
downwards in the region between the lower edges of the plates. The depressive force at the free
surface is stronger than the magnetostrictive force, resulting in a pressure P2 just below the free
surface which is greater than the pressure p
1 above it. Note that the magnetostatic analogue to
the expression (2.9a) is always non-negative, in the diamagnetic as well as in the paramagnetic case.
Quite generally, the condition P2
j~ is a necessary condition for stability.
The considerations so far have been based on the Helmholtz force density expression (2.5).
Ifwe instead start from the Einstein—Laub expression (2.6), we find immediately that no net surface
force is predicted to exist at all at the free surface, i.e. P2 = Pi~In the Einstein—Laub picture the
elevation of the liquid is interpreted to be due to volume forces of density ~V(E~P), which act in the
inhomogeneous region between the lower edges of the plates and tend to press the liquid upwards.
Integration of the vertical component of this force density over the region of inhomogeneity yields
the same result (2.7) for the elevation as before (see also the discussion in [24]). It is therefore clear
that an experimental test of the pressure distribution throughout the liquid is desirable to test
the theories. This subject will now be considered.
—
2.3. Hakim and Higham’s pressure experiment [14]
The arrangement was here a condenser with closely spaced cylindrical electrodes completely
immersed in a dielectric non-polar liquid. The liquids investigated were carbon tetrachioride and
n-hexane; the maximum average electric field between the electrodes was about 5 x 107 V/rn.
In such a strong field an appreciable amount of liquid will be drawn by electrostriction forces into
the field region and increase the density p and the pressure p locally. (Note that by p we mean the
elastic (or hydrostatic) pressure which, ignoring changes in entropy, is a function of the density p.
Thus p is the same as the pressure that would exist in the liquid at the same value of p, in the
absence of field.) The purpose of Hakim and Higham’s experiment was to measure the local
increase of pressure Ap with optical methods.
2Ve, present in (2.5) and (2.6), is negligible. The
It
should
first
be
observed
that
the
term
—
~E
permittivity increases somewhat because of the increased pressure, but when Ap ~ 1 atm as is the
case here, it was shown by Hakim and Higham that the change in e is very small. Using the Clausius—
Mossotti relation in (2.5), one thus finds in the Helmholtz case
Ap’~=
~E2p(ôe/ôp)~ = E2(~ ~) (~+ 2s
—
0)/(6E0).
Similarly, (2.6) yields in the Einstein—Laub case
2(~ se).
—
APE
=
~EP
=
~E
(2.10)
(2.11)
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
147
These two predictions for the excess pressure were compared in the following way: A light ray was
sent through the region between the electrodes, the region made inhomogeneous by the applied
static field. Because of the inhomogeneity the ray was deflected, the magnitude of the deflection
being a measure of the produced change in refractive index An. Assuming that An is due to the
produced change in density Ap, related to the corresponding change in pressure Ap through the
compressibility of the liquid, one can write
An =~-~~eAp.
(2.12)
The experiment effectively measures An. The quotient dn/dp is calculable from the Lorentz—Lorenz
relation (the optical analogue to the Clausius—Mossotti relation), The experimental value of Ap,
calculated by means of (2.12), was found by the authors to agree with the prediction (2.10) (to
within ±5%) and to disagree with (2.11).
We are indebted to a referee for pointing out that one ought to investigate whether there are
other effects besides electrostriction of importance for this experiment, notably the quadratic
electro-optic effect. The strong electric field makes the liquid optically anisotropic; the polarizability cc11 parallel to the field is different from the polarizability oc~in the transverse direction. The field
dependence of the polarizability will in turn influence the refractive index through the Lorentz—
Lorenz relation. The quadratic electro-optic effect was not discussed by Hakim and Higham.
Let us here estimate its magnitude, using results obtained in the non-linear quantum theory of the
refractive index in a simple isotropic medium. Bullough [83] considered an artificial medium
consisting of two-level atoms. His2/E~),where
result for thecc atomic polarizability in the direction of the field
can be written as cc11 = cc0/(1 + E
0 is the polarizability in the absence of field and
E0 is the constant E0 = hwS/(\/~exoj. Here 11w5 is the energy difference between the two levels
of the atom and x05 the matrix element of the position operator between the same levels. For
simple non-polar liquids it is the electronic polarization which is of interest, so we may for the
resonance frequency insert w5 —~ 1016 51, corresponding to the ultraviolet region. Further,
assuming the matrix element to be of order 10_to m, we obtain E0
1011 V/m. As a check on
this magnitude of E0, we may note that the theory ofthe hydrogen atom yields E0 = 0.73 x 1011 V/rn,
so the magnitude seems to be reasonable. If we now compare the Helmholtz increment AnH
produced by the density increment, i.e. the expression (2.12) with (2.10) inserted, with
corres1’ =the
6B/[e
ponding electro-optic increment Aflq, we find by a simple calculation that Anq/An
0E~
2,
x (Er — 1) ~ + 2)] where B is the bulk modulus. For carbon tetrachloride we have B —~ i0~N/m
2, so that Anq/AnH
10_2. The estimate thus suggests that the relative importance of the
electro-optic effect is small, although the analysis is not sufficiently accurate to permit a definite
conclusion. (The analysis is sensitive with respect to E
0 if E~were 1010 V/rn instead of 1011 V/m
one would find that the two effects were of the same magnitude.) However, from tabular works [84]
one finds for the liquids in question that Anq/AnH —~0.1, showing that the electro-optic effect does
hardly make up an appreciable fraction of the total effect. The bare fact that the observations
agree with the Helmholtz prediction to within ±5% makes it rather unlikely that the electro-optic
effect — or other non-linear effects — are essential here. One may note that the Einstein—Laub
prediction (2.11) lies about 30% lower than the Helmholtz prediction (2.10).
Additional support for the smallness of the electro-optic effect in this experiment may be given
if we consider the polarization properties of the light beam (this argument is due to J.S. Høye).
148
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
As the light used in the experiment is unpolarized, we may decompose it into two polarization
components, parallel and orthogonal to the field respectively. Of these, only the first-mentioned
component is able to sense the slight change in refractive index. When a ray of light passes through
the test cell a double refraction takes place, resulting in principle in two rays of light out from the
cell. Therefore, if the electro-optic effect were significant in the experiment, one should expect that
the difference between the deflections of the two emerging rays were of the same order of magnitude
as the deflection itself. As no double refraction was reported to take place, one may expect that
the electro-optic effect was practically insignificant.
After these considerations, although we have confined ourselves to simple estimates instead of
quantitative calculations of the possible sources of error, we think that it is legitimate to conclude
that the Hakim—Higham experiment provides a support for the Helmholtz theory. Also, the Helmholtz interpretation ofthe pressure distribution in the elevated liquid in a condenser, cf. the previous
section, is supported.*
Finally let us present a simple theoretical argument, that also supports the Helmholtz force
density expression in a non-polar fluid whose permittivity is constant. The argument demonstrates
the close relationship that exists between the Helmholtz force expression and the Clausius—
Mossotti relation. The latter is an immediate consequence of the assumption that the effective
field E’ at the site of a molecule is E’ = E + P/(380), and so gives rise to a polarization
P
=
Np
NcxE’
=
(2.13)
where N is the density of molecules and p the molecular dipole moment (a pure compound is
assumed). Now it is clear that it is in the spirit of the Clausius—Mossotti relation to expect that
each molecule is acted upon by a force (p V)E’ = x(E’ V)E’. The corresponding force density
follows after a simple calculation as
2/(68
f = Ncc(E’ V)E’ = (8 — 8~)(8 + 280)VE
0)
(2.14)
.
.
where the condition VE = 0 has been taken into account.
The theoretical expression (2.14) is seen to be in agreement with the Helmholtz expression (the
last term in (2.5), with the Clausius—Mossotti relation
in a non-polar
fluidfound
of constant
2V6 in inserted)
the Helmholtz
force is not
by the
permittivity.
On
the
other
hand,
the
term
— ~E
reasoning above. This is not to be expected in view of the simplicity of the argument.**
* The conclusion we have drawn on the basis of the Hakim—Higham experiment appears to be more definite than that drawn by
de Groot and Suttorp [12, p. 305]. These authors state that also (in our language) the Einstein—Laub prediction for the electrostatic
pressure is reconcilable with the experimental result provided one interprets the theory consistently. we cannot see, however, that
any reasonable interpretation, applicable to the Einstein—Laub case, actually exists. For under the same physical external conditions
the excess pressure, which is a definitely measurable quantity, is predicted quite differently in the two cases. And as the observations
clearly agree with the Helmholtz expression (2.10) and disagree with the Einstein—Laub expression (2.11), we think that it is legitimate
to conclude that the latter expression is inappropriate in the given case.
It should also be mentioned that the calculation given by de Groot and Suttorp [12, p. 99—100], as an application of the Einstein—
Laub expression for the excess pressure, actually has no close relationship to the peculiar properties of the Einstein—Laub tensor.
The authors calculate the total electrostatic force that acts on a solid body immersed in a liquid, and obtain a result where the electrostriction pressure is absent, i.e. the same result as one would obtain by starting from the Abraham—Minkowski tensor (without electrostriction) from the outset. This feature is, however, a general property of any tensor whose corresponding force density differs from
the Abraham—Minkowski force density only in a gradient term. For at mechanical equilibrium, such a gradient term will always give
rise to a contribution to the total force that is compensated by elastic pressures in the liquid. As both the expressions (2.5) and (2.6)
are of the kind mentioned, it is obvious that we would equally well obtain dc Groot and Suttorp’s expression for the total force by
starting from the Helmholtz expression (2.5).
* * Hakim’s theoretical derivation [30] of the full Helmholtz force, including the term — 1E2Ve, is implicitly based on the condition
F x Vs = 0 (see his eq. (18)). This condition is not generally valid.
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
149
2.4. Goetz and Zahn’s non-equilibrium experiment [15, 16, 31]
This experiment is of comparable importance to the Hakim—Higham experiment as regards
information about the electrostriction force. The principle of the experiment is to apply a high
frequency electric field E = E0 cos wt between the two plates of a condenser filled with a dielectric
liquid, polar or non-polar, and to measure the appropriate harmonic component of the attractive
force between the plates by means of a selective piezoelectric transducer fixed to one of the plates. *
We shall consider the total attractive force between the plates, assuming for simplicity that the
liquid is non-polar. We shall include also the electrostrictive contribution to the attractive force,
for the moment simply disregarding the eventual compensating force from the elastic pressure
in the liquid. Let the symbol a~in general denote the attractive force per unit surface area. One
can in the Helmholtz theory as well as in the Einstein—Laub theory calculate a~from the electrostatic tensor expressions. It is convenient to start from the electrostatic versions of(2.4) and (1.7a);
these expressions are equivalent to the force density expressions (2.5) and (2.6). We find in the two
cases
2p
= ~-E~
[8 + ~-~-— (~ e~~)(~
+ 280)] cos2wt
(2.15)
= a~+ ~E
—
2wt,
(2.16)
0)cos
where c is the permittivity of the liquid and a~= a~’= ~8E2 is the surface force density in the
Abraham—Minkowski theory.
Some remarks are called for, as regards the use of the electrostatic expressions in (2.15) and
(2.16). In this experiment, typically w
iO~s ~ and there will also be an induced magnetic field.
The magnetic contribution to the force between the plates is however negligible. We may call the
present field a “quasi-stationary” field; the small induced magnetic field H can be found from
Maxwell’s equations inserting for the electric field simply the applied field E = E
0 cos wt. In
orders of magnitude one finds in the present case H wlEE, where I denotes the linear extension
of the
plates.
The ratio
between
the magnetic
surface
the small,
order
2/cE2
ej4wl)2
(wi/c)2.
Inserting
w = iø~~and I the
= 1electric
cm we find
this forces
ratio tois beofvery
pH
about io~.
The use of the electrostatic expressions (2.15) and (2.16) is therefore justified. Note
also the omission of the partial derivative symbol on dE/dp in (2.15). For the high frequency
variations considered here it would be appropriate to use the derivative (ae/8p)
5, evaluated at
constant entropy, in the electrostriction term. When the liquid is non-polar one need not, however,
distinguish between the isothermal and adiabatic cases as the permittivity depends on the density
only, through the Clausius—Mossotti relation.
The harmonic parts of the expressions (2.15) and (2.16) vary with the frequency 2w. Choosing
2w equal to the resonance frequency of the transducer, the theoretical expressions directly comparable to observation are thus
=
a~+ ~EP
=
~E~(2e
—
8
‘~
a~(2w)= ~
+
L
—~-—(8
—
e~)(8 +
38~
a~(2w)= *E~(2e
—
8~)cos
280)1 cos 2wt
(2.17)
j
2wt.
(2.18)
* Actually, in the test of non-polar liquids a static electric field was superimposed on the varying field. This static field is however
ofno importance for our discussion, and will therefore be neglected.
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I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
With various non-polar liquids, Goetz and Zahn [15] were able to verify the correctness of (2.17)
to within ±4%.Equation (2.18) disagrees with this result.
Zahn [16] extended the measurements to polar liquids. The results were comparable also in
that case, although the uncertainties were found to be somewhat greater for the liquids of great 6
in view of the uncertainties of the state equation for 8. With nitrobenzene, the most extreme liquid
investigated (Er = 33.3), the electrostrictive alternating pressure amounted to 24 N/rn2 when
E = 106 V/rn. One may replace the Clausius—Mossotti equation by the Onsager equation or one
of its modifications [32]. A discussion ofthis point has recently been given by Lahoz [24].
Summing up so far, the important conclusion one can make on the Goetz—Zahn experiment is
that it seems to give another strong support for the Helmholtz theory. There is however an important physical effect not yet considered, viz, the elastic counter pressure from the liquid. It might
perhaps be astonishing at first to learn that also the electrostriction part of the attractive force
was included in the measurement; in a state of mechanical equilibrium the electrostriction contribution to the force is, as we have already pointed out, certainly non-detectable due to the elastic
counter pressure. The only reasonable explanation for this circumstance must be that the liquid
is actually in a state of mechanical non-equilibrium; the field is alternating so quickly that the
elastic deformations do not have the opportunity to build themselves up. The remainder of this
section will be concerned with a simple theory of the non-equilibrium problem, as we are not aware
that it has been considered before.
The mechanism for the production of elastic pressures is the following. At the same instant
as the Helmholtz surface force density a~attacks the plates, volume forces of density ~ act in the
liquid, in the inhomogeneous field region between the edges of the plates, tending to press the
liquid inwards. Let the excess elastic pressure, as a function of space and time, be called p’. Thus
= p
p°,wherep is the total pressure at any space-time and p°is the pressure in the undisturbed
liquid outside the condenser. p’ must vary with time in the same way as the surface force, i.e. it
consists of a time-independent part and a time-dependent harmonic part, denoted by p’(2w),
that varies with the same frequency 2w as the harmonic part of the surface force. It is p’(2w) that
is of interest here, as 2w is the resonance frequency of the transducer. We wish to estimate
(1) the amplitude of p’(2w);
(2) the time necessary for the produced excess pressure between the edges of the plates to propagate
inwards and create pressure equilibrium within the condenser.
These two effects will be considered successively. Figure 3 shows schematically the region around
—
z
~
s
E
~\\~‘~~\
~
i2
+
Fig. 3. Sketch of the region around one end of the condenser.
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
151
one end of the condenser. The distance between the plates is d, and the plates are assumed to be
large in the y-direction. In the experiment d —‘ 1 cm, so that the condition w c~c/d, characteristic
for quasi-stationary phenomena, is amply satisfied. We can thus form the expression for the electric
field E(x, z, t) in the inhomogeneous region simply by starting from the expression for the electrostatic field and multiply it by cos wt. The electrostatic field can be found by using the theory of
complex variables. The expression turns out to be somewhat complicated [33, p. 322] and will
not be given here. Actually, plotting the variation of the electrostatic field component E~(x,z = 0)
along the x-axis, we find that it makes a reasonable approximation to assume a simple exponential
variation proportional to exp 2x/d) outside the condenser and a constant value inside. Accordingly, we adopt a model with
(—
E~(x,z= 0,t)
E
5(x, z = 0, t)
=
E0e2~”coswt
=
E0 cos wt
(x >0)
(x <0)
(2.19a)
(2.19b)
on the x-axis. By symmetry, the x-component of the electric field is zero on the x-axis.
The expressions (2.19) suffice to give a crude picture of the pressure variation. Let v denote the
horizontal velocity of the liquid on the x-axis in the region x 0. The linearized Euler equation
reads
2 ~ (ae) e4~’cos2wt,
(2.20)
0
=
— 2E0
—
where p°is the density of the undisturbed liquid outside the condenser. We have neglected the
first term in the expression (2.5), and assumed for definiteness the adiabatic derivative of the
permittivity. The linearized equation of continuity is
~p’/at + p°äv/ôx = 0,
where p’
p
=
ô2p’
—
p°is the excess density. From (2.20) and (2.21) we find the equation
1 ~2p’
8E~
e~” cos2wt,
[(8p/~p)
=
I ô8\
=
—
where u
(2.21)
(2.22)
2is the velocity of sound. The stationary solution of (2.22) is
5]”
P
—
1E20P~P)e
(ä8•’~
~
—4x/d
L[1
+ 1 +(wd/2u)2
cos 2wt
(
2 23
.
)
Consider first this solution in the special case w = 0, corresponding to a time independent applied
field E = E
0 in the interior of the condenser. In this case the adiabatic derivative in (2.23) must
obviously be replaced by the isothermal derivative, and we get
2oP(t38/8P)re4~’.
(2.24)
P’stat
=
~E
The uniform excess pressure in the interior of the condenser is obtained by setting x = 0 in (2.24).
Note that (2.24) then becomes equal to the conventional result in electrostatics; the excess pressure
just counterbalances the electrostrictive contribution to the attractive force between the plates.
A measurement of the attractive force will not show any electrostriction effect.
152
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
Next assume a high value of w, so that the adiabatic derivative in (2.23) is appropriate. We are
interested in the dynamic part of (2.23), varying with the frequency 2w:
p’(2w)
=
~E~p (~)
e -4x/d 1 +(wd/2u)2’
(2.25)
It is seen that this dynamic component varies in phase with the force component (2.17). In order
to test its magnitude we calculate the ratio to the full static excess pressure (2.24):
p’(2w)
—
(ôE/~p)~ cos 2wt
(2 26
286/~p)T1 + (wd/2u)2’
(
For a non-polar liquid the ratio between the adiabatic and the isothermal derivatives in (2.26)
is just unity; for a polar liquid the ratio is usually somewhat less. Inserting w = iø~s’, d = 1 cm,
u = i03 rn/s we find that the amplitude of the ratio (2.26) is ~ io~.
It is therefore clear that the
dynamic pressure (2.25) can by no means compensate for the electrostrictive part of the attraction
force between the plates, and we have demonstrated explicitly why the full Helmholtz force
(including electrostriction) is measurable in the Goetz—Zahn experiment.
The above remarks were concerned with the amplitude of the dynamic pressure produced
around the edges of the plates. Finally consider briefly the time required for the elastic disturbances
around the edges to create pressure equilibrium in the liquid. This time must at least be of the
order of the time that sound waves need to propagate from the edges into the interior of the
condenser. To propagate a distance of 1 cm sound needs a time of about 10 ps, which is in fact
greater than the period 2ir/w
1 ps of the oscillations. Thus, pressure equilibrium cannot be
established in the present case. Equation (2.26) indicates that the effect of the counter pressure
should begin to become significant when wd/2u
1, i.e. w
iO~s’. This corresponds to an
oscillation period of 2m/w
60 ps, which should be a sufficient time for the sound waves to traverse
the condenser. So for frequencies w ~ iO~s’, one should not expect any longer to have the opportunity to measure the electrostrictive contribution to ax.
P~
tat
—
3. Measurement of Abraham’s force in quasi-stationary fields
3.1. The experiment ofG.B. Walker, D.G. Lahoz and G. Walker [7, 34]
In this section we shall be concerned with the total electromagnetic force or torque acting on a
solid body. As local effects are not measured, we can ignore electrostriction and magnetostriction
from the outset. Of main interest is the comparison between Minkowski’s (1.3) and Abraham’s (1.6)
force density expressions in the case where the body is isotropic and non-conducting. We repeat
the expressions here:
= —~E2Vc— ~H2Vp
(3.1)
6~ç_1 -~(Ex H).
1A
..1M
(3.2)
+
The Abraham term in (3.2) is usually not directly observable; in a high frequency field, for instance,
the term is small and will in addition simply average out. There exists however the following
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
153
possibility: Instead of using electromagnetic waves, use strong, time-varying, orthogonal electric
and magnetic fields. Then the magnitude of the Abraham term may become relatively large, and
if the time variation moreover is slow (quasi-stationary fields), so that the oscillations themselves
are subject to measurement instead of their average effect over a period, one will have the possibility to measure the term. For instance, one may imagine a cylindric shell or annular disk of
large permittivity or permeability suspended as a torsional pendulum in a vertical magnetic field.
If a radial electric field is applied between the inner and outer surfaces of the disk, it is clear that
the Abraham term has a nonvanishing azimuthal component and so predicts oscillations of the
disk about the z-axis. Minkowski’s force density, on the other hand, has no azimuthal component
and predicts no oscillations.
The idea sketched above for testing Abraham’s force seems to go back to Marx and Gyorgyi
[35]. It was taken up and elaborated theoretically in some detail in [11, p. 30], assuming a constant
electric field and a varying magnetic field. G.B. Walker, D.G. Lahoz and G. Walker have actually
carried out the experiment, using a varying electric field and a constant magnetic field, and a disk
of barium titanate (Er 3620, Pr = 1) in the torsional pendulum. The inner (r = r1) and outer
(r = r2) surfaces of the disk were coated with a layer of aluminium, and a harmonic voltage
V = V0 cos wt of the same periodicity as the mechanical pendulum was applied across the disk
(see fig. 4). The radial electric field in the disk is thus Er = Vo[r ln (r2/r1)] 1 cos wt. A simple
calculation yields the following expression for the Abraham axial torque:
N~=
~8~(8r
—
p0HV0w sin wt,
(3.3)
where L is the thickness of the disk and H the axial magnetic field. Solving the equation of motion
for the pendulum, one can express the oscillation amplitude in terms of the expression (3.3), the
moment of inertia, and the damping constant. In the experiment, oscillations were really observed
and were found to be in agreement with the Abraham prediction to within ±10%. With
B = p0H 1 T (10~gauss), V0 ‘~ 260 V and w/27r 0.4 Hz, the amplitude of N~‘~ 4 x 10 ~ Nm.
‘-~
Fig. 4. Axial cut through the barium titanate disk.
In interpreting the importance of this experiment one must be aware of its limitations: (1) The
frequencies are very low; (2) the test body is non-magnetic; (3) the H field is constant, whereas the
E field is varying. In spite of these limitations the experiment is very important, as it provides a
direct verification of the existence of the Abraham term. Minkowski’s force is contradicted by the
observed oscillations of the disk. Thus in quasi-stationary fields Minkowski’s tensor is inadequate.
It may be mentioned here that this fact does not prevent us from preferring Minkowski’s tensor
in most experiments in optics. The reason is the non-observability of the Abraham term at optical
154
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
frequencies. Minkowski’s tensor has attractive formal properties that makes it convenient to use
this tensor in most cases in optics. See the later discussion in section 4.1.
It is also of some interest to examine the Einstein—Laub tensor applied to the Walker—Lahoz
experiment. Inspection of(1.9) shows at once that the Einstein—Laub force differs from the Abraham
force only with respect to a gradient term which gives no contribution at all to the total torque.
The Walker—Lahoz experiment, being a global experiment, gives no information about the local
force density distribution within the test body. The experiment thus cannot distinguish between
the Abraham and the Einstein—Laub alternatives.
Finally, we shall consider an interesting modification of the above experiment, recently reported
in two brief notes by G.B. Walker and G. Walker [37, 38]. The intention of this experiment was
to abolish the previous restriction that H be a constant field. Both E and H were now time-varying,
at a relative phase shift of ir/2 and at a frequency w that was about 250 times the resonance frequency of the pendulum. Otherwise the experimental apparatus was the same as before. The time
average of the torque was measured. The first note [37] reported that the observations were in
agreement with an electromagnetic force density equal to p0P x H and in disagreement with
Abraham’s expression (3.2). Such a result is, however, highly improbable, and it would in fact
be incompatible with all the tensors that we are considering. The next note [38], however, gives
the explanation of the phenomenon: The mean value of the torque is due to surface forces which
act on the electric charges on the metal coatings that have been evaporated on the outer and inner
cylindrical surfaces of the disk.
Although only preliminary notes on the experiment have as yet been published, its main features
seem to be the following.*
Let the axial magnetic field be H~= H0 cos wt. This field induces an azimuthal electric field
Eq,
=
~p0H0wr sin wt.
(3.4)
Next, suppose that the applied radial electric field across the disk is proportional to sin wt:
Er
=
V0
sin wt.
r ln (r2/r1)
(3.5)
This means that the electric surface charge per unit axial length on the inner coating is
q
2ltEoErVo
=
ln (r
stn wt.
(3.6)
2/r1)
The charged surface elements of the inner coating are subjected to azimuthal forces due to the
field (3.4). The corresponding torque on the coating, per unit axial length, is qE~,r1.Adding a
similar term for the outer coating, we have for the total axial torque acting on both coatings:
2wt.
(3.7)
N~°~
=~8~T~)
~ ~.t0H0v0w
sin
The time average is’~
=
ircoc~L(r!—r~)
p
0H0V0w.
(3.8)
The time average of the torque acting on the dielectric is however, according to Abraham, equal
to zero. The average torque measured in the experiment thus seems to be just the quantity (3.8).
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
155
Note that the experiment thereby does not provide a critical test of the various tensors. What one
actually measures is the time average of electric forces acting on surface charges, and the theory
for this is unambiguous. Minkowski’s tensor for example, is also in conformity with the result (3.8).
Note that it has been assumed here that the cylindrical condenser surfaces are fixed to the
dielectric. If the radial electric field, on the other hand, is produced by external cylindrical surfaces,
as assumed in [11, p. 30], one will avoid the inclusion of the surface contribution (3.8) in the measured torque. In such a situation the average torque on the disk will vanish. If the frequency is low
enough to permit observation of the individual variations, one will in this way be able to measure
directly the Abraham torque (or the Einstein—Laub torque) also in the case of a varying H field.
The reason why the expression jtoP x H in the first note [37] was claimed to represent the
force density in the dielectric, is probably that a straightforward use of this expression (without
regard to surface forces on the coatings) gives approximately the same result as obtained above.
Actually, inserting Po~x H for the force density and integrating over the volume of the dielectric,
we obtain an average expression which differs from (3.8) only in that 6r is replaced by (Cr — 1),
When Cr = 3620, this distinction is not detectable. It is therefore understandable why jt
0P x H
at first sight might seem to be the appropriate expression for the force density.
GB. Walker and G. Walker conclude their second note [38] by stating that their experiment is
consistent with a force density equal to (P~V)E + Po~x H. This is actually the Einstein—Laub
expression (1.8a), and it is observationally equivalent to the Abraham expression.
3.2. James’s experiment [6, 39]
When Walker et al. carried out their torque experiment they were during the main part of the
work unaware of the fact that some years earlier an essentially similar experiment had been
carried out by James and reported in this thesis [6]. James’s test system consisted of two identical
ferrite toroids (Er/.Lr
102), connected axially with each other through a piezoelectric transducer.
Figure 5 illustrates a cut through one of the toroids. An azimuthal magnetic field HQ is produced
by an axial periodic current I, and a radial electric field Er is produced by a periodic voltage V
applied between the metal coatings on the inner and outer cylindric surfaces of the toroid. Again,
the main purpose was to compare Minkowski’s and Abraham’s force density expressions given by
(1.3) and (1.6). It will turn out that Minkowski’s form predicts in general an axial net force on the
toroid, as the axial forces that act in the boundary layers at the annular surfaces (at x = ±~L, x = 0
being the centre of the toroid), do not compensate each other. Abraham’s form also predicts an
axial force, but of different magnitude due to the Abraham term. A measurement of the axial force
will therefore be a critical test of the two predictions. By arranging it so that the voltages across
the two toroids are oppositely phased, one will obtain symmetrical axial oscillations. The idea of
James was to measure the corresponding axial mutual force by means of the interconnecting
transducer.
A detailed account of the experiment has now appeared, see [87].
Instead of proceeding as above, one can alternatively derive (3.8) by considering the stress tensor at the coating surfaces.
G~B.Walker and G. walker argue in the latter way. It is however unnecessary to assume a small air gap between the coatings and the
dielectric, such as these authors do. The result (3.8)is independent ofwhether or not there is a gap. For instance, following the reasoning
above it is clear that in any case the torque acting on a unit length of the inner conductor must be equal to qE en, where q and E
are given respectively by (3.6) and (3.4).
*
**
156
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
x~O
Jr
1
L
Fig. 5. Axial cut through one of the ferrite toroids.
Let us now consider the essentials of the calculation of the axial forces. In order to obtain
quantitative predictions in the two cases, it is necessary to calculate the electromagnetic fields,
including the induced corrections at the annular ends of a toroid. This necessity of taking into
account also the surface forces makes the present experiment more complicated to describe than
the experiment considered in the previous section. Let the axial current be equal to I = I~sin w1t,
where I~is a constant. This current produces an azimuthal magnetic field
~
=
2irr
sin co1t,
(3.9)
where a superscript zero has been added to the left to designate the zeroth order field. The condition for quasi-stationarity, w1L/c 4 1, is amply satisfied here. Thus the field (3.9) induces a first
order radial electric field:
1~
=
cos w
—
E~
1t
2irr
(3.10)
(cf. fig. 5). Similarly, if the applied voltage across the toroid is V
frequency w~,one has a radial zeroth order electric field
=
V0 sin w~t,varying at a different
E~°~
=
V0
5lflWvt
r ln (r2/r1)
(3.11)
and a first order azimuthal magnetic field
~ =
E0ErV0WyX cos oi,,t.
—
(3.12)
r ln (r2/r1)
The total field components in the radial and azimuthal directions are equal to
Er
=
E~°~
+ ~
Hq,
=
~
+ H~.
(3.13)
The reason why the axial surface forces at the ends x = ±~L
of the toroid do not compensate each
other is the asymmetry of E~’~
and H~Ø’~
with respect to x. The ratios E)/E~°~
and H~/H~,°~
are
very small, of the order of io~under typical experimental conditions. Nevertheless, products of
zeroth order fields and first order fields that are contained in the expressions for the surface force
densities make the surface effects of the same order of magnitude as the volume effects, which are
calculated from the zeroth order fields alone (see below).
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
157
The axial Abraham force can be calculated from an expression similar to (1.14):
F~=
—
Js~cndS
—
~-
Jg~dV,
(3.14)
where the first integral refers to a surface in the vacuum just outside the toroid. The last integral
in (3.14) involves only the zeroth order fields and is easily calculated:
j~—Jg~dV
=
~
~-
JE~0)H~p0)27rr
dr =
V010
~-
(sin w~tsin w1t).
(3.15)
Next consider the magnetic part of the surface integral in (3.14). It requires knowledge about the
tangential field H~C( ±~L)
in the vacuum just outside the toroid ends. As HQ is independent of
the presence of the toroid, and as H~q,’~
is induced by the time-varying voltage across the toroid,
one may put H~C( ±4L) = Hq,( ±4L), i.e. the internal magnetic field of (3.13) evaluated at the
toroid ends. This gives for the magnetic part of the surface integral in (3.14):
—
JS~nkdsl magn =
~Po J [H~L)
—
—
H~( iL)] 2irr dr
—
VoI0ErWv sin w1t
=
rj
cos Wvt,
(3.16)
which is seen to be of the same order as (3.15).
The corresponding electric part of the surface integral is more difficult to find because55(
of
the
±~L)
fringing
of the electric
field the
neartoroid.
the toroid
problem
is toisfind
the radialacross
field E~
in
the vacuum
just outside
Oneends.
may The
assume
that E~°~
continuous
the boundaries, but the correction E~1~
induced by the time-varying field H~°~
contains Pr as a factor and is in
the present simple theory changing abruptly when one passes from the medium to the vacuum
side. There is thus a matching problem for Er at the ends of the toroid. The result of James is obtained if one puts
~ac(±.~L) = E~°~
+ p~’E~(±~L).
(3.17)
This equation ought to be better justified, in view of the mentioned matching problem, but we
shall simply adopt it in the following without going into further considerations at this point.
With the use of (3.17) one finds for the electric part of the surface integral in (3.14):
-
J~kflk
dSlei
=
=
—~8~J{[~(~L)]2
—i-
-
[~(—~L)]2}2~r
dr
V
010w1 sin w~tcos w1t.
(3.18)
The Abraham axial force F~on the toroid is found by subtracting (3.15) from the sum of (3.16) and
(3.18). The frequencies w1 and co~are chosen such that wi wvl equals the mechanical resonance
frequency w0( 2 x iO~s~’)of the test system. Including only the component that varies with
the frequency w~one obtains
—
F~(wv,w1= w1, ±w0)
=
±
VoIo(Cr
—
1)w~sin co0t.
(3.19)
158
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
The Minkowski force is similarly calculated as
F~(wv,w1
=
w~±w0)
=
VoIo[(Cr~r
—
1)w0 ±(Cr
—
l)wv] sin w0t.
(3.20)
The measurements of James support the prediction (3.19) and disagree clearly with (3.20)*.
Although the expression (3.18) is somewhat questionable, we must nevertheless conclude that
the experiment seems to provide another support of the Abraham tensor or one of its equivalents.
An advantage of the experiment is that it covers also the case of magnetic media; its drawback
is that one has to include the surface forces in the calculation.
The Einstein—Laub tensor requires no special attention. The experiment is a global experiment,
just as the torque experiment of Walker et al., and it is therefore clear that the Einstein—Laub tensor
predicts the same result as the Abraham tensor.
4. Optical phenomena
4.1. General remarks
In the remainder of this paper we shall be concerned with phenomena in the optical domain,
and we shall mostly be dealing with isotropic and non-dispersive media. At the high optical
frequencies, obviously, a direct observation of the rapidly fluctuating Abraham term is no longer
feasible, and we have to resort to the observation of mean intensity phenomena. The relative
permeability p1 differs usually very little from unity at optical frequencies, so we shall in the following simply put Pr = 1. The refractive index is then n =
where Er pertains to the appropriate
frequency region. Consistent with the neglection of dispersion the electromagnetic stress tensor
can be constructed formally as the sum of its electrostatic and magnetostatic counterparts,
2E. Similarly, the
we
relation
between
D
and
E
being
given
by
the
high
frequency
relation
D
= s0n
shall assume that the results of the force experiments in the quasi-stationary domain, as reported
in section 3, holds also at optical frequencies.
Now consider a travelling optical wave in the homogeneous medium. The picture that we
obtain for the wave is dependent on whether we choose to include the electrostriction effect or not.
We consider the simplest alternative first.
(1) Electrostriction omitted. The electromagnetic momentum density is
~
gA~ExH.
(4.1)
The electromagnetic force density is given by the Abraham term as
fA
*
n2— 1 ~-(E x H).
(4.2)
The theoretical expressions that James compared with his observations were actually not (3.19)and (3.20) themselves, but instead
the so-called “reduced force”, defined in each case to be the arithmetic mean 1[F+(co~, (Oi = w,q +
W
0) + F~(m~,
W1 = Wy — con)].
The reduced force in the Abraham case is thus according to (3.19) vanishing, whereas in the Minkowski case it is non-vanishing. The
accuracy of the experiment was such that the observed reduced force, averaged over several observations, was less than 5°/~
ofthe
Minkowski prediction (— 1 nN) and thus in good agreement with the Abraham prediction.
One may also wonder if the construction of the reduced force eliminates the above-mentioned uncertainty in the calculation of(3.18).
However, a simple calculation shows that this isnot so: The electric part (3.18) contributes to the reduced force. By contrast, the magnetic
part (3.16) averages out.
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
159
If the medium is a fluid, the action of the force density (4.2) is assumed to be the production of
an accompanying mechanical momentum density
n
grnech =
1 E x H.
(4.3)
The accompanying momentum is assumed to be distributed in space in proportion to the Abraham
momentum. In a solid, the elastic coupling between the molecules may complicate this simple
picture.
The sum of the quantities (4.1) and (4.3) is simply Minkowski’s momentum density
H=D x
~
(4.4)
B.
Thus, when electrostriction is omitted, Minkowski’s momentum emerges as the total propagating
momentum. It may be appropriate here to write down some reasons why Minkowski’s tensor is
such an attractive alternative for the description of optical phenomena: (i) In a homogeneous
medium it is divergence-free, öVS~, = 0, so that the four components
G~M=IJS~dV
(4.5)
form a four-vector when the integration is extended over the whole field. (ii) The propagation
velocity of the energy of the wave (the “ray” velocity) u = SM/WM, transforms like a particle
velocity under Lorentz transformations. This property is not merely of mathematical significance,
as it has sometimes been stated, because there is experimental evidence that the ray velocity actually
behaves in this way under Lorentz transformations. Correspondingly, in a space-time diagram
one finds in the Minkowski picture that the world lines for the ray velocity remain fixed under
rotation of the space-time axes. (iii) The Minkowski tensor is very convenient for the description
of the Cherenkov effect. (iv) Finally, the Minkowski tensor fits naturally into canonical theory
based upon a Lagrangian. Consequently, this tensor fits well also into phenomenological quantum
theory. We shall occasionally return to some of these aspects later on. The reader may consult
several references on these and related topics, by Ginzburg et al. [22 23, 36], and by others [5, 10,
11,40,41].
(2) Electrostriction included. If we shall give the expression for the true local electromagnetic
force density, it is necessary to take the electrostriction term into account. We obtain the Helmholtz
force density
2— 1 ~ (E x H) + ~e
1H =
n
2]
(4.6)
0V [E2P dn
which replaces (4.2). At optical frequencies the dipolar and ionic contributions to the polarizability
are negligible, and we assume that the remaining electronic polarizability is taken care of by the
Lorentz—Lorenz relation. The partial derivative symbol is therefore omitted in the last term in (4.6).
Assume now that the field is a wide unidirectional wave train travelling in the x-direction. As each
field component depends on x and t through the argument (x ct/n), the operators 3/3x and
(n/c)3/ôt are interchangeable when operated on some function of the fields. We shall assume
that the last term in (4.6) has the same physical effect as the first term, namely, that it produces
—
—
160
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
an accompanying mechanical momentum at the leading edge of the wave. This electrostrictive
contribution to the accompanying momentum must die out at the trailing edge of the wave and
is otherwise assumed to be distributed in space in proportion to the Abraham field momentum.
Writing the x-component of (4.6) as
fxH
=4[n2
—1 _~~~_]~(Ex H)~,
(4.7)
we thus obtain the following expression for the density of the accompanying momentum in the
Helmholtz picture:
g~ech,x = ~T[~t
—
1
~p ~_](E
—
x H)~.
(4.8)
Adding this to the field momentum density (4.1) we obtain the momentum density of the whole
disturbance:
dn21
= g~+ grnech,x =
[n2
~p
(E X H)x.
(4.9)
~
—~-—j
—
Insertion of the Lorentz—Lorenz formula gives
= -~
[n2
—
*(n2
—
1) (n2 +
2)]
(E x H)x.
(4.10)
As far as we know, the Helmholtz force density (4.6) has not been contradicted in any experimental
test in any frequency region.
(3) The electrostrictive compensation. We have pointed out earlier that in the case of static fields
it is often possible to omit the electrostriction term from the formalism without thereby altering
the predicted observational effect. It turns out that the situation is just analogous in optics:
under conditions of pressure equilibrium in a dielectric liquid into which a test body is immersed
(usually satisfied in practice), the electrostriction effect gives no net contribution when integrated
over the volume of the test body. It seems to be worthwhile to give some remarks on this compensation, in the important special case where the immersed body is a metallic conductor. When the
conductively of the conductor is large, the penetration depth of the field is very small. We can
therefore calculate the electromagnetic surface force density ~ (averaged over a period) on the
metal by starting from (1.14) and letting the integration volume be a cylinder with base surfaces
equal to unity, one base surface lying in the liquid just outside the surface and the other lying in
the interior of the metal where the fields are equal to zero. If the conditions are stationary, the last
term in (1.14) fluctuates out. If they are non-stationary, this termcan also be omitted because of the
smallness of the penetration depth (typically a few nanometers for a good conductor). We can
therefore choose the height of the cylinder so small that the last term in (1.14) gives no appreciable
contribution, while maintaining the property that the fields vanish on the base surface lying within
the body. For a metallic conductor we thus find in general
=
—S~knk,
(4.11)
where Se,, is taken in the liquid and nk is the kth component of the normal directed from the body
into the liquid. Equation (4.11) holds for any energy-momentum tensor. Inserting the Helmholtz
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
expression in (4.11) we obtain, in vectorial form,
2pdn2/dp.
= ~A + ~C
0nE
161
(4.12)
At mechanical equilibrium, the excess elastic pressure in the liquid
= 48
2p dn2/dp
(4.13)
0E
balances the last term in (4.12), and we are again left with the Abraham—Minkowski result ~ =
for the net surface force density. Since the contribution from the last term in (1.14) vanishes, the
case is just analogous to the static case. Except from special kinds of experiments (for instance,
local pressure experiments), the electrostriction effect can simply be omitted. In the following we
shall make use of this formal simplification, when possible. It is in conformity with usual practice.
(4) The Einstein—Laub tensor. Although we have seen earlier that the predictions of this tensor
are contradicted by the Hakim—Higham experiment and the Goetz—Zahn experiment, we shall for
completeness give the expression for accompanying mechanical momentum also in this case.
Consider again the wide wave travelling in the x-direction within a homogeneous and isotropic
fluid. The Einstein—Laub force density, given by (1.9), corresponds to the following accompanying
momentum density:
grnech,x =
n2—
(E x H)~.
(4.14)
Adding this to the field momentum density g~we obtain the total Einstein—Laub momentum
density:
=
g~+
grnech,x =
n+ 1 (E x H)~.
(4.15)
Similar arguments have been given by Hans [42]. Also, Gordon [18] starts from the Einstein—Laub
force density when analysing the special case of a gaseous medium. In this case, however, when n
is close to unity, one has p dn2/dp = n2
1, and the Einstein—Laub force density becomes equal
to the Helmholtz expression. Correspondingly, (4.10) reduces to (4.15). In general the Einstein—
Laub tensor is indistinguishable from the tensors of Abraham, Minkowski, or Helmholtz in any
mechanical equilibrium-experiment measuring forces or torques on a body immersed in a liquid.
The extra term 4n(E~P) in the surface pressure, characteristic for the Einstein—Laub theory, is
balanced by the corresponding excess pressure 4E~P predicted in the liquid.*
(5) The Peierls tensor. In the optical case it becomes of interest to examine also the predictions
of this tensor. As far as the accompanying momentum is concerned, Peierls’s analysis is essentially
similar to the one we have given above. The pure electromagnetic momentum density is thus taken
to be given by the Abraham expression, cf. (1.11). For the wave considered above, the force component f
5”, given by (1.10), is imagined to produce the accompanying momentum
—
2— 1 [1
grnech,x =
*
—
a(n2
—
1)] (E x H)~.
(4.16)
n
The argument given in [11, p. 32] ignored this compensating effect from the liquid. it is therefore not correct, as stated there,
that one would obtain a critical test of the Einstein—Laub tensor by measuring the torque exerted on a test body immersed in a dielectric
liquid. The predicted net torque would be the same as for the other tensors.
162
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
Adding this to the expression (1.11), one obtains for the total momentum density component:
P
—
glOI,XgX
p
A.j
112
1-gfl
5ech,x~2Ln
1
+
~
(2
—a~n
i~21~
~
u~
j~..x
As regards the parameters a, t in Peierls’s theory, note the following. Peierls himself gave the
values a = =
If one instead puts a = ~, ~r= 0, then Peierls’s theory becomes the same as
Helmholtz’s theory, assuming that the Lorentz—Lorenz relation holds~In that case, for instance,
(4.10) and
for the
wave reduces
considered.
On Abraham—Minkowski
the other hand if onetheory
puts
2 (4.17)
1) ‘,become
~r= 0, identical
then Peierls’s
theory
to the
a(without
= —(n electrostriction).
Our purpose in the following is to give an analysis of some of the beautiful optical experiments
which are of interest for the electromagnetic energy-momentum tensor interpretation. Although
concentrating mainly on recent experiments, we shall devote the next section to some comments
on the fundamental Jones—Richards experiment from 1954.
~.
—
4.2. Jones and Richards’s radiation pressure experiment [17]
This is one of the most important experiments in phenomenological electrodynamics. A ray
of light, emitted from a 30 W tungsten filament lamp, was sent through a glass window into a
dielectric liquid and reflected normally from a metallic wall of rhodium-plated silver immersed
in the liquid. Investigating various liquids (including water and benzene), the radiation pressure
on the wall was found to vary in proportion to the refractive index n of the liquid. (Actually, the
metallic test body was a vane mounted on a torsional suspension, and the quantity subject to
measurement was the torque on the vane produced by two asymmetrical rays.) We have discussed
this experiment earlier [10, p. 30], but shall here give some supplementary comments on it.
The condition for stationarity is amply satisfied in this case. To discriminate against convection
currents, the authors made use of a chopping technique implying that the positions of the two
light spots on the vane were switched about twice per second. The pulse duration of about 0.5 s
is immensely greater than the period of time necessary to establish pressure equilibrium. Accordingly, the electrostriction forces as following from the Helmholtz theory are everywhere balanced
by elastic pressures, and the electrostriction effect will have no influence upon the observable
pressure on the wall.
Let us now calculate the radiation pressure in three different ways, concentrating mainly on
the Abraham—Minkowski interpretation. It is instructive to take into account that the conductivity
a of the metal is large but finite. For silver, a = 6.29 x iO~S/rn (S =
1), whereas for rhodium,
a = 2.21 x iO~S/rn. In both cases the inequality (a/wEm)2 >~ 1 is amply satisfied, where Cm 15
the permittivity of the metal. Further, define the constants
~
=
,.Jpowa/2,
k
=
nw/c.
(4.18)
Here 1/cx is the penetration depth in the metal and k is the wave vector for the fields in the liquid.
Under the present conditions, 1/cx is only a few nanometers. Also, the parameter k/x is small
compared to unity. For rhodium, one finds that k/~is somewhat less than 0.1. We shall include
terms of the first order in k/~in the expressions for the fields, assuming that the wave propagates
in the x-direction and impinges normally on the wall at x 0. Assuming that the incident field is
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
163
polarized along the y-direction we obtain, using the complex representation,
E
E0 [et~~_05t)
=
—
,.J~e_~ et)]
1~ — tOt) + ~
H~=
[e
E~=
(1
i)eax
—
H~= !~~[2 + (i
—
—
iö
x0
(4.19)
0,
(4.20)
e — j(~+ tot)]
et~~0t)
,
1)
~1e~ e~~~t)
x
where
R
=
1
—
2k/a,
tan 5
=
—
k/a.
(4.21)
E
0 is the amplitude of the wave and R is the reflection coefficient at the wall. We have put p
=
also in the metaL The above expressions contain some more terms than the analogous expressions
given by Peierls [13]. The terms contained in (4.19) and (4.20) are sufficient to satisfy the boundary
conditions for E~and H~at the wall to the first order in k/cc.*
The radiation pressure a~can be calculated in the following ways.
(1) The most natural method is probably to make use of (4.11), where S~is taken in the liquid
just outside the wall. As the component S~is the same in the Abraham as in the Minkowski case
we get
2E~2+ ~J1o lH~I2= ~ (1 + R)S~,
(4.22)
= S~A~
= S~= ~C~n
where use has been made of (4.19) at x = 0. The quantities in the first line in (4.22) mean averages
over a period; similarly, ~ is the averaged Poynting vector for the incident wave in the liquid.
We see that the only tensor component ofimportance for the radiation pressure is the momentum
flux density component S
0. Thus the decomposition ofMinkowski’s momentum into an Abraham
field part and an accompanying mechanical part plays no role for the theoretical prediction ofthe
radiation pressure.
(2) One may alternatively calculate the radiation pressure by recognizing that the Lorentz force
density in the metal is directed in the x-direction and has the magnitude p0crE~H2(in the real
representation). The radiation pressure is equal to this expression integrated over the volume of
the metal, for unit surface area. As the penetration depth is so small, we can integrate over x from
x = 0 to infinity, and obtain in the complex representation
=
~PoC Re
JEYHZ*
dx.
(4.23)
* There are some trivial but confusing calculation errors on pp. 507 and 508 in Stratton’s book [43]. The term 1 in the denominator
29
in his eq. (85) for tan ö1 should be replaced by 2 and in his eq. (88) for tan a factor 2 should multiply cos 0. In particular, his eq.
(87) gives twice the correct result. The corrected expression is the same as (4.21) above in the special case of normal incidence.
164
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
Inserting (4.20) in this integral we find the same result (4.22) as before. This method was employed
also by Peierls [13], who found the result (4.22) in the limiting case R = 1.
We see that the method used here has no close relationship to the electromagnetic energymomentum tensor; the method rests only on the expression for the Lorentz force density in a
non-magnetic metallic medium and that expression is unambiguous. However, there is another
possible effect not yet considered, namely the surface force density whose magnitude in the xand which is due to the difference between the permittivity Em of the
direction is ~(C Em)
metal and the permittivity C of the liquid. (Note that when the conductivity a is finite, as assumed
here, the integral (4.23) calculates the radiation pressure as a volume effect. If there in addition are
surface forces due to differences in permittivity on the two sides, they have to be considered separately.) In the present case it turns out, however, that this surface force can be omitted, because E~
is according to (4.20) of the order of k/~and its square is negligible. The corresponding magnetic
surface force term vanishes because the calculation assumes that the metallic body is non-magnetic.
Thus, (4.23) gives the radiation pressure correctly within the accuracy aimed at.
(3) Thirdly, the pressure can be found by the following argument, which demonstrates the
direct applicability of the Minkowski theory in optics. Actually, this argument was a main reason
why we in [10] advocated the idea that the Abraham field momentum density is accompanied
by the mechanical part grnech. (Similar ideas had actually also been put forward earlier, for instance
by Ginzburg [36].) Assume that the incident wave is described by some energy-momentum tensor
S~about which we know that it satisfies the relation
2
—
E~I
3XS~ +
3g~/3t= 0,
(4.24)
where g~j= iS~/c.When all fields are functions of the argument (x ct/n) it follows that
(S~ cg~/n)is a constant. As this constant vanishes when the fields vanish, we conclude that
any tensor satisfying (4.24) will also satisfy the relation
—
—
—
S~= cg~/n.
(4.25)
This relation expresses that the momentum of the incident wave travels with velocity c/n. Now
starting from Abraham’s tensor and making the assumption that the Abraham term produces an
accompanying momentum grnech, we get Minkowski’s tensor which satisfies (4.24) and hence (4.25).
The latter expression, which in the Minkowski case can be written as nS~/c,gives the rate of momentum that is taken away from the incident wave per unit cross section. A similar term nRS~/c
holds for the reflected wave, and the rate of total momentum transfer is the sum n(1 + R)S~]/c
of the two terms. When there is no complication arising from extra divergence terms in the force
density, this rate of momentum transfer will simply give the pressure on the wall. The wave is in
this case behaving essentially like a free particle in the reflection process. The agreement with
(4.22) shows that the Abraham—Minkowski case is just of this kind. The argument shows the
straightforward applicability of Minkowski’s tensor, and it shows the close relationship, rather
than conflict, that exists between Abraham’s and Minkowski’s tensors in the steady optical case.
As mentioned earlier, a direct observation of the rapidly fluctuating Abraham term is not possible
in steady optical fields. Recall the contrast with the quasistationary state, where the oscillating
Abraham term itself was subject to observation.
So far, we have considered the mechanical momentum. Some remarks are in order, as regards
the corresponding momentum flux. Besides the electromagnetic momentum flux density S~in
the incident wave there is a mechanical momentum flux density associated with the drift of particles
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
165
in the x-direction. This mechanical momentum flux, unlike the mechanical momentum itself, is,
however, vanishingly small and does not contribute to the radiation pressure. As a simple estimate,
ignoring electrostriction, we may put the longitudinal component of (4.3) equal to pv~,where v,
is the drift velocity. The average mechanical momentum density becomes
1)nC
p <vs> = -~--(n2
0E~,
(4.26)
2c
—
from which we conclude that the ratio between2/C
the mechanical momentum
flux and the
electro2,which is completely
negligible.
magnetic
momentum
flux
is
of
the
order
p
<v~
0E~
C0E~/pc
The observable radiation pressure is therefore due to the electromagnetic momentum flux only.
At this point one might be tempted to use the hydrodynamical equations to obtain more accurate
description of the density or pressure variations in the wave. For instance, Mikura [44] has
proceeded in this way. It should, however, be pointed out that the fields are varying so quickly
that local hydrodynamics cannot be expected to hold. The condition for the validity of the hydrodynamical approximation in general is that the fields period is much greater than the intermolecular
collision time. In a typical liquid the intermolecular distance is ~ 1 nm and the velocity of sound
1 km/s, giving an intermolecular collision time 1 ps. This time interval is actually much larger
than an optical period, and to speak about local pressure variations in the propagating wave does
not seem to be physically meaningful.
Next consider the effect from electrostriction. First of all, as this is a steady wave problem, we
know in general that the electrostrictive contribution to the pressure must be non-observable.
The present case is actually more simple: the electrostriction force at the wall, and hence also the
excess elastic pressure, are within the present approximation equal to zero. This is evident from
(4.13), which shows that Ap” is proportional to E2, and from (4.19), which shows that E2 for x = 0
is proportional to k2/x2 and hence negligible. Assume for simplicity that R = 1. Then (4.19) shows
that the electric field in the liquid has form of a standing wave with one node lying at the wall.
On a time average, there act electrostrictive lateral forces in the inhornogeneous outer regions of
the beam, tending to press the liquid inwards. The resulting pressure is balanced by mean longitudinal electrostrictive forces within each half-wavelength of the beam, i.e. between two successive
nodes.
Note that one will not be able to measure the electrostriction effect even by using a very short
laser pulse, corresponding to non-equilibrium conditions in the liquid. It is not difficult to obtain
non-equilibrium conditions. In the Jones—Richards experiment the beam width was probably
~ 1 mm; as sound waves need 1 ps to traverse this distance, one should obtain non-equilibrium
with a laser pulse lasting for instance 50 ns. Also in such a case, however, the electric field practically vanishes at the wall and thus gives vanishing electrostriction pressure. In the present case
the non-observability of the electrostriction effect thus has no bearing on the equilibrium conditions of the liquid. Note in this respect the essential difference from the Goetz—Zahn experiment
considered in section 2.4.
We round off these remarks on the Jones—Richards experiment by considering the alternative
energy-momentum tensors. Characteristic for the Einstein—Laub tensor is the existence of the
extra term ~V(E P) in the force density, cf. (1.9). This term obviously gives no contribution to the
pressure on the wall, so the Einstein—Laub tensor will not be given further attention. Of more
interest is the Peierls alternative: Does the Jones—Richards experiment test the values of the
parameters a, r? The answer is no. As regards the component S~in (1.12), the first and second
>
166
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
terms on the right hand side vanish because E~= 0, Hx = 0, while the third term is negligible
because E2 is negligible at the wall. The remaining fourth magnetic term is the sole term responsible
for the radiation pressure, and we evidently obtain the same result (4.22) as in the Abraham—
Minkowski case.*
It is worthwhile to point out here that the method (3) above, which made use of the total momentum in the wave,in general ceases to apply for arbitrary values of a and r. We know that the relation
8XS~~
+ 3g~j)~/at
= 0
(4.27)
holds for the incident wave, where ~
is given by (4.17). Similarly, the relation (4.25) will also
hold in the Peierls case. However, we will no longer have the property that the direct pressure
of the field on the wall equals the rate of total momentum transfer 2g~~c/n (we assume R = 1).
When calculating the expression for S~,we will find non-vanishing cross terms between incident
and reflected wave, giving a net contribution equal to ~C
2— 1) [1 + a(n2
1)]. This inter0E~(n
action term must be added to the rate of momentum transfer 2g~~c/n in order to give the pressure
S~x.This kind of behaviour is typical not only in the Peierls theory, where a = ~, but actually in
any theory for which a is different from —(n2 — 1)
Thus also the Helmholtz theory, corresponding to a = ~, shows the same feature. Only in the special case where a = —(n2
1) ‘ does
the cross term above vanish, so that the radiation pressure becomes calculable as the rate of
momentum transfer. With this value of a, and with ~ = 0, the Peierls expression (1.12) reduces
to the Abraham—Minkowski expression.
The analysis given by Peierls was actually somewhat different from that above. He considered
the case where a wide finite electromagnetic pulse impinges on the wall. One merit of his paper
is that it points out in such a case the existence of an extra mechanical momentum “deposited”
in the medium, due to the cross terms between incident and reflected wave in the force density.
When the pulse has left, this deposited momentum propagates in the medium according to the
laws of acoustics. If ~° denotes the total energy of the incident pulse per unit area, one finds by
integration over negative x and all t that the total deposited momentum per unit area is
.
—
.
~.
—
J
f~’dx dt
=
—
n
1 [1 + a(n2
—
1)].~’
(4.28)
(R = 1 assumed). Thus, in addition to the surface pressure a~= S~exerted by the field directly
on the wall, one must in principle consider also the fate of this deposited momentum. Note that
the deposited momentum is a transient effect, due to the finite longitudinal extension of the field,
for in the case of a steady wave the Peierls force density at an arbitrary position fluctuates out when
integrated over an optical period. Now it turns out, however, that only for ultrashort pulses is
there any possibility for the deposited momentum to play a significant role for the pressure. For
the momentum deposited by a pulse is distributed over a region which is of the same order of
magnitude as the pulse itself. If the extension of the pulse is much greater than the extension of
the real experimental apparatus, then the pulse will only be able to deposit a tiny fraction of the
momentum that would have been deposited if the liquid were extending to minus infinity. For
instance, a pulse of duration 1 ms would require dimensions immensely greater than those present
in a real experiment. If the pulse is very short, of duration 1 ns, say, the situation is different. The
*
It should be stressed that this argument assumes normal incidence. Actually, as we shall see in the appendix, the case ofoblique
incidence provides an opportunity to test r.
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
167
pulse is then about 20 cm long, and the deposited momentum may take place within the liquid
in the experiment. Now one may assume, as the liquid is an elastic medium, that half of the deposited
amount (4.28) will travel in the positive and half in the negative x-direction. If the wall is great
enough, and if it acts as a perfect reflector for the sound wave, the total momentum given to the
wall from the sound wave simply equals (4.28). The total momentum given to the wall becomes
thus the amount (4.28) plus the amount 2n~’/c(i.e. the integral of (4.22) over the history of the
pulse, for R = 1). Comparison with (4.17) shows that the net result is the same as if the rate of
momentum transfer 2g~~c/n were simply integrated over the history of the pulse. Thus, using
a very short pulse, an experimental detection of the deposited momentum might be possible.
-
4.3. Ashkin and Dziedzic’s experiment on the radiation pressure on a liquid surface [8]
To provide some background for this experiment (cf. also section 1.1), consider first the situation
where a stationary optical wave, travelling in the horizontal direction, is sent obliquely through
a glass plate which is suspended in a torsional string so that it is able to rotate about the vertical
axis when subjected to a torque. This is the main feature of the Barlow experiment [9]. One generally observes a torque on the glass plate when the wave propagates through it. One should
recognize here that any alternative for the electromagnetic energy-momentum tensor will give
the same prediction for the observable torque. For under stationary conditions the ith component
of the force density can be written asJ = — 3,S,,. If (i, k, t) is a cyclic combination of indices, the lth
component of the torque on the test body can be written as
N,
=
,f (x,fk
—
xkf~) dV
=
—
(X,S~7~—
xkS~)n~
dS.
(4.29)
surTace
Here the isotropy of the test body has been taken into account. The last integral in (4.29) is taken
over a surface in the vacuumjust outside the body, and we see that the predicted torque is the same
for any alternative S~kfor the electromagnetic energy-momentum tensor reducing, as it must,
to the unambiguous form S~outside the body. The experiment thus cannot distinguish between
the various tensors. In particular, it gives no information about the direction of the local surface
force. In the Abraham—Minkowski theory the surface force density is ~-e0(n2— 1)E2, and is predicted to be directed from the medium towards the vacuum side. The great merit of the Ashkin—
Dziedzic experiment is its locality: it really shows that a narrow light beam impinging normally
on a free liquid surface acts upon the surface by an outward pull, in accordance with the prediction
of the simple Abraham—Minkowski theory. The theory of this experiment will now be considered
in some detail.
Figure 6 shows the geometry and the shape of the beam. We let the undisturbed free surface
be located in the plane z = 0, and let R denote the cylindrical radial coordinate. Ashkin and
Dziedzic used 20 pulses per second of single transverse mode doubled neodymium: yttrium—
aluminium—garnet laser radiation focused on the liquid (water) surface from above. The wavelength
of the incoming radiation was
= 0.53 pm, the peak power P°~
was 3 kW (low enough to make
nonlinear effects negligible), and the duration T of each pulse was about 60 ns. The radius of the
beam at the waist was reported to be only w
0 = 2.1 pm.
As the time periods during which the physical quantities in this experiment (intensity of the
beam, or elevation of the free surface) change appreciably are of the order of nanoseconds, it follows
168
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
Fig. 6. Shape of beam focused on the free liquid surface.
that a local hydrodynamical description of the liquid should be appropriate. For we recall from
the previous subsection that the mean time between two molecular collisions in the liquid is only
of the order of a picosecond, which is much less than the experimental time scale. We can therefore
introduce the concept of a local hydrodynamical pressure p. Moreover, we observe that the
dimensions are such that the time required for the beam to traverse the liquid is only about 0.01 ns.
This enables us to make the assumption that the electromagnetic forces start to act simultaneously
throughout the liquid. The motion of the free surface is determined by the electromagnetic surface
force, the electrostriction force, gravity, and surface tension.
Assuming that the beam is switched on at the instant t = 0, we obtain the following picture
for the distribution of forces that govern the motion of the liquid. In those interior regions of the
liquid where the field is inhomogenous, electrostrictive compressive forces immediately arise. The
produced elastic deformations propagate inwards at the speed of sound u. In addition, at the free
surface there act electrostriction forces tending to press the surface downwards. The magnitude
of this force, per unit surface area, is
1) (n2 + 2),
(4.30)
—~s0E2pdn2/dp = —*C0E2(n2
where the minus sign signifies that the force acts downwards. The electric field E is the rms-value
of the net field at the free surface, thus equal to the field in the transmitted wave when we ignore
the influence from reflections at the lower boundary of the liquid.
Apart from the surface force density (4.30), the free surface is subject also to the Abraham—
Minkowski surface force density acting upwards:
—
a~’= a~’= ~C
2(n2— 1).
(4.31)
0E
This result is found by integrating the z-component of the force density (1.3) over the boundary
layer of the free surface. In the Abraham case, the additive Abraham term appearing in (1.6)
describes a volume effect which obviously gives no contribution after integration over the boundary
layer. The Abraham and Minkowski surface force expressions are thus identical. It is necessary
to stress this point, because statements to the contrary have recently appeared [45].
Just after the onset of the pulse the free surface is subject to the combined effect of the forces
(4.30) and (4.31). Note that the negative force (4.30) is stronger than the positive force (4.31), so
that the net electromagnetic force in reality acts downwards to begin with. (Recall the analogous
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
169
effect in the electrostatic case considered in section 2.2.) This situation prevails until the elastic
disturbances produced by the electrostriction forces in the interior regions of the liquid reach the
free surface and begin to counteract the surface force (4.30). We expect that the time required for
this compensation is of the same order of magnitude as the time needed for sound waves to traverse
the beam. As the beam width at the waist was reported to be about 4 pm, and as sound needs
1 ns to traverse this distance, it follows that after a few nanoseconds one should not expect that
electrostriction plays any significant role for the motion of the free surface. This period of time
required to create electrostrictive pressure equilibrium in the liquid, of duration a few nanoseconds,
may be called the first time scale in the problem. There is also a second time scale here, namely
the time duration of the electromagnetic pulse itself. As mentioned above, this was about 60 ns
and therefore considerably greater than the first time scale. (The motion of the free surface actually
needs hundreds of nanoseconds to develop.) As the two time scales in the problem are so different
it is clear that, apart from small transient phenomena just after the onset of the pulse, one may
describe the motion of the free surface in terms of the Abraham—Minkowski force (4.31), together
with gravity and surface tension. We shall now proceed in this way and give a local hydrodynamical
theory of the experiment with omission of the electrostriction term (4.30). This means actually
that we regard the liquid as incompressible. In the latter part of this section we shall return to a
calculation of the electrostriction effect.
Incompressible theory
The assumption about incompressibility corresponds formally to setting the velocity of sound u
equal to infinity. The theory given below is based upon the theory recently given by Lai and
Young [19]. Neglecting viscosity, we can introduce a velocity potential i~t and write the local
velocity as v = V1. The equation of continuity V21 = 0 is automatically satisfied by the expansion
~(R, z, t)
=
Idk k crs(k,
t)
ekz J
0(kR),
(4.32)
where cylindrical symmetry has been assumed. We let h(R, t) mean the elevation of the free surface.
The focal length of the surface lens was observed to be f ‘~ 25 pm, corresponding to a radius of
curvature ~ ‘~ 6 pm. With w0 2 pm this corresponds to h ‘~ 0.3 pm, which is much less than
the diameter of the beam. It should therefore be appropriate to calculate the pressure from the
linearized Euler equation [29, p. 37]:
p
=
—p8D/3t
—
(4.33)
pgz.
We put p
= 0 above the liquid. The boundary condition at the free surface is obtained by setting
the pressure (4.33), evaluated at z = h, equal to the net surface force density acting downwards.
This surface force density is
composed
of xtwo
parts.
The tension
first part
is the surface
pressure
given
by
2h(R,
t), where
is the
surface
coefficient
between
air and
water.
Laplace’s
formula
as
xV
This
expression
supposes
that the curvature ofthe surface is small. The second part is the Abraham—
Minkowski surface force density (4.31), acting upwards. Neglecting the slight curvature of the lens
surface we shall evaluate this part as if the surface were plane and horizontal. We thus obtain the
following condition at the free surface:
—
p3~1/ôt+ pgh
—
c~V2h= ~C
2 1)E2
—
0(n
(4.34)
170
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
(recall that E is the rms-value ofthe net tangential field. This field is to be calculated asif the surface
were kept fixed in its initial position z = 0). The equation (4.34) agrees essentially with that obtained
Lai and Young, although their reasoning seems to be quite different. (They started from Gordon’s expression for the electromagnetic force density [18], and assumed this expression to hold
also over the inhomogeneous boundary layer at the free surface.) Using the expansion
by
h(R, t)
=
Jdk k i~(k,t) J0(kR)
(4.35)
2, and using the relation kc!i
=
3i~/3tthat holds for small amplitude
and
a similar
expansion
for E
waves,
we obtain
from (4.34)
32/~~k
t’
2
‘
k
+ Q2i~(k,t)
=
E
2
—
1)—
E2(k, t),
(4.36)
0(n
where ~ = [gk + ~k3/p]112is the frequency of the capillary gravity waves. One expects that the
most important values of k are of the order k w~1 0.5 pm~, which corresponds to
~ 10~
To solve (4.36) with respect to we have to model the pulse. We write
~
E2(R,
t)
=
E2(R) T(t)
(4.37)
where T(t), normalized to a maximum value of unity, describes the time development of the pulse.
We follow Lai and Young in taking
2
T(t)
where e
=
=
=
=
+e2
a,(t/t~)2exp (— t/t,),
(4.38a)
2.718~and
17.5 ns,
a
50 ns,
1
a2
=
=
0.9
0.25.j
(4 38b)
The a~are dimensionless. The following behaviour of this function should, however, be noted.
Plotting T(t), as given by (4.38), as a function of t we find that theform of the curve agrees reasonably
well with the form of the experimental curve given by Ashkin and Dziedzic. The two curves are,
however, displaced with respect to each other; the theoretical curve has its maximum at t = 40 ns
whereas the experimental curve has its maximum at t = 100 ns. Reasonable overlap is obtained if
we redefine the experimental origin of time (the onset of the pulse) and displace the experimental
intensity curve by 60 ns to the left along the time axis. Accordingly, for the comparison between
theory and experiment we shall simply displace all experimental curves given by Ashkin and
Dziedzic by 60 ns to the left.
2(R) in (4.37). Lai and Young representconsiderby
theachoice
of an
appropriate
foraEcut-off parameter. In contradistinction
ed Next
the intensity
Fourier
integral
over k,function
involving
to this, we shall in the following assume a Gaussian intensity distribution; this distribution is a
good approximation for many lasers and it is moreover an ideal, that would be realized by a
perfectly constructed laser. We shall write the distribution at the free surface as
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
E 2 (R)
=
2
2 exp
2P°~
(n +4 1) iu~
0cw0
\
/
—
—~—j,
2R2\
W0
171
(4.39)
/
where P°~
is the total energy flux of the incident radiation and w0 the beam radius at the waist.
In this expression we have taken into account
that
part of
theintegrating
incident flux
reflected
at the
2 in
thealiquid
and
overisthe
cross section
surface:
forming
Poynting’s
vector
S
=
e0ncE
of the beam we find that the transmitted energy flux is 4n(n + 1)- 2 p(i) i.e. the transmission coefficient at the surface multiplied by the incident energy flux. From (4.39) and the relation
-
exp(_~2)
=
~w2o1dkkJO(kR)exp(_*k2wo2)
we find
E2(k, t)
=
E2(k) T(t)
=
4
P(’)T(t)
(n + 1)2 21rE
2w~).
0c exp (—*k
(4.40)
-
When (4.40) and (4.38) are inserted in (4.36) and the equation solved subject to the boundary
condition h = 0, öh/3t = 0 for t = 0, we obtain
~(k, t)
n
1 ke2P~~
n + 1 4irpc [exp (—*k2w~)]~ a.r~K,(Q,t),
—
=
(4.41a)
where we follow Lai and Young in introducing the abbreviation
K,(1l, t)
=
(1 + Q22)-3 ~2(1
+ e_t1t1[2(3
—
—
f~2t?)
3f~2t~) ~t
t
—
2(3
—
+ 4 —(1 + ~2r~) +
f~2t~)cos Qt
—i-
(1 +
Q2~?)2]}.
(4.41b)
Now h(R, t) can be found by inserting (4.41) in (4.35). Ashkin and Dziedzic gave a graphical representation of the reciprocal of the focal length f of the ideal lens which best fitted the observed
beam shape. For the comparison with experiment we therefore wish to calculate
ln
11
nlV2h
(4.42)
The first of these expressions assumes that the surface of the liquid is spherical, whereas the second
assumes that the curvature is small. Actually, the above calculation shows that the free surface is
not spherical, for the factor J
0(kR) appearing in the integral in (4.35) will necessarily appear also
in (4.42) and make the radius of curvature ~t depend on the coordinate R. We expect that a reasonable approximation is obtained if we choose to evaluate the curvature at the centre R = 0 of the
bulge. We 2thus
J insert (4.41)
2 J in (4.35), and thereafter calculate (4.42) making use of the general
relation V 0(kR) = k 0(kR). Finally setting R = 0, we get
2e2P~
1f = (n
1)
8(n + 1)irnpc ~ a~t~
,
4K,(cI, t) exp (—*k2w~).
(4.43)
1[dk k
—
—
172
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
The integral in (4.43) must be calculated numerically. For the input parameter w0 it is natural
first to try the value w0 = 2.1 pm, i.e. the beam radius given by Ashkin and Dziedzich. Because
of the exponential in (4.43), the numerical integration can be terminated at kw0
10. We also
insert ~ = 0.073 N/m, n = 1.333, p(i) = 3 kW. The result that we obtain in this way is that the
theoretical prediction is in striking disagreement with observation; the calculated value of 1/f
is about 9 times as great as the observed one and even violates the small amplitude assumption
upon which the calculation was based. Moreover, the theoretical maximum for 1/f turns out to
occur already at t ‘~ 150 ns whereas the observed maximum occurs much later, at t
350 ns.
Possible explanations for this discrepancy can be given as follows.
(1) The intensity distribution of the incident beam may correspond to a greater value of w0
than the 2.1 pm reported by Ashkin and Dziedzic. For the same value p(i) of the incident energy
flux, the distribution of the flux over a broader surface area would clearly correspond to a smaller
elevation of the surface, and hence give a smaller value for 1/f We suggest that this explanation
gives the main reason for the discrepancy. We have calculated 1/f with various input values for w0.
The result turned out to be that 1/f is rather insensitive with respect to variations in w0 up to
w0 ~ 2.5 pm, whereafter it decreases rapidly with increasing values of w0.
(2) As mentioned above, the expression (4.43) corresponds to calculating the curvature at the
centre R = 0 of the bulge. If (4.43) were calculated for a different non-vanishing value of R, one
should expect a smaller result because the extra factor J0(kR) suppresses the integral. However,
when the real bulge is to be approximated by a spherical surface, it appears most plausible to choose
the centre R = 0 for the evaluation of the curvature. This is also in agreement with the treatment
of Lai and Young.
(3) The calculation assumes that the free surface of the liquid is smooth. One must, however,
expect that the surface is more or less irregular and covered by fine scale ripples that are responsible
for the observed surface scattering of the incident light. Ashkin and Dziedzic report that the
scattered light is observed mostly as a broad halo in the forward direction. Because of the irre-
gularities in the surface one should expect that the elevating power of the beam is somewhat
lowered.
(4) There are uncertainties in the experimental determination of the incident energy flux p(i),
and perhaps also noticeable uncertainties in the value of the surface tension coefficient c~depending
on the degree of impurities in the water. We expect, however, that the influence from these factors
on the value of 1/fis relatively small. Also, the variation of x because of heating seems to be negligible as the absorption of light in water is small.
(5) The assumption of a Gaussian intensity distribution for the beam may be inadequate.
We may take all the above effects into account by introducing an “effective” beam radius w0,
while otherwise keeping the formalism and the remaining parameters unchanged. Trials indicate
that the choice w0 = 4.5 pm gives reasonable agreement between theory and experiment. Figure 7
shows the theoretical curve calculated from (4.43) with this value for w0. Comparison with the
experimental curve of Ashkin and Dziedzich, displaced by 60 ns to the left, shows good agreement
in the region about maximum, whereas the experimental curve is seen to be somewhat broader
than the theoretical one.
We have calculated also the theoretical elevation h(0, t) of the centre of the bulge, assuming
w0 = 4.5 pm. The result is that the surface is predicted to rise to a maximum of 0.9 pm at t ~ 450 ns,
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
173
4(m~)
10
100
200
300
400
500
600
700
Fig. 7. Time development of the inverse focal length 1/f Full line gives the theoretical result calculated from (4.43) with the effective
beam radius w
0 = 4.5 jnn. Broken line gives the experimental result of Ashkin and Dziedzic [8] displaced by 60 ns to the left.
whereafter it falls relatively slowly. At t ~ 1000 ns the height will have decreased to one half of its
maximum value.
It is worthwhile to summarize the points where the above calculation differs from the pioneering
calculation of Lai and Young:
(i) For reasons given above, the form (4.38) for T(t) implies that we have displaced Ashkin and
Dziedzic’s curves by 60 ns to the left.2 appears in the main result (4.43) because of the fact that E
A correction
factor
4(n + field
~ at the surface. The result of Lai and Young corresponds to
in (ii)
(4.34)
means the net
tangential
putting E in (4.34) equal to the field in the incident beam.
(iii) Whereas the expression for E2(R) was above taken to be the Gaussian (4.39), the corresponding expression E2(R)Ly in Lai and Young’s theory can be written as
E2(R)Ly
=
4
p(i)
(n + 1)2 21rE
0c Idk k J0(kR) exp
k
3/2
[—
(~~_)
]
(4.44)
(we have inserted an extra factor 4(n + 1)- 2 in front to make the expression directly comparable
to (4.39)). Here k0 is a frequency cut-off parameter that was chosen equal to 0.5 pm 1, i.e. equal
to the inverse value of the beam radius given by Ashkin and Dziedzic. The reason for the choice
(4.44) was to make the integral in (4.43) amenable to analytic evaluation. *
Finally, the following point should be noted. Above we encountered difficulties in reconciling
theory with the Ashkin—Dziedzic result w0 = 2.1 pm for the beam. It is therefore at first somewhat
surprising that Lai and Young obtained reasonable agreement with observation using the Ashkin—
Dziedzic input data in an apparently straightforward way. The explanation for this agreement
seems, however, to be the following: Their choice 0.5 pm~’for the value of the cut-off parameter k0
implicitly determines the beam radius to be actually greater than 2 pm. To show this, we calculate
the ratio of the expression (4.44) to the square of the field at the symmetry axis R = 0 (i.e. the
relative intensity):
2(R)Ly
3 ~ dk k J
312]
445
EE2(0)Ly
0(kR)
exp
(k/k0)
2k~f(4/3)
—
—
[—
.
(
.
)
~It ought to be mentioned that the exponent 10/3 in the denominator in the last term in the parenthesis in Lai and Young’s eq. (15)
should be replaced by 5/3.
174
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
The variation of this ratio with R is shown by the full line drawn in fig. 8. The broken line in the
same diagram represents the relative intensity exp 2R2/w~)following from the Gaussian distribution (4.39), assuming w
0 = 4.5 pm. We see that the intensity profiles are in fact nearly the same
in the two cases. Thus Lai and Young’s assumption that k0 = 0.5 pm~ implies that their beam
becomes approximately as wide as assumed in our theory. That the two theories give results for
1/f and h that are in reasonable agreement with each other is therefore what we might expect.
(—
~R(~tm)
Fig. 8. Relative intensity of laser beam
2/w~),
as wfunction
ith w of R. Full line gives the Lai—Young result (4.45), with k0
gives the Gaussian result exp (— 2R
0 = 4.5 ~m.
=
0.5 ~m
~.Broken line
Both theories indicate that the laser beam was in reality somewhat broader than what Ashkin
and Dziedzic reported.
Electrostriction: Calculation of the transient pressure variation
Although we have indicated above that electrostriction has no influence on the gross time
development of the height of the free surface, it is of physical interest to consider the onset of the
electrostriction force and its gradual compensation by elastic pressures in some detail. The order
of magnitude analysis given above indicates that the time needed for equilibration is roughly
the time that sound needs to traverse the beam at its waist, i.e.
1 ns, which is small compared
to the time scale of the motion. We must expect, however, that the elastic counter pressure in the
physically interesting region near the free surface will need somewhat longer time to develop,
because it is due not only to elastic waves produced by nearby forces but also to elastic waves
that have propagated much longer in the liquid and hence used longer time. We are not aware that
any treatment of the equilibration process has been given before, and will therefore in the following
give a description of it, assuming a simple model.
We shall consider the following situation. The undisturbed liquid is assumed to extend from
z = 0 to z =
x. A Gaussian laser beam impinges normally on the free surface at the instant
t = 0 (cf. fig. 6), and the intensity is thereafter held constant. The time function T(t) appearing in
(4.37) can thus in the present case be written as T(t) = 0(t), where the step function 0(t) equals 1 or 0
according as t > 0 or t <0. Because the velocity of light is immensely greater than the velocity
of sound, we may assume that the electrostriction forces start to act simultaneously throughout
the liquid.
Concerning the intensity distribution, we recall that (4.39) holds only at the beam waist. For
—
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
175
general values of z we have for a Gaussian beam [46]
4
(n +
2
E (R, z)
where
w(z)
2P~
1)2 JEE
cw2(z) exp
2R~1
[—w2(z)j’
[
(4.46a)
2/l~]~2
(4.46b)
0[1 + z
is the beam radius and 1R = irw~/~.
is the Rayleigh length. Here ,~is the wavelength in the liquid.
The Rayleigh length is a measure of the distance over which the beam may be kept collimated.
Insertion of w
1R ~ 35 pm, whereas w
1R ~ 160 pm. In the first
0
= 2.1 pm yields
0
= 4.5 pm yields
case 1R is about 8 times the diameter at the waist, while in the second case the ratio is about 18.
Is is therefore likely that in both cases the broadening of the beam, as given by (4.46), plays only
a minor role for the development of the pressure, but as a reassurance we will keep the broadening
effect in the formalism below.
Next consider the equation of motion for the elastic waves in the liquid. Still restricting ourselves
to first-order wave theory we may study the electrostriction force independently of the other
forces acting on the liquid in the real experiment: the Abraham—Minkowski surface force density,
gravity (which is a very small effect), and surface tension. Thus, ignoring all other effects than
electrostriction we can write the force density in the liquid as V~,where
=
w
2pdn2/dp = *C 2(n2 — 1) (n2 + 2)
(4.47)
0E
0E
may be called the electrostriction potential. Apart from the volume forces in the interior of the
liquid the surface force density, given simply by (4.47), is a constant for t > 0.
The linearized Euler equation reads
x
=
~e
p dv/dt
=
—Vp + V~.
(4.48)
Taking the curl of both sides, we obtain V x v = constant. As (4.48) must hold also before the
arrival of the beam, it follows that V x v = 0. Also in the present compressible case we have thus
the opportunity to introduce the velocity potential c1. Insertion of v = VD in (4.48) yields
p
=
—p 3c1/ôt +
(4.49)
x.
Let us calculate the time development of the electrostriction pressure p(0, t) at the origin, R = 0,
z = 0. The pressure follows immediately from (4.49) when D is known at the origin. The differential
equation for D follows by combining (4.49) with the linearized equation of continuity. We obtain
— -~
s—?
=
—
—~
~,
(4.50)
where u = [(äp/3p)~]1~’2
is the velocity of sound. Note that in the limiting case u —+ x, (4.50)
immediately yields Laplace’s equation, which is characteristic for incompressible phenomena.
Equation (4.50) must be solved with appropriate boundary conditions at the free surface. This
task can be accomplished in an approximate way by the following simple trick. It is reasonable
to assume that the rise of the elastic pressure near the free surface is not much influenced by the
small motion of the surface, induced from t = 0 onwards by the constant surface force density
(4.47). We may therefore calculate the pressure as if the free surface were fixed in its initial position
176
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
z = 0. The boundary condition for ‘F on the plane z = 0 is 3D/3z = 0, i.e. the Neumann boundary
condition. This again enables us to use the method of images and calculate D from the fictitious
situation in which the liquid, together with the beam, extends from z = — cc to z = cc. Since the
fictitious situation is completely symmetric about the plane z = 0, it is clear that the normal
velocity 31/äz vanishes on this plane. The real and the fictitious situations must give the same
result for the potential in the physical region z <0 as they correspond to the same differential
equation and the same Neumann condition on the plane z = 0. The solution of (4.50) at the
origin in the infinite liquid is simply the particular integral
D(0,
t)
~[~
4irpu2 J~~x(r’;,~’)/t3t’
=
— t’ —
d3r’ dt’,
(4.51)
where r’ is the radius vector to a source point. The solution (4.51) gives the potential at the origin
as a superposition of spherical waves emitted at the retarded time t’ = t — r’/u. As the time dependence of x is given by the step function 0(t), satisfying dO/dt = ö(t), we have
1(0, t)
1~:~)
~(~
— uj d3r’.
\
1 2
4irpu j r
=
(4.52)
We insert (4.47) and (4.46) in this integral, and transform it to cylindrical coordinates (R, z) using
the relation r’ = (R2 + z2)~2.We obtain
1(0 t)
—— (n —n1) +(n21 + 2)
2P~
3irpu2cw~
Jf~ 1
dz
~“
R dR
+ z2/l~ (R2 + z2)~2
—~
J
0
x ex~[_ W~(l2~
2/l~)]~[t
- ~(R2 + z2)h/2]
(n
=
—
[
2 + 2) 4p(i)
dz
ex
1)
(n
n + 1
3irpucw~ 1 + z2/l~ P[
J0
1
2(u2t2 —z2)
w~(1+ z2/l~)]~
(4 53)
Differentiating this relation with respect to t and inserting the result in (4.49), we obtain the following expression for the pressure p(O, t) at the origin:
Ut
p(O, t)
=
x(O) ~
u2t2
4ut
dz
+ u2t2 + w~J(l + z2/l~)2exp [
—
2(u2t2 —z2)
w~o(1+ Z2/l~)]}
(4.54a)
Here
x(O)
=
(n
—
1) (n2 + 2) 3ircw~
4p(i)
n+1
(4.54b)
is the constant space part of the electrostriction potential (4.47) at the origin.
The integral in (4.54a) must in general be calculated numerically. Figure 9 shows the calculated
relative pressure p(O, t)/~(0)as a function of t. The lower time scale assumes that the beam radius
at the waist is w
0 = 4.5 pm. It is apparent that there is a transient effect following the onset of the
pulse; the relative pressure increases to about 1.3 when t is about 3 ns, whereafter it decreases
asymptotically towards the equilibrium value of unity. The figure indicates that when t ~ 8 ns
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
177
140
~
tIns)
Fig. 9. Time development of the relative pressure p(0, t)/y(O) at the origin, when a uniform beam is switched on at the instant t = 0.
Lower time scale corresponds to the value w
0 = 4.5 pm for the beam radius at the waist, whereas upper time scale corresponds to
w0 = 2.1 pm. The pressure approaches the equilibrium pressure x(O) for large values oft.
the pressure deviates only slightly from its equilibrium value. This result is in satisfactory agreement with, and justifies, the conventional order of magnitude analysis given above. After the
transient stage just following the onset of the pulse, electrostriction generally plays no role for the
elevation of the free liquid surface.
The upper time scale in fig. 9 assumes that w0 = 2.1 pm. In this case the equilibration time is
obviously smaller, about 4 ns.
It is useful to consider the simplified expression for the1R pressure
obtains
if the We
curvature
in (4.54a) one
equal
to infinity.
obtain
of the beam is neglected. This case corresponds to setting
p(0, t)
x(O) 2\/~utF(~1’~14t),
=
(4.55a)
where the function F is Dawson’s integral
F(x)
=
e
x~ J
e~2ds.
(4.55b)
This integral is tabulated, for instance, in [47]. From the tabulated values it is easy to estimate
when the curvature of the beam has negligible influence on the pressure. For it turns out that
x F(x) is at maximum when x ~ 1.5, which in dimensional units means that Ut ~ w
0. Similarly,
it is reasonable to expect that the expression
(4.54a)
is
greatest
when
Ut
is
not
very
differentfrom
w0.
1R in the integral can be assumed to be small ifthe maximum
The
influence
from
the
Rayleigh
length
value of z2/l~in this case, i.e. u2t2/l~~ w~/l~,
is much smaller than unity. As ‘R = irw~/)~,
the
condition can be written as
22
~ ir2w~.
Similarly, for the first term in (4.54a) we obtain the same condition.
(4.56)
178
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
The inequality (4.56) is amply satisfied under the present experimental conditions, and we
therefore expect that (4.55a) is a good approximation for (4.54a). Actually, it turns out by explicit
calculation that the differences between these two expressions are too small to be visible in the
graphical representation in fig. 9.
Alternative tensors
We conclude our considerations on the Ashkin—Dziedzic experiment with some remarks on
the tensors of Einstein—Laub and Peierls. The Einstein—Laub tensor need not be considered in
detail, as the gradient term ~V(E~P) in (1.9) clearly cannot play any significant role for the free
surface elevation. This term behaves qualitatively in the same way as the electrostriction term
V~considered above, and its action on the liquid will therefore be balanced by an elastic counter
pressure only a few nanoseconds after the onset of a pulse. The Einstein—Laub tensor thus gives
the same elevation as the Abraham or Minkowski tensors. The Peierls tensor, on the other hand,
ought to be considered. We can calculate the surface force density according to this tensor from
the momentum flux densities in the liquid and in the vacuum just above the free surface. In the
liquid, according to (1.12),
(4.57)
S~ = 4E0[n2 + 1 — a(n2 — 1)2]E2,
where E is the field in the transmitted wave. (Alternatively, this expression can be constructed as
(— c/n), using (4.17).) On the vacuum side we have
= ~
2 + 1)E2,
(4.58)
0(n
-
and the total surface force density exerted by the field becomes
a~’= S~ —
2 — 1)2E2.
(4.59)
0a(n
For a positive a, this expression is negative, corresponding to a downward force. Does this mean
that the Peierls prediction runs into conflict with the observed effect, which is an elevation? The
answer is actually no. The reason is that the term in Peierls’s force density responsible for the result
(4.59) has the form of a gradient and is hence of the electrostriction type. It is instructive to write
the relation between the Peierls and Abraham force densities in the form
JP
fA + cO~k[r(n — 1)2E
2— 1)(1 + a(n2 — 1))E2].
(4.60)
1E~]+ ~E0~,[(n
=
—+E
As the last term is a gradient term, its action on the liquid is balanced by a corresponding elastic
pressure almost immediately after the wave has entered the liquid. The single term in Peierls’s
force that can be tested in an experiment of the kind considered here is the term containing x in
(4.60). At normal incidence, however, where E~= 0, it is seen that this term gives no contribution
to the surface force. We conclude that the Ashkin—Dziedzic experiment is unable to distinguish
between the various tensors.
Finally, we mention that Wong and Young [45] have recently given a theoretical derivation
supporting the Peierls expression. As their arguments are of microscopical nature, they will not be
considered in this paper.
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
179
4.4. Torque experiments. Remarks on the quantized theory
The remainder of this paper will mainly be devoted to less extensive remarks on some further
experiments and experimental possibilities in optics. Obviously we cannot survey all experiments
which are of interest for the application of the electromagnetic energy-momentum tensor. We hope,
nevertheless, that it may be useful to present and comment upon some experimental situations
that we are aware of. The present section discusses angular momentum effects, and also some
aspects of the quantized theory of the electromagnetic field.
Roosen and Imbert’s experiment [48]
Consider the situation sketched in fig. 10: Light from an argon laser source, linearly polarized
in the plane of the figure, passes through a uniaxial anisotropic crystal whose optic axis is lying
in the plane of the figure. The beam, entering the crystal at I and leaving at I’, thus traverses the
crystal as an extraordinary ray and exerts a torque in the z-direction that in principle can be
measured. This is the situation investigated experimentally by Roosen and Imbert. Varying the
light intensity, and testing successively two different uniaxial crystals, they were able to verify
the theoretical prediction for the torque to within less than 10%.
OPTIC AXIS
~j~~4IIII~IIIIII1I1j__’
~J
.~
Fig. 10. Anistropic crystal traversed by an extraordinary ray.
Does this experiment distinguish between the different energy-momentum tensors? The answer
is no. Again, this is a point that should be stressed because conflicting opinions have been stressed
in the literature. Let us calculate the torque N on the crystal. The torque is in general equal to
the rate of change of total field angular momentum, with the opposite sign. Thus, if (i, k, 1) is a cyclic
combination of indices, we have for any energy-momentum tensor
N,
=
(x~fk—
X~f~+
5ik
— 5kt) dv,
(4.61)
the integration being taken over the whole crystal, including the boundary layer. This expression
shows that the local torque is predicted differently by the various energy-momentum tensors.
In the Minkowski case the non-vanishing torque component N~is composed of the following two
parts:
(1) A volume part due to the non-symmetry of the tensor. The torque density component in the
z-direction is S~— S~ = E~D~
— END, = e 2E2 sin r cos r (cf. fig. 10), n denoting the extraordinary refractive index and E the field in 0n
the crystal. If reflections from the surface are suppressed by means of an antireflection film, then the condition that the x-component of Poynting’s
vector in the crystal be equal to Poynting’s vector in the incident wave yields the relation
E = E~~~/(\/~
cos r), where ~ is the incident field. Integrating over the volume we find that the
volume part of the total torque is equal to (n/c)P~°b,
where Pt1~is the incident energy flux and b the
lateral shift.
180
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energ v-momentum tensor
(2) There is a surface contribution to the torque resulting from the forces acting in the boundary
layer. At the surface element I where the wave enters the crystal the surface force density is directed
in the negative x-direction and is equal to a~t= E0(1 — n)E~2. At I’, the surface force density is
equally large but oppositely directed. The contribution to the torque from the surface forces is
((1 — n)/c)P~~~b.
Adding this to the amount given under (1), we find for the total torque
1°b.
(4.62a)
N~= ~ P~b+ 1
n P~’~b
=
P
The characteristic feature of the Minkowski picture is that the momentum density vector D x B
is collinear to the wave vector k. The direction of the momentum density is thus oblique to the
direction of propagation of the wave.
Now consider the Abraham case. Because of the tensor symmetry the torque is interpreted as
being due to the forces only. Again, we can construct the total torque as the sum of a volume part
and a surface part:
(i) From (1.5a), under stationary conditions we can write the Abraham force density as
EkD). Using this, the contribution to the z-component of the torque from
the homogeneous interior region of the crystal (where fM = 0) turns out to be (n/2c)P”~b.
(ii) The surface force density
in the
is the sameatasI’. in
Minkowski
2atcomponent
I and an equal
butx-direction
opposite expression
In the
addition,
there
case:
a~
= a~’= C0(1 — n)E~
are tangential surface force density components, equal to a~’= —~C
tan
2 r at I and equal
but opposite at I’. The contribution to the total torque arising from a~0nE~~
and a~is ((2 — n)/2c)P~b.
Adding the contribution found under (i), we obtain the total torque
— ~k(EDk
1A =
N~=
—
-~- Pt°b+
2c
2
—n P°~b=
2c
c
P°~b,
(4.62b)
which is the same as (4.62a). Although the torque distribution is predicted differently in the two
cases, we find the same expression for the total torque. This is actually evident beforehand, for if we
insert the relation fE = — ~
applicable under stationary conditions, in (4.61) and perform
partial integrations, we obtain the expression to the right in (4.29). From this expression the result
(4.62) follows immediately. For the reasons given in the beginning of section 4.3 it is clear that the
result is independent of which energy-momentum tensor we use. Inclusion of extra electrostriction
terms clearly leads to the same result.
The experimental situation studied here has been discussed also in [49—51].
Q uantal
aspects. Proposal of measuring the angular momentum of circularly polarized waves
The phenomenological theory of electromagnetic fields within homogeneous media can conveniently be brought into a quantized form. For the case of isotropic media the theory has been
developed by Ginzburg [52], Jauch and Watson [53], Ryazanov [54], Brevik and Lautrup [40],
and others. In the theory the refractive index n plays the role of an input parameter.
As Maxwell’s equations in the medium can readily be obtained from a Lagrangian, it is natural
to use canonical formalism for the construction of the quantum theory. The characteristic features
of the canonical formalism is that its results are intimately related to the results obtained if one
uses Minkowski’s tensor from the outset. Thus the canonical expressions for energy, momentum
and angular momentum for the field are equivalent to the Minkowski expressions. (This is so
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
181
even in the more general case where the medium is in a state of uniform motion.) Let us consider
the angular momentum operator for a free radiation field. As discussed in [40], this operator can
naturally be divided into an orbital angular momentum part and a spin angular momentum part,
each of which is concerned. It turns out that circularly polarized photons carry spin ±h, just as in
the vacuum case (n = 1).
We are not aware that any measurement of the photon spin angular momentum in a dielectric
medium has been made. We suggest the following possibility, which is a simple extension of the
well known Beth experiment [55]: Let a monochromatic circularly polarized wave, of positive
helicity say, be passing from the vacuum into a dielectric liquid where it impinges normally on a
quarter-wave plate which transforms the wave into a state of linear polarization. (Alternatively,
one may imagine that the wave is simply absorbed by the plate.) We assume for definiteness that
the incident wave travels in the z-direction, and that the refractive index n~of the plate in the
x-direction is greater than the refractive index ny in the y-direction. The thickness d of the quarterwave plate is
d=
1
—
(4.63)
ny 2w
Note that d is independent of the refractive index n of the surrounding liquid. According to Mmkowski’s tensor each positive helicity photon in the liquid carry the spin h in the z-direction.
Neglecting reflections from the plate, we can write the z-component of torque N~as h times the
that per unit time flows into the plate. Here p(i) is the incident energy
number of photons P~t~/hw
flux in the liquid. Thus
=
p(i)/w
(4.64)
Assuming for simplicity that the beam enters the liquid from the vacuum through an antireflection
film, P°~
will be equal to the energy flux in the vacuum. Suspending the plate in a torsional string,
one can in principle measure the torque (4.64). Note that the torque is predicted to be independent
of the refractive index of the surrounding liquid.
As the result (4.64) does not contain h, it is clear that one can derive it without using quantum
theory. It is convenient to give the classical derivation, also in order to include the correction
from reflections at the surface. The field components in the incident wave in the liquid are
~
=
E
0 cos (nwz/c
—
wt),
~
=
E0 sin (nwz/c
— wt),
(4.65)
where E0 is the constant amplitude. In the crystal we have
2E0
E~= n~*+ 1 cos (n~wz/c— wt)
(4.66)
2E0
E~= n~*+ 1 sin (n~wz/c— wt),
where n~= n~/n,n~= np/n are relative refractive indices. (Following Beth, we have taken into
account the Fresnel coefficients at the boundary z = 0 where the wave enters the crystal, but have
for simplicity ignored reflections from the boundary z = d.) Integrating the torque density over
the volume of the crystal, we obtain in the Minkowski theory
182
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
N5
=
J(D x ~
dV
=
T~PW/w,
(4.67a)
where
T
=
2(n~+ n~)/{(n~
+ 1)(n~’ + 1)}
(4.67b)
is the transmission coefficient at the surface z = 0. This coefficient is usually near to unity, so that
we recover practically the expression (4.64).
The calculated torque is very small. For instance, p(i)
1 W, w
3 x 1015 s’ give
N~—~ 3 x 1016 Nm. This is the order of magnitude of the torque that Beth measured, using a
very delicate equipment. As the torque is inversely proportional to the frequency, one can increase
the effect by using microwave frequencies. Carrara [56] proceeded in this way. He used Ptt~ 7 W,
w ‘~ 6 x 1010 s~,corresponding to N~ 10_b Nm, which was easily detectable. It is likely,
therefore, that if the experiment is to be repeated with a dielectric liquid surrounding the plate,
it would be advantageous to use microwave frequencies. (It may also be mentioned that Allen
[57] has more recently measured the angular momentum of microwaves by observing the rotation
of a small electric dipole suspended in a circular waveguide. Such an arrangement does not seem
to be convenient in the present case, however, because of the viscous drag from the liquid.)
The discussion above was based on Minkowski’s tensor, as this is most natural. That Abraham’s
tensor leads to the same expression for the torque is clear when we use the relation f1 = —
to write the expression (4.61) as
—~
N,
=
—
J
(x,S~’Jl— xkS~’)n,dS.
(4.68)
surface
This formula is similar to (4.29), but the stress tensor S~’is now taken in the liquid just outside the
crystal. As the stress tensor in the liquid is the same in the Abraham case as in the Minkowski
case, it follows that (4.68) makes no distinction between the tensors.
It ought to be mentioned, however, that the form (4.68) is inconvenient for the evaluation of the
torque. In fact, if we insert in (4.68) the plane wave expressions for the fields, we find that the
integrand vanishes. This is an effect analogous to that encountered if one calculates in classical
field theory (for n = 1) the angular momentum of a wave along its direction of propagation: the
angular momentum is classically a boundary effect, associated with the finite width of the wave.
If one calculates the basic angular momentum integral $ (r x g)~d V along the direction of propagation z with the free field expressions used in the momentum g, one will obtain the result zero.
Conventionally one takes the finite width into account by transforming the integral by means of
partial integrations into a volume integral where the integrand does not contain r explicitly.
In this transformed integral the plane wave expressions for the fields are applicable (cf. for instance,
the discussion in [58,sect. 2—5]). In the present problem, if one wishes to calculate the torque on
the crystal starting form Abraham’s force density, one should simply insert the Abraham expression
(1.5a) in the general torque expression (4.61) and by means of partial integrations transform the
integral into the Minkowski form to the left in (4.67a), in which the plane wave expressions for
the fields can be inserted.
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
183
On the uncertainty relation
When dealing with the quantal formulation of the theory, it is of interest to discuss a generalization of the Heisenberg two slit-experiment. The generalization consists in imagining that the
whole equipment is imbedded in an isotropic dielectric medium of refractive index n. Slit experiments of the Heisenberg type have earlier been considered by Jones and Richards ([59], and private
information) and by Ratcliff and Peak [60]. In the usual case where n = 1 the situation, as discussed
for instance by Schiff [61,
sect. 4], is the following: A plane electromagnetic wave, travelling in the
x-direction, falls normally on a screen in which two slits are cut in the y-direction. Let d be the
distance between the slits. The diffraction pattern is observed on another screen separated a great
distance from the first one.
Quantum mechanics states in general that it is impossible to measure through which slit a
photon is going without destroying its interference pattern. This principle obviously cannot be
changed in the generalized case where the equipment is imbedded in a medium. The question is
now: Does the mathematical formulation of this principle, i.e. the Heisenberg uncertainty relation,
follow in the case of a medium as well as in the case of a vacuum? The answer turns out to be yes,
one assumes that de Brogue’s relation for the photon momentum
Px
=
h/2
=
hkx
(4.69)
holds also in the medium case. Here 2 is the wavelength in the medium. Let us verilS’ the statement
by considering the accuracy with which one can simultaneously obtain information about the
vertical momentum and position of a photon. Assume, like Schiff, that a set of photon indicators
have been placed behind the first screen. Let 0 be the angle between the forward direction and the
direction to the first diffraction minimum. From experiment, or from optical wave theory, one
has for the case d >~ 2 that 0 = 2/2d. The uncertainty of the vertical momentum component p~
of a photon after its encounter with an indicator is
Ap~
0p~= h/2d.
(4.70)
The uncertainty 1~zof the position of the photon indicator that recoiled is t~z d. Thus
~~p~iXz
h,
(4.71)
which is the usual Heisenberg uncertainty relation. We have thus verified that this relation emerges
if (4.69) is assumed.
Quantization of the electromagnetic field in the Minkowski picture generally yields the fourmomentum p,~= hk~,of a photon. This is a consequence of the fact that Minkowski’s tensor is
divergence-free, leading in the first place to constant values of the total field momentum GM
and energy ~M and in the second place to four-vector transformation properties of the combination (GM, i~2M/c).Minkowski’s theory thus fulfils the relation (4.69), and hence also the uncertainty
relation in its usual form (4.71), in the two-slit arrangement considered above. This fact provides
another example of the applicability of Minkowski’s tensor. Within the Abraham picture, the
quantization of the field turns out to be inconvenient [62].
184
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
4.5. Further experiments in optics. Measurement of the integrated effect of the Abraham term?
Optical levitation by radiation pressure
The stable elevation of a transparent sphere (dielectric solid sphere or liquid drop) in a laser
beam is another recent impressive experiment on the force action from optical fields on dielectric
matter. This experiment was also pioneered by Ashkin and Dziedzic [20,
21], and has been repeated and elaborated by Roosen et al. [63, 64]. It has been found possible to levitate and manipulate
freely charged and neutral spheres in the diameter range from about 1 pm to about 40 pm. This
levitation technique should be of importance in various areas, such as cloud physics, aerosol
science, fluid dynamics, and optics.
In our context the important question is: Does the experiment distinguish between the electromagnetic energy-momentum tensors? The answer is no. This is clear from (1.15), which shows that
any component of the force on the sphere can be calculated from the spatial part of the energymomentum tensor in the vacuum just outside the boundary, where all tensors are equal.
Let us consider briefly the theory of the elevation phenomenon. To describe the axial and
transverse radiation pressure force on the sphere, for arbitrary values of incident wavelength 2°~,
sphere radius a, and beam radius w, seems to be difficult. There are two special cases, however,
that can be handled relatively easily:
Case 1. a ~ w; size parameter q = 2ira/2~unrestricted. This is the case where the width of the
incident beam is sufficiently great to permit neglection of the transverse intensity variation over
the dimensions of the sphere. We may thus utilize the theory for the case where a plane wave is
diffracted by a sphere. The theory for this case was given by Debye [65].
Assuming that the beam
is directed in the z-direction, and that the sphere is non-absorbing, one can write the axial force
as [66]
=
(4.72a)
~ S@ira2Q,,~,
where
5(i)
is the incident Poynting vector and ~
—
Qpr = Qsca[1
the quantity
<cos 0>],
(4.72b)
Q~5being the normalized scattering cross section. If a, and b, are the expansion coefficients of the
scattered field in vector spherical wave functions, one can write Qsca as
2].
q i=i
=4
~
(21 + 1) [Ia,12
+ Ib,~
In the limiting case of very great spheres, i.e. for q
is the so-called asymmetry factor, given as
Qsca
<cosO>
r~q~2~
4
=
2
q
~
Qsca
‘‘
I
L
‘
/
(4.73)
1, one has Qsca
~s
21+1
Re(a,a,*vi + b,b,*+
1) + ,~,
+ 1
t~,t
~
=
2. Furthermore, <cos 0>
1
Re(a,b~’)
~.
(4.74)
+
The transverse force on the sphere is evidently zero. As anticipated above, this plane wave theory
should be applicable to the case of very small spheres in the laser beam. The experimentalists used
a TEM00
at ~ to= hold
0.5145
pm. Ashkin
reported
thata an
energy
t~argon
0.2 mWlaser
wasbeam
sufficient
a pure
glycerol and
dropDziedzic
(n = 1.47)
of radius
—~ 0.5 pm.
flux
P~
The drop was then probably situated at the waist of the beam. The radius of the waist, estimated
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
185
from the figures, is w0 ~ 10 pm. Now levitation occurs if F~= mg, where m is the mass of the sphere.
We assume that the drop is situated on the symmetry axis. For a Gaussian beam, the relation
between the incident energy flux P~’and the Poynting vector ~(i) on the symmetry axis is
p(i) = ~irw~S~.
We obtain
w~= ~
2irpagc
(4.75)
where p is the mass density of the drop. In this case, q = 6.1. Insertion of Q~,1= 1.1 (taken from the
3,
curve
pertaining
to
n
=
1.50
in
Irvine’s
paper
[66]
for
q
=
6.1),
P~’~
=
0.2
mW,
p
=
1260
kg/m
a = 0.5 pm yields w
0 = 7.5 pm, which is a reasonable result compared with the estimate mentioned
above. The plane wave theory thus proves useful for small spheres in the laser beam.
Case 2. Size parameter q >~s 1; beam radius w unrestricted. The theory for this case was worked
out by Roosen et al. [63, 64]. The smallness of the wavelength incomparison to the sphere diameter
implies that the geometrical optics approximation is applicable. Quantitatively, this occurs when
q ~ 100, which implies a ~ 8 pm in the case of an argon laser beam. The force components can
be calculated from the formula (1.15). Each ray of light is subject to multiple reflections at the
surface of the sphere, and the fractions of light transmitted in each case can be found from the
Fresnel coefficients. Summing over all multiple reflections, and afterwards integrating over all
incident rays, one arrives at the integral expression given in [64].
The expressions are somewhat
complicated, and the reader is referred to the original articles for details. (Note that eq. (2) in [63]
contains errors that are corrected in [64].)
Typically, if n = 1.50 and p(i) = 1 W, the maximum
value of the axial force F~for a symmetrically situated sphere occurs ‘at a/w = 1.06 and is about
0.7 nN. The transverse force depends in addition on the ratio a0/w, where a0 denoted the horizontal
offset of the sphere. If n = 1.50, p(i) = 1 W, and a/w = 1, the maximum transverse force occurs
at about a0/w = 0.75 and is about 0.5 nN. The measurements of Roosen et al., and also by Ashkin
and Dziedzic, support the theoretical predictions.
Frequency shift measurements
Consider the situation illustrated in fig. 11: A monochromatic ray emitted from a radiation
source at A is sent through a tube filled with a dielectric liquid and thereafter absorbed at B.
The front surface I is at rest, while the rear surface I’ moves to the right with uniform velocity v.
Liquid is continuously supplied through the inlet at C. If the refractive index n of the liquid is
greater than unity, as usual, one observes a frequency shift at B toward lower frequencies, i.e.
w1<w.
~
~
--~-~r
Fig. ii. Monochromatic ray propagating through tube filled With dielectric liquid. Right hand boundary moves to the right with
uniform velocity v.
186
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
The situation sketched here is similar to the one discussed theoretically by Lucas [67]. An
experiment of this type has been carried out by Grodzins and Phillips [68].
Their technique was
to move wedge-shaped solid bodies (Lucite) transversely to the direction of propagation of the
14.4 keY Mössbauer gamma radiation emitted from a 57Co source. (The purpose of the experiment
was to measure the refractive index of Lucite. At this high photon energy the refractive index is
actually slightly less than unity, but this aspect has no bearing on the present discussion.)
Two remarks are in order as regards experiments of this kind. In the first place, the experiment
is purely kinematic: The observed frequency co~is determined entirely by the four-vector transformation property of the four-dimensional wave vector k,
5 = (k, iw/c) under Galilean transformations (in the present case, = v/c ~ 1). This property is a consequence of the fact that the phase
of a wave is invariant with respect to a change of inertial system, and it is independent of which
energy-momentum tensor we use.
The relation between w and w1 can be derived as follows. Under stationary conditions the
laboratory system frequency w is unchanged when the wave propagates into the moving liquid and
gets dragged along with it. The frequency w’ in the inertial system S’ moving to the right with
uniform velocity v is w’ = w(1 — n/I); this frequency will not change in the system 5’ during the
passage of the wave through the boundary I’ because I’ is at rest in 5’. After the passage through
the liquid the frequency in 5’ is thus a!, and in the laboratory system it is co~= w’(l + /3)
= w(1 — n/J)(1 +
= w[1
— (n — 1)/I] to the first order.
The second remark that we wish to make is that this experiment, although basically kinematic,
may also be described in dynamic terms. This brings the energy-momentum tensor into consideration, and shows again the applicability of Minkowski’s tensor. Assume for simplicity that the
surfaces I and I’ are covered with antireflection films. If a wave packet of initial total energy ~*~‘
in the laboratory system is transmitted through I and thereafter dragged along with the moving
liquid, the total energy remains equal to A°all the time because the Minkowski force density within
the liquid is equal to zero. Transforming Minkowski’s four-momentum G~’= (GM, iA~M/c)in the
moving liquid to the inertial system 5, we deduce that ~~M’ = ~*‘(1— n/i). This is also equal to
the field energy after the transmission through I’ in 5’ because the field exerts no work on a surface
which is at rest. We obtain for the final energy ~*‘~ in the laboratory system ~ = ..~*‘[1 — (n — 1)/I],
which is in agreement with the formula for w1 given above in view of the proportionality between
energy and frequency.
This possibility of giving a straightforward dynamic description of the experiment is evidently
a consequence of the fact that the Minkowski four-momentum G~’is proportional to the photon
four-momentum k~:G~’= Nhk~,where the photon number N is a constant. A dynamic description
in terms of the other energy-momentum tensors would be more complicated.
/3
/3)
Transformation of the propagation velocity of the energy in a light wave
Consider a light wave propagating in the x-direction within an isotropic and homogeneous
non-dispersive dielectric medium. The propagation velocity of the light energy, the so-called ray
velocity, is equal to c/n. Then consider the situation in another inertial system in which the medium
moves with uniform velocity v in the x-direction. What is the magnitude of the ray velocity u in
this new inertial system? Making the explicit assumption that the ray velocity transform like a
particle velocity under Lorentz transformations, we obtain
=
(c/n) + V
1 + v/nc
(4.76)
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
187
To the first order in v/c,
(4.77)
u = c/n + (1 — 1/n2)v = c/n + civ,
where ~ is the drag coelflcient.
The assumption leading to the’ prediction (4.77) can be tested by observation. The noticeable
fact is that the theoretical expression for the drag coefficient has been verified with great accuracy.
Even the magnitude of the dispersive correction to ~ has been verified. The first experiment of this
type was carried out already in 1851 by Fizeau, who used moving water as dielectric medium.
In recent years, the experiment has been repeated with the use of ring lasers. See the paper by Bilger
and Zavodny [69],
and further references therein. This experimental fact ought to be reflected
in a simple way in the electromagnetic energy-momentum tensor, intended to describe the total
propagating wave. The conditions under which the quantity S/ W transforms like a particle velocity
under Lorentz transformations, have been examined by von Laue [4]and Møller [5]. It turns
out that S/W has the required transformation property if the quantities
R~
5= ~
+ ~ s,~~u5u~
(4.78a)
are equal to zero in some inertial system. Here
U~=
—
(S/W, ic).
222
(4.78b)
It is sufficient to calculate R,55 in the ipertial system where the medium is at rest. A simple calculation
shows that R,45 = 0 in the case of Minkowski’s tensor, whereas R,55 ~ 0 in the case of the tensors
of Abraham, Einstein—Laub, and Peierls.
The fact that Minkowski’s tensor satisfies the transformation criterion, in contradistinction to
the other tensors, is another demonstration of the applicability of this tensor for the description
of the whole propagating wave. The reason why the other tensors mentioned do not satisfy the
transformation criterion is that their force densities are non-vanishing, and give rise to accompanying mechanical momenta that have to be taken into account before the criterion may be
satisfied. In this context it is physically instructive first to express the mathematical condition in
another form. As shown in [11,
p. 54], one can replace (4.78) by the condition
=
514ek
—
=
~
(4.79a)
in the rest inertial system, e being the wave normal. When the wave travels in the x-direction,
the only non-vanishing component of (4.79a) is
S~ =
-~
g,.
(4.79b)
It should be noticed that this is actually a weak way of stating the transformation criterion. It
does not even require that the force density be equal to zero.
Minkowski’s tensor evidently satisfies (4.79). If we choose to include electrostriction and use
the Helmholtz tensor, the condition (4.79) is also satisfied ~fthe accompanying mechanical momentum
is taken into account so that g~in (4.79b), in the case of a wave travelling in the x-direction, means
188
1. Brevik, Experiments in phenornenological electrodynamics and the electromagnetic energy-momentum tensor
the total momentum density ~
given by (4.10). Similarly, the Peierls tensor satisfies (4.79b) if
the total momentum density ~
given by (4.17), is inserted for g~.These are satisfactory properties of the extended Helmholtz and Peierls tensors. The agreement with the transformation criterion
is actually what one should expect beforehand for a tensor being able to givea reasonable description of the whole wave.
The Einstein—Laub tensor is an exception, as the condition (4.79b) is violated even if ~
given by (4.15), is inserted for g~.The reason for this is the particular form (1.7c) for the energy
density WE. If W’~were replaced by the usual expression (1.ld), the transformation criterion would
be satisfied also for the extended Einstein—Laub tensor.
Transverse drag. The discussion so far has been concerned with longitudinal drag. Jones [70]
has made accurate measurements of the transverse drag experienced when light passes through a
dielectric medium moving at right angles to the incident direction of the light. He projected a
beam of light through a rotating disk (rotation range ±1500 rpm), and measured the transverse
displacement (—~1 nm) of the beam. Apart from dispersion effects, the observations were found
to agree with the formula
2)v,
(4.80)
= (1 — 1/n
where v is the velocity of the medium, assumed to move in the x-direction, and u
5 is the corresponding ray velocity component.
One remark is in order, concerning the transverse drag experiment. This kind of experiment
is subject to the restriction that, to the first order in v/c, it can test only the Galilean, not the relativistic, addition formula for the ray velocity. This can be seen as follows. The relativistic addition
formula for the x-component of the ray velocity is
U~=
1
i/2~
(4.81)
where u~pertains to the inertial system 5’ which moves together with the medium. By
considering
2. Insertion
in
the
light
transmission
through
the
disk
in
5’,
one
easily
deduces
that
u~
= — v/n
(4.81) shows that the denominator deviates from unity only to the second order in v/c, and so we
obtain to the first order the Galilean addition formula u~= u~+ v = (1 — 1/n2)v, which is the
same as (4.80). Note in this respect the contrast with the longitudinal case, where also the denominator in the relativistic addition formula (4.76) contributes to the first order approximation (4.77).
There is a fundamental distinction between the two cases.
Measurement of the integrated effect of the Abraham term?
Among all experiments considered so far in this paper, only the two quasi-stationary experiments treated in section 3 have been able to demonstrate the existence of the Abraham term appearing in (1.6). As pointed out earlier, the fields are varying so quickly in the optical region that a
direct measurement of the Abraham term is not feasible. It would, however, be of interest to detect
the existence of this term in optics also. It is natural then to search for experimental possibilities
which may show the existence of the Abraham term as an integrated effect, i.e. when averaged over
many periods. In the following we shall discuss three possibilities for detecting the Abraham
accompanying momentum when a solid slab is transmitted by an electromagnetic wave.
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
189
(1) Horizontal displacement ofslab — ultrashort pulse. Consider a long dielectric slab of length L
and dielectric constant n, situated parallel to the x-axis and transmitted by a short electromagnetic
pulse of duration t and initial energy
By arranging it so that the end surfaces are covered with
antireflection films of refractive index ~ the energy of the pulse in the interior of the medium will
also be equal to
We assume first a transient situation where the pulse length 1 = ct/n in the
medium is much less than L. For instance, taking t = 1 ns, n = 1.5, we have 1 = 20 cm, so that
transient state should be practically achieved if L is a few meters long. The pulse needs the time
Ln/c to propagate through the slab. We shall find the small displacement Ax of the slab when the
pulse has transmitted. The pulse leaves the impulse
,~.
~°.
Gsurf = ~°(1
— n)/c
(4.82)
at the front surface of the slab. The impulse left at the rear surface when the pulse leaves the slab
is equal and opposite. Next consider the accompanying momentum in the medium. When the
whole pulse is contained in the medium, one has according to the Abraham theory
2— 1)/nc.
(4.83)
G~ech= ~*‘(n
Assume now for simplicity that the medium is absolutely rigid. Let M denote its mass and v its
uniform velocity. In the Abraham picture both (4.82) and (4.83) contribute to the mechanical
momentum:
MVA
=
Gsurf + G~eeh=
.*‘(n — 1)/nc,
so that the Abraham displacement AxA
AxA
=
2~L
(n — 1).
=
(4.84)
vALn/c becomes
(4.85)
Note that the inclusion of the electrostriction effect makes no change in the predicted displacement
(4.85). For when the whole pulse is contained in the medium, the longitudinal electrostriction
forces at the leading edge and at the trailing edge of the pulse are equal and opposite so that there
is no net accelerating force on the slab. Similarly during the period when the pulse enters the
medium: the electrostriction surface force at the front surface of the medium just balances the
electrostriction force at the leading edge of the pulse. The result (4.85) therefore holds also in the
Helmholtz theory. By contrast, the orthodox Minkowski case, where gM is regarded as a pure field
momentum, is different as only the surface term (4.82) then contributes to the momentum MvM
of the slab. We find in that case
AxM
=
2~’/L
n(1 — n).
(4.86)
The expressions (4.85) and (4.86) differ in magnitude and also in sign. Ignoring any complication
arising from the elastic coupling between the molecules, and the finite velocity of sound, one may
thus in principle test the existence of the Abraham term (being responsible for G~~~h)
by verifying
(4.85) experimentally. The effect is, however, very small; one needs great energy ~*‘ and small mass
per unit length M/L. Assume, as an example, that the incident energy flux is 1 GW. For t = 1 ns,
this gives .*‘ = 1 J. If the cross section of the slab equals
2 imn2 (glass fiber), one finds
M/L
102 g/m. This gives Ax
1 pm, which is non-observable. One might use a succession
of pulses to increase the effect.
190
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
The kind of experiment considered here was in essence discussed in [11,
p. 24]. Other discussions
have been given by Gosta de Beauregard [71], and Arnaud [72].
(2) Horizontal displacement of slab — long pulse. Consider the same situation as above, except
that the beam is now assumed to be switched on at some instant and thereafter held constant for
some time t. Because of the impulse transfer to the front surface when the wave enters the medium,
and also the accompanying momentum in the medium, there is a small longitudinal drift of the
slab. As the displacement increases in proportion to time, one can accumulate the effect by choosing
r large. The momentum of the slab is given by the expression (4.84), where now ~° = P~Ln/cis
the field energy in the medium, p~being the incident energy flux. The resulting displacement
becomes according to Abraham
AXA
=
c2M/L (n — 1),
(4.87)
and according to Minkowski
AxM
=
P~0t
c2M/L n(1 — n).
(4.88)
Taking p(i) = 1 kW, n = 1.5, M/L = 10_2 g/m, we see from (4.87) that a pulse duration oft = 2s
is required to give AxA = 1 nm. The experiment would probably be difficult to realize.
Notice that although a long pulse is here assumed, the displacement of the slab is physically
due to a transient effect, viz, the mechanical momentum produced during the entrance period.
When the pulse has passed, the medium is brought back to rest.
(3) Intensity modulation of stationary beam. Instead of transmitting a single pulse, or a succession
of pulses, one may imagine to use a stationary beam, modulated in intensity so as to make the time
derivative term in (1.14) varying slowly. Consider the same slab as above, with end surfaces situated
at x = 0 and x = L, and let the incident energy flux have the form
p~)= p
2(-~k
0 cos
0x— -4w0t).
(4.89a)
Here k0 = co0/c, w0 being the low modulation frequency of the harmonic part of the energy flux,
and P0 being the unmodulated energy flux averaged over an optical period. Assuming antireflection
films at the end surfaces, the energy flux in the slab becomes
2(~-nk
P = P0 cos
0x— +w0t).
(4.89b)
From these expressions the longitudinal force F~on the slab can be calculated, for instance with
the use of (1.14). We obtain in the Abraham case
=
~
1 sin (~nk0L)sin (~nk0L— w0t)
(4.90)
and in the Minkowski case
=
(1 — n) sin (~nk0L)sin (~nk0L— w0t).
(4.91)
In both cases a harmonic force, varying with the frequency w0, is predicted. Again, the Abraham
term is responsible for the difference between the two predictions.
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
191
To test the predictions (4.90) and (4.91), one may imagine the following arrangement. Let a long
non-absorbing glass fiber, of length L and refractive index n (assumed constant) be wound up on
a cylindrical drum of small weight. The drum is thereafter suspended as a torsional pendulum so
that it is able to oscillate about its symmetry axis (the z-axis) with eigenfrequency w0. When the
energy flux (4.89a), whose harmonic part is modulated with the same frequency w0, is transmitted
longitudinally through the fiber (antireflection films assumed), an axial torque N~is produced
that can in principle be measured. If R denotes the mean radial distance from the z-axis to the
fiber coating on the drum surface we have, using the approximation sin (3nk0L) ~ ~nk0L,
=
N~’=
2c
2c
(n — 1)RLP0w0 sin (~nk0L— w0t)
(4.92)
n(1 — n)RLP0w0 sin (~nk0L— w0t).
(493)
3 + : YAG laser operating at 1.06 jim is used to produce
Assume
thatbeam,
a continuously
Nd
the
incident
and assumeworking
that a maximum
energy flux P
0 = 1 kW can be transmitted through
the fiber. For the fiber length we take L = 100 m (a greater length
canobtain
be used
losses
areamplitaken
1, we
forifthe
torque
into
account).
If
we
further
take
n
= 1.5, R = 10 cm, w0 = 10 s
tudes N~= 2.8 x 10_13 Nm, N~’= 4.2 x 10-13 Nm. This is less than the torque in the experiment
of Walker et al. [7] (10 10 Nm) and also in the experiment of Roosen and Imbert [48] (10 12 Nm);
it is of the same order of magnitude as the torque in the Jones—Richards experiment [17] and
is actually greater than the torque in the Beth experiment [55] (10 16 Nm). Realization of the
experiment seems to be difficult but not impossible.
4.6. Further remarks and conclusion
There are still many topics related to the electromagnetic energy-momentum problem that we
have not touched upon in the preceding. For instance, we have not considered the existence of the
Cherenkov sound in a solid as predicted by Peierls [13]. We have not discussed the artificial
dielectric (a meandering bifilar line) introduced by Arnaud [73], nor Arnaud’s proposal [74]
of measuring the transverse tunneling of light propagating in a thin anisotropic film into an absorbing disk that can slide on top of the film. Another topic not being touched upon is the relation
between the electromagnetic energy-momentum tensors and the theory of Brownian motion as
discussed by Skobel’tsyn [75]. From the literature published some years earlier we may mention
the “cube-stuff” model introduced by Shockley [88]. The model consists of material cubes separated by tiny vacuum spaces. The formal developments of Shockley’s paper will, however, not be
discussed here.
A special kind of argument that has been put forward is the socalled “principle of virtual power”
(POYP), as can be found developed in the book by Penfield and Hans [76]. This principle was
considered to some extent also in the recent review papers by Robinson [25] and Pao [77]. As
we have pointed out earlier [11, pp. 60, 78], the POVP unfortunately rests upon the improper
application of the formalism of special relativity to noninertial frames. Criticism against the POVP
has been raised also in papers by Lo Surdo [79], Cavalleri [80], and in a joint note [81], and the
topic will not be further considered here.
Another aspect of the energy-momentum problem, to be considered in some detail, is to exploit
192
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
the conservation laws for momentum and centre of mass velocity for the closed system (matter plus
field) in order to obtain the expression for field momentum. This way of approaching the problem
has gained some popularity. What we ought to stress in this context is that such a method is unable
to give an unambiguous expression for the field momentum: The final result is fixed by physical
assumptions implicitly brought into the formalism at an earlier stage. Use of the conservation
theorems for total momentum and centre of mass velocity means essentially a consistent use of
the integrated forms of the differential conservation equations ÔVS~= 0 for the closed system,
and the global arguments mentioned can yield no more physical information than that already
contained in these differential equations.
The point of view expressed here runs into conflict with that expressed, in particular, by
Skobel’tsyn [26],
and also by Thornber [82].
Skobel’tsyn derived Abraham’s momentum density
by using the global conservation laws, and emphasized that Minkowski’s result is unacceptable
because it contradicts these fundamental laws. It is therefore desirable to analyse the two basic
Gedanken experiments given by Skobel’tsyn. The first example is identical to one of those considered in the previous section, viz, a short pulse of energy .~*‘ transmitted through a solid slab
of mass M, whose end surfaces are covered by antireflection films. Again, assume the slab to be
absolutely rigid. The volume integral of the differential equation ~
= 0 (taken in the longitudinal x-direction) immediately asserts that the initial field momentum .~*‘/cequals the total
momentum when the pulse has penetrated into the medium. Skobel’tsyn writes the latter quantity
as the sum of the unknown field momentum G and the momentum Mv of the slab, so that
=
G + Mv.
(4.94)
Next, the centre of mass theorem immediately follows by multiplying the fourth equation ~
= 0
by x and then integrating over the volume. If dXtot/dt is the velocity of the centre of mass for the
total system, we can write the result as
(.~°+
Mc2) dXt0t/dt
=
~
dv.
(4.95)
As the expression on the left hand side of this equation is a constant, we only have to examine the
expression on the right hand side before and after the pulse enters the medium. Before the -arrival
of the pulse, the right hand side obviously equals .~°c.
Putting this expression equal to the sum
of the electromagnetic and the mechanical parts after the entrance of the pulse, Skobel’tsyn writes
=
.i~f’c/n+ Mc2v.
(4.96)
Equations (4.94) and (4.96) yield G = ~°/nc, which is Abraham’s momentum expression.
Why is Abraham’s result obtained by this argument? The reason can be traced out in various
ways. One way is the following. Equation (4.94) puts the total momentum of the slab equal to Mv,
whereas eq. (4.96) puts the volume integral of the energy flux density of the slab equal to Mc2v.
As the quotient between these expressions is equal to c2, it means that the formalism assumes
Planck’s principle of inertia of energy to hold for the slab. As Planck’s principle necessarily holds
for the total system, because of the symmetry of the total energy-momentum tensor, it follows
that this principle holds also for the electromagnetic subsystem. The electromagnetic momentum
density can thus be found as g = S/c2, which is Abraham’s result. It is in ~ccordance with the
relation G = .$f/nc found above*.
The reason why Minkowski’s momentum does not follow from the above formalism in a natural
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
193
way, is of course the particular feature of this momentum that it includes also the mechanical
momentum that accompanies the wave. Formally, Minkowski’s momentum follows from the
formalism if (4.96) is maintained, and the term ~$f(n2— 1)/nc subtracted from the right hand side
in (4.94).
The second Gedanken experiment considered by Skobel’tsyn [26, p. 395] is the following.
Imagine a cylindrical capacitor (of unit length, say) almost completely filled with a dielectric
solid medium. Between the capacitor surfaces and the medium there are small air gaps. The outer
capacitor surface and the dielectric medium can rotate about the symmetry axis (the z-axis) freely
and independently. The inner and the outer capacitor surfaces carry electric charges q and — q,
respectively, and the whole capacitor is imbedded in a uniform magnetic field H in the z-direction.
When the magnetic field is switched off at some instant t = 0, there is a transfer of the original
axial field angular momentum in the dielectric to measurable mechanical angular momenta in the
dielectric and in the outer capacitor surface. Let us apply Abraham’s and Minkowski’s tensors
to the angular momentum balance.
Consider first the angular momentum Mr” transferred to the outer capacitor surface when the
magnetic field is switched off. The induced azimuthal electric field in the dielectric during the
transient period is EQ = — ~p
0I~r,and the torque on the outer surface is — qE0R, where R denotes
the radius to this surface. Integration over time yields for the resulting angular momentum
2.
(4.97)
= —-~qp0HR
This result does not depend on which energy-momentum tensor is used.
Now use Abraham’s tensor to calculate the resulting angular momentum M~.’~’
of the medium.
We obtain
M~’= JdtJrf~ dV
=
~(1 — 1/e~)qp
2,
(4.98)
0HR
when (1.6) is inserted. The inner capacitor surface has simply been assumed to be a line, so that
Er = q/(2irsr). According to (4.98) the medium is predicted to start rotating when the field is turned
off. The angular momentum conservation is easily checked: Adding (4.97) and (4.98) we obtain
M~+ M~’= M~,where
—4
2
(4.99)
0HR
is the Abraham field angular momentum originally residing in the medium.
Now focus the attention on Minkowski’s case. As mentioned above, (4.97) holds also now.
Iffurt her the angular momentum given to the medium were given by (4.98) also in the Minkowski case,
then one would run into a conflict with the angular momentum conservation since M~’is different
=
*
J
rErH dV
=
—
~-
qp
The explanation given here is similar to that given in [ii, p.
2611, where we
considered in detail a situation of the same kind as
above. A curious point in Skobel’tsyn’s article is its violent attack on the present author, who was accused, for instance, to have left
to the reader to “guess at” the complete form of the momentum balance. The mentioned passage in [11] contained however nothing
else than a demonstration ofwhy conservation equations for momentum and centre ofmass velocity, written in the same form as (4.94)
and (4.96), necessarily lead to Abraham’s result.
194
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
from M~:M5M
6rM~This is the tenor of Skobel’tsyn’s argument. The solution of the problem is
however very simple. In the Minkowski case there are no azimuthal forces on the medium, only
radial forces at the boundaries, and no resulting angular momentum of the medium is therefore
predicted. The angular momentum balance in the Minkowski case reads simply Mr” = M~.
The initial Minkowski angular momentum M~’is predicted to be transferred in extenso to the
outer capacitor surface. There are thus no consistency problems here. Rather, the real interesting
feature of the experiment is that the physical motion of the medium is predicted differently in the
two cases. An observation of the predicted rotation of the medium, as given by (4.98), would
therefore be another verification of Abraham’s force. Such an experiment would be of the same
type as the experiment of Walker et al. considered in section 3.1.
A brief conclusion of our work may be formulated as follows.
(1) There exists one simple consistent phenomenological theory that in the case of isotropic
fluid media accounts for all known experimental facts, namely, the Helmholtz theory. The spatial
part of the energy-momentum tensor is in this case given by (2.4), whereas the remaining tensor
components are identical to the corresponding components in the Abraham case, given by (1.4b, c).
The excess pressure predicted by the Helmholtz force has been measured in the Hakim—Higham
experiment (section 2.3). Similarly, the Goetz—Zahn experiment (section 2.4), carried out at
mechanical non-equilibrium, agrees with the Helmholtz theory. In both these experiments, the
Einstein—Laub expression (1.9) is contradicted. The theory of Helmholtz seems to hold for all
frequencies considered in this paper, in the static case as well as in the quasi-stationary and optical
cases, and we tend to conclude that the Helmholtz force expression is physically correct. More
experimental information is desirable.
(2) Often one may simply omit the electrostriction (or magnetostriction) terms from the formalism without running into difficulties with the experimental predictions. This turns out to be the
case when the total force or torque acting on a test body at pressure equilibrium is calculated.
When the formalism is simplified in this way, the Helmholtz tensor reduces to the Abraham tensor,
given by (1.4). The tensor competing with this one has historically been the alternative given by
Minkowski, cf. (1.1). In the case of an isotropic body, the difference between the Abraham and
Minkowski force densities is given by the “Abraham term”, cf. (1.6). The merit of the experiments
of Walker et al. (section 3.1) and of James (section 3.2) is that they have definitely verified the
existence of the Abraham term. Minkowski’s tensor is unable to describe these observations.
Note that these experiments are global, not local experiments, and it is therefore irrelevant for the
final result whether we choose to include electrostriction and magnetostriction terms in the formalism or not. That means, the tensors of Helmholtz and Abraham are for the purposes of these
experiments equivalent. It is only when we wish to know the physically correct local force in the
medium, that it is necessary to take the electrostriction or magnetostriction terms into account.
(3) At optical frequencies the situation is different, as a direct observation of the rapidly fluctuating Abraham term cannot be made. Under all ordinary circumstances, including all existing experiments that we are aware of, one will obtain the same results with the Minkowski tensor as with the
Abraham tensor. (See (1.14), where the last term fluctuates out.) It is usually best to use the Mmkowski tensor, because of its formal advantages. See the discussion of this point in section 4.1.
The straightforward applicability of Minkowski’s tensor under all ordinary circumstances in
optics was the reason why the first part of our earlier work [10] was devoted to this tensor. Yet,
the Abraham term describes after all a real effect, and section 4.5 discusses some possibilities for
a detection of the accompanying momentum due to this term.
=
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
195
(4) Some attention has been given to the use of Peierls’s tensor (1.12) in optics, although a proper
discussion of this tensor ought to be made using microscopic theory. The experiments considered
above, including the Jones—Richards experiment (section 4.2) and the Ashkin—Dziedzic experiment (section 4.3) actually do not distinguish the Peierls tensor from the tensors of Abraham and
Minkowski. (See, however, the appendix.) More experimental information is desirable.
(5) No arguments based on global conservation theorems (for momentum, angular momentum,
or centre of mass motion) can be used to give an unambiguous derivation of the correct field
momentum density in matter (section 4.6).
Acknowledgements
I am indebted to P.C. Hemmer, J.S. Høye and other colleagues at NTH for discussions, and to
S.O. Kanstad for valuable information about the literature. A. Strand has assisted in the drawing
of the figures. I have also benefited from a wide correspondence; in particular, I am grateful for
the correspondence with V.L. Ginzburg, R.V. Jones, D.G. Lahoz, R. Peierls and G. Roosen.
This work was supported in part by The Norwegian Research Council for Science and the
Humanities.
Appendix
Recently, it has come to our attention that Peierls has published a second paper [85] on the
electromagnetic momentum problem. The physical situation discussed is that of the Jones—
Richards pressure experiment, generalized to the case of oblique incidence for the light beam.
This case is interesting, as it implies that the electric field is now different from zero at the surface
of the reflector, even if this is a perfect conductor. Moreover, the case of oblique incidence has
already been tested experimentally: Jones and Leslie [86] have just published a detailed report
on a repetition of the Jones—Richards experiment, which is not only a generalization of the old
experiment to the off-normal case, but also gives a more than tenfold improvement in precision.
We think therefore that it is appropriate to make the reader aware of these new developments,
along with giving some comments on the ability of the experiment to discriminate between the
theories.
As in section 4.2, let the reflecting wall be lying in the yz-plane and let the x-axis be pointing
normally into it from the surrounding liquid. We shall simply assume that the reflector is an
infinitely good conductor, so that its reflection coefficient R = 1. (Actually, the situation in the
experiment was somewhat more complicated, as the mirror had layers of dielectric materials
covering it, thereby giving high total reflectivity and low absorption. We shall return to this point
later.)
Assume now that a light pulse travels through the liquid and impinges on the wall at an angle
of incidence cx. We shall write down the expression for the observable action from the field on the
wall, as it follows from the Peierls theory, and compare with what follows from the standard
Helmholtz theory which we gave preference to earlier. For completeness, we shall consider both
the two cases amenable for calculation, viz. (i) a very short pulse, and (ii) a long pulse (stationary
beam).
196
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
Short pulse
The term “short” means that the duration of the pulse is much less than the time required for
the elastic pressure to equilibrate throughout the liquid. The implication of this assumption is
that divergence terms in the expression for the electromagnetic force density are not ignorable
when calculating the observable momentum given to the wall. Therefore, the Abraham and
Minkowski theories are not applicable; it is necessary to use the complete Helmholtz force. We
assume that the electric field vector lies in the plane of incidence. There are the following two
contributions to the observable impulse given to the wall:
The first contribution is the direct action ofthefield. The normal stress exerted by the field is
simply S~.The total impulse given to the wall from a pulse of incident total energy ~*‘ follows
by integrating over the history of the pulse and over the surface of the wall:
12n cos cx — 2(n2 —1) [1 + (cr + 2x) (n 2 — 1)] sin2cx)> .Jt°.
(A 1)
JI ~xx dS dt = <—
~c
nc
coscxJ
I
x0
For details, see Peierls’s paper. He calls (A 1) the “prompt” impulse on the wall.
The second contribution is from the momentum deposited in the liquid. This is due to the cross
terms between the incident and the reflected pulse, and for reasons explained in section 4.2 it is
necessary to take this effect into account when the pulse is very short. The deposited momentum
is obtained by integrating the x-component of the force density over time and over the volume of
the whole liquid:
Jf
dv dt
=
n
1 {_[i
+ ~(n2 — 1)]coscx + 2[1 +
(~
+ 2x)(n2 — 1)] sin2~}
~.
(A2)
The deposited momentum propagates in the liquid according to the laws of acoustics. In the special
case of normal incidence, (A 2) is seen to reduce to (4.28)*.
Note that (A 2) can have both signs; it is negative for small values of cx and becomes positive
for large values ofcx. In any case we assume, following Peierls, that one half of the deposited momentum propagates in the positive x-direction and one half in the negative x-direction. Assuming that
the wall reflects the incident sound wave, the momentum transmitted to the wall by this wave is
simply equal to (A 2).
The total momentum transmitted to the wall is therefore the sum of (A 1) and (A 2):
Js~. dS dt +
Jf
dv dt
=
-~-[n2
+ 1 — a(n2 —
1)2]
~‘
cos cx.
(A 3)
Just as in the case of normal incidence, cf. the end of section 4.2, this result can also be calculated
as the rate of total momentum transfer in the x-direction, namely 2g~ (c/n) cos cx, integrated
over the area of the pulse and over its history. Here g~is given by the expression (4.17).
Thus, if the incident pulse is so short that (1) the electrostrictive counter pressure does not come
into play, and (2) the volume of liquid is sufficiently large to contain the deposited momentum so
that half of it is able to propagate towards the large wall and become reflected there, it follows
that one can in principle test the value a = ~ in Peierls’s theory by comparing (A 3) with observa‘
* A factor n 1 has been left out in eq. (5.4) in Peierls’s paper. Therefore his eq. (5.5) is incorrect: the first term in the curly bracket
in that equation should be replaced by —n 1(n2 — 1)[1 + r(n2 — 1)] cos a. This brings the equation in agreement with (A 2)above.
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
197
tion. The corresponding expression in the Helmholtz case, assuming the Lorentz—Lorenz relation
to hold, follows simply be setting a =
It ought to be pointed out that the result (A 3) is independent of the polarization ofthe incident
wave. This is seen most easily from the fact that (A 3) is calculable as the appropriate integral of
the rate of momentum transfer 2g~ (c/n) cos cx: obviously, the momentum density g~ is not
affected ifthe plane of polarization ofthe incident wave is changed. Therefore, our initial assumption that the wave be polarized in the plane of incidence has no importance for the expression
(A 3). By contrast, both the two terms (A 1) and (A 2) from which (A 3) is constructed turn out to
depend on the state of polarization. Thus, changing the incident wave from parallel to perpendicular polarization wechange both the “prompt” impulse on the wall and the deposited momentum
in the liquid, but leave the total observable momentum of the wall unchanged. The detailed
expressions for the two component terms in that case will not be given here.
‘
Long pulse
A physically more important case is when the incident pulse is so long that the electrostriction
forces have established pressure equilibrium in the liquid. As we have seen in connection with the
Ashkin—Dziedzic experiment, this will be the case even for a short laser pulse, if it is narrow
enough. Actually, the new Jones—Leslie experiment falls well into this class. Although this experiment made use ofa laser source, the theory ofthe experiment can be worked out as if the wave were
stationary.
Let us therefore assume a stationary beam at oblique incidence, with the electric field lying in
the plane of incidence, as before. In this case, no account has to be taken of the momentum deposited in the liquid, so that there is no contribution from terms of the type (A 2) in the observable
pressure on the wall. The determination of the expression for the observable pressure on the wall
requires some care. In the first place, the direct action of the field on the wall would lead to the
surface force density ~~x’ This is not the pressure that one can actually observe, however, because
the electromagnetic field produces elastic (hydrostatic) pressures in the liquid that partly compensate for the direct action ofthe field. Let us in this context return to eq. (4.60), which gives a general
comparison between the Abraham and the Peierls force densities. It is apparent that the last term
on the right hand side, as it is a gradient term, gives rise to an elastic excess pressure in the liquid.
Therefore, when calculating the observable pressure on the wall, we may simply omit this term.
Next, consider the middle term (the t-term) to the right in (4.60): will this term be compensated
also? The answer turns out to be no. The reason for this is that the only type of electromagnetic
force being able to produce an elastic pressure in the liquid is one that can be expressed as the
gradient of a scalar function. The t-term in (4.60) is not of this kind; the ith component of the
force can merely be written as a divergence ofthe three-dimensional tensor t(n2 — 1)2ElEk. Therefore, if this force were to be balanced by an elastic force, ofthe form akaa say, then the mechanical
stress tensor aa would generally have to contain non-diagonal terms also. This is not possible in
a liquid. What then is the physical action ofthe t-term in the interior ofthe liquid? If we for simplicity assume the beam to be homogeneous and of finite width, then the i-term may give rise to
local transverse momenta of the fluid elements in the boundary region of the beam, i.e. where
the fields are inhomogeneous. After the pulse has passed, there are left local transverse momenta
in the liquid. These momenta must eventually be expected to disappear by dissipation processes
(see also the discussion by Peierls [85]). We do not here have to describe the local momenta;
the essential point in our context is that in the interior ofthe liquid the i-term is unable to produce
198
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
an elastic pressure compensating for the direct action of the field on the wall. Accordingly, we calculate the predicted observable pressure a~from the first two terms on the right hand side of
(4.60):
=
S~— e0r(n2 — 1)2E~=
[~
cos cx
—
4t(~~c_1)2 sin2~]p(i)
(A 4)
where P°~
= 50) cos cx is the mean incident energy flux upon unit surface area of the wall. The
remarkable fact here is that the expression (A 4), in principle directly comparable to experiment,
is able to test the value of the parameter r. The Helmholtz case corresponds to i = 0, 50 that the
second term in the parenthesis in (A 4) is then vanishing, and we have o~ = (2n/c)PW cos cx. Note
that, irrespective of which energy-momentum tensor is used, the Helmholtz pressure is always
obtained if the plane of polarization is changed such that the electric field vector becomes perpendicular to the plane of incidence. This follows from the fact that E~= 0 in the latter case, cf. (A 4).
Obviously, the Helmholtz result is the same as the Abraham—Minkowski result in case of the
stationary beam assumed here.
In view of the polarization dependence of the pressure predicted by (A 4) it is instructive for a
moment to look back to the method (3) considered in section 4.2. We saw there that, on working in
the Abraham—Minkowski picture and so disregarding any other term in the electromagnetic
force than the Abraham term, one may derive the correct expression for the pressure in a very direct
way as the rate of momentum transfer (2n/c)S~(assumed normal incidence, and reflection coefficient R = 1). In the present case at oblique incidence, analogous reasoning would yield the expression (2n/c)P~°
cos cx, i.e. only the first, polarization independent term in (A 4). What is the reason
why the method (3) turns out to be inapplicable at oblique incidence, for a general value of i?
The explanation is that the method (3) holds only if the extra terms in the electromagnetic force
give rise to elastic stresses that in turn compensate for the direct action of the field. This condition
is satisfied in the Helmholtz case, where the electrostriction term is a gradient function, and we see
that the method (3) yields the same result for the pressure as found above in the Helmholtz case.
The general Peierls case is more complicated due to the i-term in (4.60), for which no elastic
pressure compensation takes place, and the method (3) therefore ceases to apply.
Let us now set ‘r = the value predicted by Peierls for media satisfying the Clausius—Mossotti
relation, and let us write down the Helmholtz/Peierls radiation pressure ratio for the case that
the polarization (i.e. the electric field vector) is parallel to the plane of incidence:
~-,
=
[1 —~(n— 1/n)2 tan2cx]~.
(AS)
As explained above, the Peierls theory predicts this expression to be equal to the ratio between
the radiation pressures for perpendicular and parallel polarization. The latter ratio was measured
by Jones and Leslie, for various liquids and various angles of incidence. Thus, if the Peierls theory
is correct, their measured ratios should agree with the expression (A 5). Tables 1 and 2 reproduce
Jones and Leslie’s observations, and compare them with the theoretical prediction (A5).*
The values given for n for various liquids denote the phase refractive index. From table 1 it
* The theoretical expression that Jones and Leslie compared with observation, called by them the Peierls ratio, was actually different
from (A 5) above. The reason for this is that they did not omit the gradient term in the expression for the force density, i.e. the last term
in (4.60k when calculating the radiation pressure on the wall In view ofthis circumstance, it is desirable to repeat briefly the comparison
between theory and experiment here, adopting (A 5) as the theoretical prediction.
I. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
199
Table 1
Ratio of radiation pressures at 6.4°incidence, perpendicular!
parallel polarization to plane ofincidence. Temperature 21.5°C
Liquid
n
Observed pressure ratio
Eq. (A 5)
methanol
acetone
ethanol
isopropanol
1.3275
1.3563
1.3606
1.3756
0.9996 ±0.0013
1.0003 ±0.0015
0.9995 ±0.0015
0.9996 ±0.0013
1.0017
1.0019
1.0020
1.0021
1.458 1
1.4921
1.4974
0.9989 ±0.0012
1.0014 ±0.0017
1.0026 ±0.0011
1.0030
CCI
4
toluene
benzene
1.0034
1.0035
Table 2
Ratio ofradiation pressures in benzene up to 20°incidence, perpendicular/
parallel polarization to plane of incidence. Temperature 24.3°C,
n = 1.4956
Angle of incidence, deg
Observed pressure ratio
Eq. (A 5)
0.00
3.05
5.51
7.65
9.40
11.50
13.11
14.76
15.88
17.10
18.33
19.37
1.0000
1.0037
1.0018
1.0019
1.0003
1.0036
1.0013
1.0006
1.0046
1.0011
0.9970
0.9973
1.0000
1.0008
1.0026
1.0050
1.0076
1.0115
±0.20
±0.10
±0.08
±0.08
±0.08
±0.08
±0.08
±0.08
±0.08
±0.08
±0.08
±0.08
±0.0031
±0.0026
±0.0030
±0.0027
±0.0028
±0.0022
±0.0029
±0.0022
±0.0031
±0.0027
±0.0033
±0.0026
1.0151
1.0194
1.0226
1.0266
1.0310
1.0350
±0.0000
±0.0001
±0.0001
±0.0001
±0.0002
±0.0002
±0.0002
±0.0003
±0.0003
±0.0003
±0.0003
±0.0003
is seen that the theoretical Peierls expression (A 5) gives slightly higher values than the observed
ones. The difference is, however, not so conclusive as originally stated by Jones and Leslie; it
amounts to about one standard error for most liquids although it becomes as large as 3.4 times
the standard error for carbon tetrachloride. Table 2 gives the values for one liquid, benzene,
when the angle of incidence is increased to about 20°.The discrepancies between the observations
and the Peierls theory are here more pronounced for the larger values of cx. At cx = 19.37°,the
theoretical ratio (A 5) disagrees with that observed by about 14 times the standard error.
In view of this, it is natural to arrive at the same conclusion as Jones and Leslie: the observed
independence of the radiation pressure ofthe plane of polarization is sufficiently accurate to agree
with the Helmholtz (or Abraham—Minkowski) prediction and to disagree with the Peierls prediction. There is, however, one complicating factor here, which ought to be considered (Peierls,
private communication). This is the fact that the mirror used in the experiment had a coating
consisting of layers of refractive materials, the thickness being of the order of a quarter wavelength.
Does this complication change the conclusion above? The answer appears to be no. For in the
case of an ideal metallic reflector one conventionally calculates the pressure by integrating the
force density over a cylindric volume of small height and base surfaces equal to unity, the right
hand surface lying within the metal where the fields vanish, and the left hand surface lying in the
200
1. Brevik, Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor
liquid just outside the metal. This volume integral may be transferred into a surface integral over
the latter surface. If now the mirror instead has a dielectric coating, one may imagine that the left
hand surface is drawn in the liquid just outside the coating. The dielectric mirror assumed to be
a perfect reflector, the fields on the latter integration surface must on the average be the same as
in the case of a perfect metallic mirror. Therefore, the volume integral of the force density, i.e. the
pressure on the mirror, appears to be the same whether or not there is a dielectric coating.
Finally, the reader is referred also to the recent discussion of radiation pressure given by Jones
[89].
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