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Obtaining Maxwell's equations heuristically
Gerhard Diener, Jürgen Weissbarth, Frank Grossmann, and Rüdiger Schmidt
Citation: Am. J. Phys. 81, 120 (2013); doi: 10.1119/1.4768196
View online: http://dx.doi.org/10.1119/1.4768196
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Obtaining Maxwell’s equations heuristically
€rgen Weissbarth, Frank Grossmann,b) and Ru
€diger Schmidt
Gerhard Diener,a) Ju
Technische Universit€
at Dresden, Institut f€
ur Theoretische Physik, D-01062 Dresden, Germany
(Received 13 February 2012; accepted 2 November 2012)
Starting from the experimental fact that a moving charge experiences the Lorentz force and
applying the fundamental principles of simplicity (first order derivatives only) and linearity
(superposition principle), we show that the structure of the microscopic Maxwell equations for the
electromagnetic fields can be deduced heuristically by using the transformation properties of the
fields under space inversion and time reversal. Using the experimental facts of charge conservation
and that electromagnetic waves propagate with the speed of light, together with Galilean
invariance of the Lorentz force, allows us to finalize Maxwell’s equations and to introduce
arbitrary electrodynamics units naturally. VC 2013 American Association of Physics Teachers.
[http://dx.doi.org/10.1119/1.4768196]
I. INTRODUCTION
When teaching electrodynamics one is faced with the
question of whether to postulate Maxwell’s equations, as one
postulates Newton’s laws in a classical mechanics course, or
whether they should be justified from experimental evidence
(Coulomb’s law, Faraday’s induction law, Ampère’s law,
and the nonexistence of magnetic monopoles). Depending on
the choice made, the didactic approach is then either deductive (axiomatic) or inductive.
Here, we give a heuristic derivation of the microscopic
Maxwell equations. This derivation is based on the principles
of simplicity (lowest order in space and time derivatives),
linearity (superposition principle), and the transformation
properties of the fields under space inversion (~
r ! ~
r ) and
time reversal (t ! t). The starting point of the derivation is
the experimental fact that a (moving) charge experiences the
Lorentz force. In addition, in order to obtain the final form of
the Maxwell equations and to introduce electrodynamic
units, we use the experimental evidence of charge conservation, the fact that electromagnetic waves propagate at the
speed of light c, and the requirement of Galilean invariance
of the Lorentz force for low velocities.
Several attempts to deduce (or derive) Maxwell’s equations have been published.1–4 The approach presented below
is unique because it does not make use of another dynamical equation, such as the time-dependent Schr€odinger
equation1 or Newton’s law,2 as a starting point. Our derivation may serve as valuable background information
for the lecturer using either an inductive or deductive
approach to teaching electrodynamics. It can also be used
as an a posteriori justification after the Maxwell equations
have been postulated and/or obtained from experimental
evidence.
The presentation is structured as follows: in Sec. II,
we briefly review the classification of vectors and scalars
by their behavior under space inversion and time reversal. In Sec. III, we present the deduction of the structure
of the Maxwell equations from the principles of simplicity and linearity, with five undetermined multiplicative
constants. Three of the five constants are determined in
Sec. IV from well-established experimental facts together
with the requirement of Galilean invariance of the Lorentz force. Fixing the final two constants leads to the
natural introduction of three commonly used systems of
units in electrodynamics.
120
Am. J. Phys. 81 (2), February 2013
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II. POLAR AND AXIAL VECTORS AND TIME
REVERSAL PROPERTIES
In a standard physics curriculum starting with classical
mechanics, the necessity of studying electrodynamics can be
motivated by the implications of the particle’s charge, be it
fixed or moving. Experimental evidence shows that the Lorentz force on a test particle with electric charge q, moving
with velocity ~
v, is given by
~ ¼ q ðE
~ þ g~
~ Þ;
F
vB
(1)
where g is a constant. This equation postulates the existence
~ r ; tÞ and a local magnetic field
of a local electric field Eð~
~
Bð~
r ; tÞ at the position of the particle ~
r , generated by all
charge-carrying, field-generating particles except the test
particle itself. A particle with a magnetic charge has never
been observed; thus, the fields are generated only by electric
charges with (static) charge density q and moving charges
with (electric) current density ~
j.
Due to the multiplicative connection between charge and
fields in Eq. (1), their units cannot all be chosen arbitrarily.
For example, an arbitrarily defined charge unit [q] fixes the
dimension of the electric field [E] and that of the product
½gB, with an arbitrary constant g that finally defines [B] and
thus fixes the relative dimensions of both fields.
Of central importance for what follows are the transformation properties of vectors with respect to a spatial inversion
(~
r ! ~
r ):5–7
•
Vectors ~
p , that transform according to
0
~
p ! ~
p
p ¼ ~
ð~
r !~
rÞ
•
(2)
under inversion are called polar vectors (or simply
vectors).
Vectors ~
a , that transform according to
0
~
a
a ¼~
a ! ~
ð~
r !~
rÞ
(3)
under inversion are called axial vectors (or pseudovectors). They need a (right-) hand rule for their
definition.
C 2013 American Association of Physics Teachers
V
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120
long time;9 an early appearance and critical discussion in the
literature can be found in Ref. 5.
For future reference, we state that the field generating charge
density, because of the discrete nature of charge, is defined by
q limDV!0 DQ=DV, where DQ is an infinitesimal amount of
charge and DV is an infinitesimal volume (such that DV > 0).
The charge density is idealized as a continuous scalar field with
qð~
r ; tÞ ¼ qð~
r ; tÞ and qð~
r ; tÞ ¼ qð~
r ; tÞ, and thus the field
generating current density
~
~ð~
jð~
r ; tÞ ¼ qð~
r ; tÞ V
r ; tÞ;
Fig. 1. An example for the transformation properties of polar vectors (position ~
r and momentum ~
p ) and an axial vector (angular momentum
~¼ ~
L
r ~
p ) under spatial inversion.8
As an example from classical mechanics, we explicitly
depict the (spatial-inversion) transformation properties of
~ in Fig. 1.
position ~
r , momentum ~
p , and angular momentum L
The vector product of polar and axial vectors can lead to
either polar or axial vectors according to the following rules:
~
p1 ~
p2 ¼ ~
a;
(4)
~
a1 ~
a2 ¼ ~
a;
(5)
~
p1 ¼ ~
p:
a1 ~
(6)
Furthermore, just as there are vectors and pseudovectors,
we can define scalars S as transforming according to
0
S ! S ¼ S;
ð~
r !~
rÞ
(7)
under (spatial) inversion, and pseudoscalars S as transforming according to
0
S ! S ¼ S :
ð~
r !~
rÞ
(8)
Taking the scalar product of polar and axial vectors leads
to both scalars and pseudoscalars according to:
~
p1 ~
p 2 ¼ S;
(9)
~
a 2 ¼ S;
a1 ~
(10)
~
p1 ~
a 1 ¼ S :
(11)
We can now identify the vector and scalar character of the
quantities appearing in the Lorentz force, together with their
behavior under the reversal of time:
~
(i)
Because the mass m is a scalar, we see from F
2
2
¼ md ~
r =dt
that force is a polar vector with
~
~ Therefore, because q in Eq. (1) is a scaFðtÞ
¼ FðtÞ.
~ must be a polar vector with
lar, we know that E
~ r ; tÞ ¼ Eð~
~ r ; tÞ.
Eð~
(ii)
Because g is a scalar and the velocity ~
v is a polar
vector with ~
vðtÞ ¼ ~
vðtÞ, the vector product in
~ is an axial vector with
Eq. (1) tells us that B
~
~
Bð~
r ; tÞ ¼ Bð~
r ; tÞ.
This difference in character of the two field vectors of
electrodynamics has been known and used in teaching for a
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Am. J. Phys., Vol. 81, No. 2, February 2013
(12)
~ð~
with V
r ; tÞ the velocity field of the charge density, is a polar
vector with ~
jð~
r ; tÞ ¼ ~
jð~
r ; tÞ and ~
jð~
r ; tÞ ¼ ~
jð~
r ; tÞ.
III. GENERATING THE MAXWELL EQUATIONS:
THE BASIC IDEA
The classic textbook by Jackson6 contains a discussion
showing that the Maxwell equations do indeed equate quantities with the same transformation behavior under time reversal
and space inversion. The idea presented here is that the transformation behaviors, together with the Lorentz force and the
heuristic demand for simplicity and linearity, allows us to
deduce the structure of the Maxwell equations. The principles
of simplicity and linearity, together with the symmetry properties of the fields, have already been employed by Migdal10 to
derive the homogeneous curl equations. The procedure we follow here is to equate field generating quantities (q and ~
j) with
time or space derivatives of the fields up to at most first order
and in such a way that the fields and the inhomogeneous terms
fulfill the superposition principle.11
First, as already mentioned, there is no experimental evidence for magnetic charges so there cannot exist any fieldgenerating pseudoscalars. The only pseudoscalar [see Eq.
(11)] that can be generated and obeys linearity is to take the
first-order spatial derivative of an axial field vector. Thus,
we quickly conclude that
~ B
~ ¼ 0:
r
(13)
We note in passing that there is a simple physical argu~ if this divergence
ment for the vanishing divergence of B:
were non-zero, then by analogy with the electric field case, a
magnetic field proportional to ~
r would exist and the Lorentz
force would contain terms proportional to d~
r =dt ~
r . In other
words, there would exist terms proportional to the angular
momentum that would lead to an out-of-plane acceleration.
However, such trajectories have never been observed for a
charged particle in a pure magnetic field. To conclude our
discussion of Eq. (13), we mention that mixed terms of the
~
~ B,
~ would violate
correct symmetry, such as ~
j @ B=@t
or E
the superposition principle.11
It is worth mentioning that, due to an idea proposed by
Dirac, magnetic monopoles may explain the discrete nature
of the electric charge.12 Therefore, the search for magnetic
monopoles remains an active area of research. In addition,
magnetic monopoles may also be helpful as a didactic tool.13
~ E
~
~ [see Eq. (6)] are the only axial
Second, @ B=@t
and r
vectors without sign change under time reversal so they must
appear in the same equation. Because a field generating axial
vector does not exist, the heuristic demand for simplicity and
linearity leads us to the equation
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121
~
~ E
~ þ v @ B ¼ 0;
(14)
r
@t
where v is an undetermined constant.
Third, the only scalar that can be generated from first
derivatives of the fields without a sign change under time re~ E.
~ This quantity must be determined by the only
versal is r
scalar source term q and therefore we can write
~ E
~ ¼ a q;
r
(15)
with a another undetermined constant. Again we note that
~
mixed terms of the correct symmetry, such as ~
j @ E=@t
for Eq. (15), would violate the superposition principle,11
whereas a term proportional to @ 2 q=@t2 violates the use
of maximally first order derivatives.
Finally, the only polar vectors that change sign under time
reversal that can be generated from the fields by first order
~ B.
~
~ These terms must therederivatives are @ E=@t
and r
fore appear in one equation together with ~
j, giving
~
~ B
~ þ j @E ¼ b ~
r
j;
(16)
@t
where the constants j and b are still to be determined.
The homogeneous [Eqs. (13) and (14)] and the inhomogeneous [Eqs. (15) and (16)] “Maxwell equations” constructed
in this way contain four undetermined constants: v; a; j, and
b. Together with g, appearing in Eq. (1), there are five constants that need to be determined.
IV. DETERMINING THE CONSTANTS AND
CHOICE OF UNITS
Thus, for the purpose of determining v it is sufficient to
consider a Galilean transformation: ~
r !~
r0 ¼~
r ~
v 0 t;
~
v 0 ¼~
v ~
v 0 . When applied to Eq. (1) we find
~ ¼ qðE
~ þ g~
~ Þ ¼ qðE
~ þ g~
~ þ g~
~Þ
F
vB
v0 B
v0B
~ 0 Þ:
~0 þ g~
v0B
¼ qðE
For the fields, the postulated invariance of the Lorentz
force then gives
~0 ¼ E
~ þ g~
~;
E
v0 B
For the five unknown constants we therefore have three independent equations, meaning there is a degree of ambiguity
in this system. This ambiguity will be resolved with the discussion of three frequently used systems of units, the metric
system (SI units), the Gaussian system (cgs units), and the
Heaviside-Lorentz system.
A. Charge conservation
Taking the divergence of Eq. (16), together with Eq. (15),
leads to
~
~ ~
~ @ E ¼ ja @q :
br
j ¼ jr
(17)
@t
@t
~ ~
Therefore, the continuity equation @q=@t þ r
j ¼ 0 will
be automatically fulfilled if we require
ja ¼ b:
(18)
B. Invariance of the Lorentz force
For small velocities (v c), the Lorentz invariance of the
Lorentz force, to leading order, gives Galilean invariance.
122
Am. J. Phys., Vol. 81, No. 2, February 2013
~0 ¼ B:
~
B
(20)
Using a calculation similar to that in Ref. 14, we find, on
the other hand, that
~ Eð~
~ 0 Eð~
~ r ; tÞ ¼ r
~ r 0 þ~
r
v 0 t; tÞ
~ r ; tÞ ~ r 0 þ~
@ Bð~
v 0 t; tÞ ¼ @ Bð~
~ 0 ÞB
~
v0 r
0 ð~
@t ~r
@t
~
r
~ r 0 þ~
@ Bð~
v 0 t;tÞ ~ 0 v0 B
~Þ;
¼
0 þ r ð~
@t
~
r
(21)
(22)
~ 0 means to differentiate with respect to
where the notation r
the primed coordinates, and j~r reminds us to hold the vector
~
r fixed. Using these results in Eq. (14) leads to
~
~
@B
@B
0
~
~
~
~
~
¼ r ðE þ v~
v 0 B Þ þ v rEþv
@t
@t ~r 0
The undetermined constants must be chosen so that Eqs.
(13)–(16) satisfy additional constraints. We will use the following three experimental facts:
(A) Charge conservation—the field generating quantities
(q and ~
j) must satisfy the continuity equation,
(B) Invariance of the Lorentz force—the Lorentz force must
be invariant under a Galilean transformation for v c,
where c is the speed of light,
(C) Wave equation—the universal propagation speed c must
be contained in the wave equation derivable from the
equations obtained in Sec. III.
(19)
~0
~0 E
~ 0 þ v @ B ¼ 0;
¼r
@t
(23)
from which we see that
~ þ v~
~
~0 ¼ E
v 0 B;
E
~0 ¼ B:
~
B
(24)
Comparison with Eq. (20) then shows that
v ¼ g:
(25)
C. Wave equation
The wave equation for vanishing inhomogeneities q and ~
j
follows by taking the curl of Eq. (14) and then using
~ ðr
~ EÞ
~
Eqs. (15) and (16). Using the identity r
2
~ r
~ E
~ Þ r E,
~ this leads to
¼ rð
2~
~
~ @ B ¼ r2 E
~ r
~ E
~þ v r
~ vj @ E ¼ 0:
~ Þ r2 E
rð
@t
@t2
(26)
Finally, making use of Eq. (25) we find that
gj ¼ 1
;
c2
(27)
where the velocity c ¼ 2.9986 108 m=s, is determined by
measuring the velocity of electromagnetic waves in vacuum.
It is worthwhile to note that a velocity with the same
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122
numerical value can be determined by pure electrostatic and
magnetostatic measurements (“c equivalence principle”).15,16
In this way, the units are introduced naturally by fixing two
remaining constants appearing in the Maxwell equations and
the Lorentz force.
D. Final choice of the system of units
The constant v is now fixed by Eq. (25) and for j; a; b,
and g we have Eqs. (18) and (27) that lead to a=ðgbÞ ¼ c2
(see also Ref. 3). Two of these constants can be chosen arbitrarily; we discuss three common choices below.
(i)
SI units: Choose g ¼ 1 to get j ¼ 1=c2 , and b ¼ 4p
107 N=A2 l0 to get a ¼ l0 c2 1=e0 , and thus a
Lorentz force
~ ¼ q½Eð~
~ r ; tÞ þ~
~ r ; tÞ;
F
v Bð~
(28)
and the rationalized Maxwell equations (without
explicit appearance of p)
(ii)
~ r ; tÞ
~ Eð~
~ r ; tÞ ¼ @ Bð~
r
@t
(29)
~ r ; tÞ
@ Eð~
~ Bð~
~ r ; tÞ ¼ l0~
jð~
r ; tÞ þ l0 e0
r
@t
(30)
r ; tÞ
~ Eð~
~ r ; tÞ ¼ qð~
r
e0
(31)
~ Bð~
~ r ; tÞ ¼ 0
r
(32)
cgs units: Choose g ¼ 1=c to get j ¼ 1=c, and b ¼
4p=c to get a ¼ 4p and a Lorentz force
~
v ~
~
~
r ; tÞ ;
(33)
F ¼ q Eð~
r ; tÞ þ Bð~
c
where the electric and magnetic fields have the same
units. The Maxwell equations then read
(iii)
123
~ r ; tÞ
~ Eð~
~ r ; tÞ ¼ 1 @ Bð~
r
;
c @t
(34)
~ r ; tÞ
1 @ Eð~
~ Bð~
~ r ; tÞ ¼ 4p ~
jð~
r ; tÞ þ
;
r
c
c @t
(35)
~ Eð~
~ r ; tÞ ¼ 4pqð~
r
r ; tÞ;
(36)
~ Bð~
~ r ; tÞ ¼ 0:
r
(37)
Heaviside-Lorentz units: Choose g ¼ 1=c to get
j ¼ 1=c, and b ¼ 1=c to get a ¼ 1. This choice
again gives the Lorentz force in Eq. (33) but with different field units (although the electric and magnetic
fields again have the same units), and leads to rationalized Maxwell equations
~ r ; tÞ
~ Eð~
~ r ; tÞ ¼ 1 @ Bð~
r
;
c @t
(38)
~ r ; tÞ
1 @ Eð~
~ Bð~
~ r ; tÞ ¼ 1 ~
jð~
r ; tÞ þ
;
r
c
c @t
(39)
~ Eð~
~ r ; tÞ ¼ qð~
r
r ; tÞ;
(40)
~ Bð~
~ r ; tÞ ¼ 0:
r
(41)
Am. J. Phys., Vol. 81, No. 2, February 2013
V. SUMMARY
Starting from the Lorentz force as an experimental fact
and using the principles of simplicity and linearity, we
showed that the structure of the Maxwell equations for the
electromagnetic fields in vacuum can be deduced. These are
two (pseudo)scalar equations for the divergence of the respective fields and two (pseudo)vector equations for the curl
of the fields, in accord with the fundamental theorem of vector calculus. The vector character (polar or axial) of the fields
and their transformation properties under time reversal are at
the heart of our approach. Charge conservation, the Galilean
invariance of the Lorentz force, and the fact that electromagnetic waves propagate at the speed of light enabled us to fix
three of the five undetermined constants in the Lorentz force
and the four deduced Maxwell equations. As an additional
useful result of the calculation, arbitrary systems of units can
be introduced, which is didactically more pleasing than to
postulate the units a priori.
ACKNOWLEDGMENTS
Helpful comments on the manuscript by Larry Schulman
and valuable discussions with Klaus Becker are gratefully
acknowledged.
a)
Deceased on August 17, 2009
Electronic mail: [email protected]
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Adapted from <http://commons.wikimedia.org/wiki/File:Angular_
momentum_circle.png>.
9
~ and B
~ is apparent also in the construction
The different nature of E
of the electromagnetic field tensor in the covariant formulation of
electrodynamics.
10
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11
The superposition principle for an inhomogeneous, partial differential
equation with linear differential operator L states that with two solutions
obeying Lu1 ¼ g1 and Lu2 ¼ g2 , the sum u1 þ u2 is also a solution for the
inhomogeneity g1 þ g2 .
12
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b)
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123