Download Electromagnetic induction in magnetic rod moving with high velocity

Electromagnetic induction in magnetic rod moving
with high velocity
Prof. E.G. Cullwick, O.B.E., M.A., D.Sc, C.Eng., F.I.E.E., F.R.S.E.
Indexing terms:
Electromagnetism, Relativity
Abstract
An account is given of the relativistic component-field electromagnetic theory of moving bodies, which extends
orthodox relativity theory by recognising sources of the field, and it is shown to have advantages over orthodox
theory in aiding physical comprehension. It is then applied to the determination of the electromagnetic field in a
cylindrical rod of nonretentive ferromagnetic material, which can be either a conductor or a dielectric, in motion
with high velocity through crossed fields. Particular cases investigated include the electromotive force induced in
the rod by its motion through a magnetic field, it being confirmed that this is independent of the relative permeability, and the magnetisation of the rod by its motion through an electrostatic field. Expressions for the polarisation and magnetisation of the moving rod are also obtained. All the results are considerably simplified if the
velocity of the rod is small compared with the velocity of light. For low velocities the theory can be presented in
simple terms without reference to the theory of relativity, and for this reason, and also because of its physical
basis, it is considered to have considerable advantages.
List of symbols (SI units)
B
D
E
e
F
Ft
H
M
P
v, v
c
Mo
1
magnetic flux density
electric flux density
electric field intensity
electromotive force (e.m.f.)
Lorentz force
electromagnetic force
magnetic field intensity
magnetisation (dipole moment per unit volume)
polarisation (dipole moment per unit volume)
velocity
=
velocity of light in vacuo
(i-t)V)-" 2
permittivity of free space, or primary electric constant
permeability of free space, or primary magnetic constant
relative permittivity
relative permeability
Introduction
The phenomenon of electromagnetic induction, discovered
by Michael Faraday in 1831, is the rock on which electrical engineering has been built, while the experimental and theoretical investigation of the electromagnetic effects of moving bodies has been
fundamental to the development of modern physical theory in the
evolution of aether theories and the emergence of the theory of
relativity. It was indeed the latter which, in providing a purely mathematical specification of the electromagnetic effects of moving
bodies, swept out of respectable science the current physical theories
and postulated explanations of such important practical phenomena
as the electric field produced by a moving magnet.
Engineers and physicists who seek descriptions of natural phenomena in terms of physical concepts, however, have available in the
relativistic component-field theory of electromagnetism a theory of
the fields of moving bodies which is not only equivalent to the mathematical relativistic theory but also provides much physical insight
into the phenomena, and, as will be shown, physically consistent
accounts of the action. The development of relativistic electromagnetism largely as a province of mathematics has had the unfortunate result that electromagnetic theory in physical terms, as taught
in our universities and colleges, has too often been confined to the
prerelativistic form developed by H. A. Lorentz on the foundation of
Maxwell's theory, which is commonly known as Maxwell-Lorentz.
This is entirely satisfactory and sufficient for the majority of practical
applications, but in unusual problems involving moving magnets its
application can sometimes be complex and ambiguous. An example is
the case of a magnetic cylinder moving concentrically along a currentcarrying conductor,1 and an old problem, considered by Faraday in
1851,2 which has been the source of much discussion is that of a
rotating cylindrical magnet.3 In the case of electromagnetic induction
in a body which is both magnetic and a dielectric moving through a
magnetic field, prerelativistic theory actually gives an incorrect
result.4 All these problems are readily analysed and solved by means
of the component-field theory.s
In this paper the theory will first be explained and compared with
orthodox theory, and then used to find the field measures in a
magnetic cylindrical rod, which may be either a conductor or a
dielectric, moving transversely with high velocity through crossed
electric and magnetic fields. This analysis also provides a specific
proof, which so far appears to have been lacking, that when a wire
moves through a magnetic field the induced e.m.f. is the same whether
the conductor is nonmagnetic or magnetic. This fact, first discovered
and very simply demonstrated by Faraday in 1831,6 may be inferred
from the general laws of electromagnetic induction and the conservation of energy, but it will be proved here directly by finding the
magnitude of the electromagnetic field within the moving wire. From
the same analysis it will appear that the induced e.m.f. is given by the
simple flux-cutting law, even up to velocities comparable with the
velocity of light. Another particular case which will be analysed is that
of the magnetisation of an iron rod by its motion through an electrostatic field, a phenomenon sought but not detected by Faraday in
1832.7
2
Electromagnetic component-field theory of moving
media
2.1
Maxwell's equations and Lorentz transformation
Maxwell's equations for the field in free space:
curl* = - J -
(1)
at
and
curlff=^
(2)
at
define relationships between the resultant or total measures of the
electromagnetic field at a point fixed in the inertial reference system
in which measurements are made and the equations used. They do not
relate the field to its sources. The principle of relativity of inertial
reference systems, first enunciated by Henri Poincare in 1904,8
requires that these equations should be valid, and therefore the velocity
of light in vacuo equal to c, in all such systems, and this requirement is
met if the measures of space and time defining a given physical point
event, as observed in two relatively moving inertial systems, 5 and S',
are related by the Lorentz transformation:
x = Q(x — vt),
= y,
z = z, t' =
fiit-^
(3)
and, reciprocally,
, t =
t+7x
where v is the velocity of the system 5 ' relative to S along the x-axis
andj3 = ( l - ^ / c 2 ) - 1 / 2 .
It then follows that the electromagnetic field measures at corresponding points of space and time in the two systems must be related
by transformations such as
Paper 7957S, first received 10th January and in revised form 24th June 1977
En = P(E'n-vxB'),
Prof. Cullwick is Emeritus Professor, University of Dundee, and Honorary
Professor of Electrical Engineering, University of Kent at Canterbury
E - E'
E'n =P(En+vxB)
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(4)
1105
the suffixes n and v denoting components normal and parallel to the
velocity v, respectively.
Eqn. 4, and similar transformations for H, B, and D, provide a
specification of the electromagnetic field of moving bodies as required
by the special or restricted theory of relativity. The specification is
obtained by the purely mathematical application of the Lorentz
transformation to Maxwell's equations, and the relations such as
eqn. 4 are functions of the resultant field measures at a point, and are
not related to the physical sources of the field.
2.2
2.2.1
and
where M' is the magnetisation.
The resultant fields in the rest system S' are then:
01)
E' = E'o + E's
(12)
D' = D'0+D's
(13)
Component-field theory
H' = H^+Hp
(14)
Transformation of field measures
B' = B'0+B'p
(15)
In the component-field theory we extend relativistic theory
to take account of field sources. We suppose a body, which may
consist of conductors and dielectrics, nonmagnetic or magnetic, to be
stationary in an inertial reference system S' which is moving with
constant rectilinear velocity v, in the direction of the x-axis, in a
second inertial reference system S. For our purpose, we may identify
S as the reference system of the laboratory in which the body is
moving and in which observations of the electromagnetic field, at
points fixed in the reference system, are made.
Let Eo and Do = €o'Eo be the components of the electric field
measures, and Ho and Bo = n0H0 be the components of the magnetic
field measures, assumed constant and uniform,* due to sources
external to the moving body and as observed in the system S. The
corresponding fields as observed in the system S1, at points fixed in
that system, are given by eqn. 4 and similar transformations. To
simplify the expressions we shall suppose that the fields Eo etc. are
perpendicular to v. The unchanged parallel components of field, if
any, can then be added in any particular case. These external fields
then exert polarising and magnetising forces on the moving body
which in its rest system S' appear as a polarising field
+vxB0)
E'o =
(5)
In the system S the selfcomponents of field E's etc. are observed to be
increased by the relativistic factor |3 and also to cause 'motional' fields
by the motion of their sources in the moving body. These are:
(a) electric motional field
Em = -0(vxB'n)
(16)
(17)
(b) magnetic motional field
Hm = p(vxD's)
(18)
(19)
It follows that the relativistic field transformations can be expressed
in two forms, the first of which in the following equations is the
standard orthodox transformation of relativity theory, and the
second is obtained by collecting the three components of field, i.e.
external, self and motional, as given previously:
E = 0(£" -vxB')
= EQ + Q(E't -vxB'p)
(6)
(20)
(21)
Dand a magnetising field
H'o =
P(Ho-vxDo)
vxE0
D'S=D'P+D'q
(9)
where E'p, Dp are due to the polarisation of the dielectrics and E'q, D'q
are due to the displaced free charges on conductors, and
Dp = e0Ep +P', D'q = e0Eq
(10)
where P' is the polarisation, and
u'
a'
"p.
tt
p
due to the magnetisation,
•If the external field varies with time or in space the selfcomponents of field in
the moving body will vary with time in S' and will induce field components.
These will, in many cases, be negligible, but can readily be included if necessary.
See Reference 5, chap. 7, for a more extensive account of the theory.
1106
p +vx D's)
(22)
(23)
(8)
We may note that E'o is merely the Lorentz force, or the force on a
unit charge moving with velocity v through the fields Eo and Bo, but
modified by the relativistic factor j3, since it now appears in S' as a
force on a stationary unit charge, i.e. an electric field intensity. H'o is
its magnetic 'dual' or force on a unit magnetic pole stationary in S'.
D'o and B'o are related to E'o and H'o as in free space. So if we are concerned merely with a theory of moving bodies with low velocities, in
which v^.c and 0 can be taken as unity, we can deduce eqns. 5—8
independently of the theory of relativity from the physical laws for
the force on a charge and a magnetic pole when moving through an
electromagnetic field, and by transferring our viewpoint from one
reference system to the other.
The field measures at a point inside the body, as observed in its
rest system S', consist of the above 'external' field components, given
by eqns. 5—8, together with 'self field components arising from the
state of electrification, polarisation, and magnetisation of the body,
brought into existence by the action of the polarising and magnetising
fields. The selfcomponents of field, which will depend on the shape of
the body and its orientation to the external fields, may be expressed
as
D'8 = eQE's +P' ,
H = 0(H' + v x D') = Ho
(7)
The second form of these equations can also be obtained by substitution of the values of E' etc. given by eqns. 12—15 in the orthodox
transformations and the use of eqns. 5—8.
2.2.2
Magnetisation and polarisation
The transformations for the polarisation P and magnetisation M are, in symmetrical form in which M = B —
and
P = D-
M = ${M'-Ho(vxP')}
(24)
P = 0{P' +eo(vxM')}
(25)
or, if the unsymmetrical form Mh = B/n0 — H is used, then
Mh = (S(M'h-vxP')
(26)
(27)
2.2.3
Time
In these relations all unprimed quantities are those observed
in the system S at an instant f, and all primed quantities are observed
in the system S' at an instant t', where t and t' are related by the
Lorentz transformation, eqn. 3. This applies to all expressions relating
S and S' measures, but in the use of the theory in problems of electromagnetic induction involving e.m.fs of the first order only, and
tThese symmetrical definitions of M and P are preferred to the alternative unsymmetrical form in which B = n0 (H + M). The dependence of H on the concept of magnetic poles, which is symmetrical with the dependence of E on the
concept of electric charge, is inherent in Poisson's classical analysis of a
polarised magnetic medium, which remains valid and basic to the theory of magnetism even if we now accept that it is spinning electrons that cause M. The
symmetry, moreover, leads to symmetrical transformations as seen in eqns. 24
and 25. SI units provide for both forms: electromagnetic moment, magnetic
moment ( A m 3 ) ; magnetisation (A/m); magnetic polarisation (T); magnetic
dipole moment (N m 2 /A or Wb m). The latter two conform to the symmetrical
form of the definitions of M and P. For a physical explanation of the apparent
magnetisation of a moving polarised dielectric, and the apparent polarisation of
a moving magnet, see Reference S, chap. 3 and 4.
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velocities small compared with c, the distinction between t and t' can
be ignored. With constant and uniform fields, of course, the time
transformation is of no significance.
2.3
Comparison of orthodox and component-field theories
The logic of the component-field theory is very simple: the
electric and magnetic fields caused directly by the motion of magnetised and electrified matter arise from the motion of the constituent
electric charges and magnetic dipoles (if the pole concept of magnetism is used) in the moving bodies and of nothing else. These motional
fields can be expressed, for rectilinear motion, by the vector products
of the parent fields and velocities of the bodies, but if the total
resultant field in the interior of a moving body is due partly to the
body itself and partly to external sources, the true motional field of
the body must clearly be a function of the selfcomponent of field
alone, and not of the resultant field in the body. This is consistent
with Lorentz's extension of Maxwell's theory, in which he regarded
electrons, moving freely in a fixed and immobile aether, as the sole
sources of the electromagnetic field,9 and in contrast to the earlier
theory of Hertz, in which it was supposed that matter in motion
carried the enclosed aether along with it,10 This hypothesis of Hertz
was found by A. Eichenwald in 1903-04 11 and H.A. Wilson in 190512
to be inconsistent with experiments which were in accord with
Lorentz's theory.
To compare the use of the two forms of the relativistic field transformations given in eqns. 20—23, consider the simple case of a solid
iron sphere, of very high relative permeability, moving with constant
rectilinear velocity transversely through an external field Bo with
Eo = 0. The magnetising flux density acting on the sphere in its rest
system S' is, from eqn. 8,
B'o = $Bo
that
E = -$13(vxBQ) = B x v
(38)
as in eqn. 36. This, however, is all that we do find. There is no suggestion in eqn. 38 of any displaced charges and the two parts of E are
obscured. We cannot derive the induced e.m.f. from the result, and if
we attempt to do so we may conclude that there is none. Tofindthe
e.m.f. we must go back to the reason for E' being zero and accept the
idea of two field components cancelling each other, and so are led to
the component-field theory. The orthodox transformations, being the
result of a purely mathematical operation on Maxwell's equations, are
a poor and indeed misleading guide to physical interpretations. It
follows that the component-field theory is not just an alternative
method of obtaining relativistic results: each term of the equations
has a recognisable physical meaning and the theory brings physics into
the transformations where previously it was absent.
A further advantage of the component-field theory is that it
provides a low-velocity (§ = 1) theory of moving media which can be
taught and used without reference to the theory of relativity at all,
and which, contrary to prerelativistic Maxwell-Lorentz theory which
is still prevalent, is consistent with the principle of relativity and
experiment in all known cases, including that of induction in a moving
magnetic dielectric.
E
o°rBo
(28)
and the resultant field in the sphere, in its rest system, is given by
E' = 0
(29)
since the iron is a conductor, and
B' = 3B'O
(30)
very closely.
To find the electric field E inside the moving sphere as observed
in the system S, we may use either form of the field transformation
given in eqn. 20. First, consider the component-field form:
E = E0+P(E's-vxB'p)
(31)
The polarising electric field acting on the sphere in its rest system S'
is, from eqn. 5,
E'o = P(vxB0)
(32)
and this displaces charges in the sphere, which set up a selfcomponent
of field E's cancelling the external component E'o, i.e.
E'8 = -E'o
=
-P(VXB0)
(33)
Then, since
B' = B'o+B'p = 3B'O
Fig. 1
Cross-sectional view of cylindrical rod at rest in a transverse electric
field EQ or a transverse magnetic field BQ
(34)
3
we have
Bp = 2B'O
E =
-i'
The use of eqn. 31 shows clearly that the resultant electric field E is
made up of two parts with different physical origins: the first, of
magnitude one-third of the whole, being the field of the charges
displaced in the conducting sphere, while the second, of magnitude
two-thirds of the whole, is the motional electric field produced by the
motion of the magnetised iron.
The extent to which charges are displaced in the sphere is a
measure of the induced e.m.f.; so it follows that this, induced by the
motion of the sphere through a field Bo, is proportional to Bo and not
to the resultant internal field B.
In comparison, consider the orthodox form of the field transformation:
E = $(E'-vxB')
electromagnetic field
(35)
so that, since Eo — 0, eqn. 31 gives
(37)
We have E' = 0 and B' = 3B'O and so, using eqn. 28, we find at once
Magnetic rod moving with high velocity through an
3.1
Field components in cylindrical rod
3.1.1
Transverse fields
Fig. 1 depicts the circular cross-section of a solid long
straight rod of magnetic material, of relative permeability /i r , which
may be either a dielectric of relative permittivity e r or a conductor. It
is stationary in a transverse electric or magnetic field due to external
sources which contribute to the resultant field, at an internal point,
the 'external' field components:
E0,D0
= eoE'o
or
H'o,B'o =
(39)
Primed symbols are used since the rod is considered to be stationary
in- an inertial reference system S', and unprimed symbols will then
denote the field measures as observed in the system S in which the rod
is moving. These would be the field measures if the rod were removed,
assuming that its removal did not disturb the external field sources:
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1107
Dielectric rod
By the usual methods of potential theory it may be shown that the
field measures at an internal point are given by:
E' =
2E'O
er+l
The polarisation or magnetisation will be
P' = (er-\)D'o
= D's or M' = (ptr-l)BQ
= B'p
(51)
or H' =
(40)
Conducting rod
If the rod is a conductor, the electric field will be given by eqns. 45
and 46, and the magnetic field will be the same as in a dielectric rod.
or
(41)
3.2
Magnetic conducting rod moving across a transverse magnetic
field
3.2.1
Fields for internal equilibrium
and
D' =
and the selfcomponents (eqns. 9—11) of field are:
E'q =0,E's
= Ep
=
-
^
D'q = 0, D's = Dp = - ^ — D'o
er + 1
or
or
I'o (43)
The rod, Fig. 2, is assumed to be long and to move with
constant velocity v transversely through a constant and uniform
magnetic field Bo due to external sources. The rod is on 'open circuit',
carrying no current, and so for internal equilibrium, from eqns. 45,
46 and 5:
E'
The polarisation or magnetisation will be
= 0, E's = -E'o
=
-P(VXB0)
(52)
and from eqns. 43 and 8:
(44)
or
Conducting rod
If the rod is a conductor the magnetic field will be the same as
before, but the electric field measures will be
£" = 0,
D' = 0,
/>' = 0
Hr + 1
= 0,
D'p = 0,
3.1.2
E't = E'Q--=-E'o
D's = D'Q= -D'o
(45)
E=
-p
= E'o
or
(54)
and
r
Tangential fields
E'
(53)
Mr+1
(46)
(55)
Mr + 1
The Lorentz force is
Dielectric rod
For a dielectric rod parallel to the external field, the resultant field measures at an internal point are
D' = ejDo or
Mr +
since f0 = 0. Then, from eqns. 20 and 23, we find that
and the selfcomponents will be
4
o — p
H' = H'o
(47)
B' = (irB'o
(48)
F = E + vxB
E's = 0
or
H'p = 0
(49)
Al = (e r -l)^>o
or B' = (u
(50)
(56)
confirming the state of internal electrical equilibrium.
3.2.2
and the selfcomponents are
= 0
E.M.F. induced in rod
The electromagnetically induced e.m.f. along an open path in
a conductor is the line integral along the path of the electromagnetic
force, Fh which is defined as the force on a unit charge situated at a
point on the path and stationary relative to the body, induced by
electromagnetic action alone and excluding the electrostatic component of force due to true charges displaced from their neutral
positions in the body.13 To find the induced e.m.f. in the present
case, therefore, we remove the component E's from the expressions for
E and B given by eqns. 54 and 55 and find for the fields to be used in
the expression for Ft:
(57)
and
(58)
Bt =
so that
/•j
—
£/
i = vxB0
(59)
If the length of the rod is L, the induced e.m.f. is
e = B0Lv
Fig. 2
Cross-sectional view of cylindrical rod moving with velocity v perpendicularly to transverse electricfieldEo or to transverse magnetic
field Bo
1108
(60)
The e.m.f. is thus the same as in a nonmagnetic conductor. The result
may be obtained in an elementary way by assuming that the magnetised rod moves through the external component of flux density Bo
alone, carrying its selfcomponent Bp with it. It is indeed obvious
that the electromagnetic force Ft must be independent of the magnetisation of the rod, since, in the rest system of the latter, the magnetisation causes no force on a stationary unit charge, so there can be no
such force in any other inertial reference system in which charge and
rod are co-moving. We may briefly sum up these conclusions in the
statement that a magnet moving with constant rectilinear velocity
cannot induce an e.m.f. in itself.
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3.3
vxH,
Magnetic conducting rod moving across transverse electrostatic field
The long rod, Fig. 2, moves transversely with constant
velocity v through a constant and uniform electrostatic field EQ due
to external sources. For internal equilibrium, from eqns. 5 and 46:
Eg = ~E0
= —&EQ
so that, from eqns. 20—23
(62)
H = 02
Ean. 62 also gives the magnetisation Af' in the rest system 5 ' sihce
Hp = 0. Since Bo = 0, we then find from eqns. 20 and 23:
E=
vxB0
(69)
2 \vxE0
(70)
"'
j
B = 02
vxEQ
-PQlr-
2
* - ( P "
(61)
while from eqns. 8 and 50:
B'p =
(68)
D=
2er\vxH0
2
T\ c r° I
(63)
-f-^
vxD0
-?£il*-'-
(71)
(72)
From eqns. 69 and 70 the Lorentz force is
and
vxE0
(64)
3.4.2
Thus, the magnetic flux density in the rod consists of two parts:
(a) that of the magnetisation, (iBp
(b) the motional magnetic field of the charges induced by Eo on the
conducting rod.
The Lorentz force in the rod, F = E + v x B, is zero.
In any practical case, the magnetisation will be extremely small.
For example, a rod moving at one-tenth the velocity of light through
an electrostatic field of 104 V/m, with relative permeability Mr = 100>
would be magnetised to the extent of only about 10"s T.
2
F=E+vxB=
(73)
-(Eo+vxBo)
er+T
Comparison with conducting magnetic rod
To compare the case of a magnetic dielectric rod moving
through an electrostatic field with that of a conducting rod, Section
3.3, we put Bo = 0 in eqn. 70, so that
(74)
This value of B is smaller than that given by eqn. 64, since the selfcomponent E's is smaller when the rod is a dielectric. The magnetisation is the same in both cases. This difference is shown more
clearly by expressing the results in the form:
(a) conducting rod
(75)
(b) dielectric rod
er —
l)vxE0
(76)
In eqns. 75 and 76 the term in (fir — 1) is the magnetisation of the
rod, while the second terms give the magnetic field produced by the
motion of the charges induced on the rod.
3.4.3
Magnetisation and polarisation
The magnetisation and polarisation of the rod may be
obtained either by using eqns. 69 to 72 in the relations
\
M =
P =
B -
(77)
D-e0E
or from the S' measures given by eqn. 44 and the transformations in
eqns. 24 and 25. The resulting expressions are lengthy, but if v«c
they reduce to the simpler forms:
x.
(78)
M = (urand
P —
Fig. 3
Cross-sectional view of cylindrical rod moving with velocity v perpendicularly to transverse electricfieldEo and longitudinal magnetic
3.5
field Bo
3.4
3.4.1
Magnetic dielectric rod in a transverse electric field and a
tangential magnetic field, moving perpendicularly to both
fields
Electromagnetic field in rod
3.5.1
2(e r -l)Ux/y 0
2(er~l)
Magnetic dielectric rod in tangential electric field and a
transverse magnetic field, moving perpendicularly to
these fields
Electromagnetic field in rod
The situation is that of Fig. 3 with Eo and Bo interchanged.
Following the same procedure, we obtain:
The situation is shown in Fig. 3. The rod is long, moving with
constant velocity v, and the fields are constant and uniform. From
eqns. 42,50,49,43 and 5-8, we find:
E[ = E'p = -
(65)
)Ur - 1 \ V X E o
B = 01
H=
PROC. IEE, Vol. 124, No. 11, NOVEMBER 1977
(80)
Eo-
)vxD0
(66)
(67)
(79)
D = 02
t?
2
2 \vxHp
Mr+1
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(81)
(82)
(83)
1109
Eqns. 80-83 can also be derived from eqns. 69-72 by interchanging
electric and magnetic field measures, er and jur, and changing the sign
of the vector product terms.
5
Acknowledgment
The author wishes to express his indebtedness to the referees
of the paper for helpful comments and suggestions.
The Lorentz force is
F = E + vxB
3.5.2.
- Eo +vxB0
(84)
Magnetisation and polarisation
As in Section 3.4.3, we find, if v«c,
6
that
(85)
and
(86)
P = (eP
These expressions can also be derived from eqns. 78 and 79 by
appropriate interchange of symbols and signs.
4
Conclusion
The foregoing analysis provides proofs of the motional e.m.f.
in a magnetic rod and its magnetisation by motion through an electrostatic field. In addition, rigorous expressions are obtained for the
electromagnetic field in a magnetic dielectric rod moving with high
velocity through constant and uniform crossed fields. These results
satisfy the orthodox relativistic field transformations given in eqns.
20—23, as may be confirmed by substitution. Expressions for the
magnetisation and polarisation are limited to the case in which v « c,
but the rigorous expressions for high velocities can readily be obtained
by the use of eqn. 77 and the rigorous expressions for the field
measures given by eqns. 69—72 and 80—83.
References
1 CULLWICK, E.G.: 'An experiment on electromagnetic induction by linear
motion',/. IEE, 1939, 85, pp. 315-318
2 FARADAY, M.: 'Diary' (Bell, 1934), Vol. v, p. 397
3 CRAMP, W., and NORGROVE, E.H.: 'Some investigations on the axial spin
of a magnet and on the laws of electromagnetic induction', / IEE, 1936, 78,
pp. 481-491, and Discussion ibid, 1936, 79, p. 344
4 LORENTZ, H.A.: 'Lectures on theoretical physics' English translation by
L. Silberstein and A.P.H. Trivellis. (Macmillan, 1927-31). Vol. 3,
pp. 303-304
5 CULLWICK, E.G.: 'Electromagnetism and relativity' (Longmans Green,
1959, 2nd edn.), chap. 9-11
6 FARADAY, M., 'Experimental researches in electricity', Second series, 6,
Dec. 21st, 1831, at the Royal Institution, paragraphs 194-195, reproduced
in TRICKER, R.A.R.: 'The contribution of Faraday and Maxwell to electrical science' (Pergamon, 1966), pp. 154-155
7 FARADAY, M.: 'Diary' (Bell, 1934), Vol. i, pp. 425-426
8 POINCARE, H.: 'L'e"tat actuelle et l'avenir de la physique mathe"matique',
Bull Sci. Math., 1904, 28, pp. 302-324 and 306 (English translation,
'Monist', 1905, 15, p.l)
9 LORENTZ, H.A.: 'Versuch einer Theorie der electrischen und optischen
Erscheinungen in bewegten KOrpern' (Leiden, E.J. Brill, 1895)
10 HERTZ, H.: 'The fundamental equations of electro-dynamics of moving
bodies', Wiedermann's Annalen, 1890,41, p.369
11 EICHENWALD, A.: 'Uber die magnetischen Wirkungen bewegter Korper im
electrostatischen Felde', Ann. Phys., 1903, 11, pp. 1-30, 421-441, and
1904, 13, pp. 919-943
12 WILSON, H.A.: 'On the electric effect of rotating a dielectric in a magnetic
field', Philos. Trans. A, 1905, 204, pp. 121-137
13 CULLWICK, E.G.: 'Electromagnetism and relativity' (Longmans Green,
1959, 2nd edn.), p. 126
Correspondence
NUMERICAL METHODS IN CONFORMAL TRANSFORMATION
Indexing terms: Conformal transformations, Electrostatic
fields, Integration, Magnetostatic fields, Numerical methods
00
f =
(«i + i "
w =
dw =
-(I/A
yields the result:
fcTJi
•'o
Abstract
Numerical methods can usefully extend the application of
conformal transformation. Various approximations have been
proposed to improve the calculation of integrals containing
singularities, but approximations can be avoided in the manner
described.
[u- 4. f(l//3,)
<
fe it"i +
(2)
In many practical cases of electromagnetic-potential field
analysis, conformal transformation can facilitate accurate solution.
Methods for dealing with polygonal contours have been given,1'2 but
the difficulty of integration between limits that are isolated singular
points has persisted. Some approximations have been proposed and
their inaccuracies noted. However, the singularities of integrals can be
completely avoided without approximation, as follows:
The integrals concerned have the form
at =
c\
n
(w-
where j3fe is always positive. This integral, is the sum of two integrals
of the same form, but, respectively, with the limits:
(i) between ut and $(u,- + M1+, )
(ii) between £(M,- + ui+1) and MI+1 .
Using the following changes of variable:
(i)
f = (w w
= u.
No singularities appear, because in the first integral the (i)th singular
point is avoided, and in the second the (i + l)th singular point is
avoided.
When the variable is at the integration limits, the modulus of the
whole expression in eqn. 2 is obtained by replacing the parentheses
in the integrals by the corresponding moduli, because eqn. 2 preserves the same arguments of the complex value.
24th March 1977
PROF. A. NICOLAIDE
Department of electrical Engineering
University of Brasov
2200 Brasov
Romania
References
1 BINNS, K.J.: 'Numerical methods of conformal transformation', Proc. IEE,
1971,118, (7), pp. 909-910
2 BINNS, K.J., and LAWRENSON, P.J.: 'Analysis and computation of electric
and magnetic field problems' (Pergamon, 1973)
SPTC99S
1110
PROC. IEE, Vol. 124, No. 11, NOVEMBER 1977
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