Eur. J. Phys. 19 (1998) 451–457. Printed in the UK
PII: S0143-0807(98)95317-X
Magnetic fields in moving conductors:
four simple examples
Paul Lorrain†, James McTavish‡ and François Lorrain§
† Department of Earth and Planetary Sciences, McGill University, 3450 rue Université,
Montréal, Québec, Canada, H3A 2A7
and
Institut d’Astrophysique de Paris, CNRS, Paris, France
‡ School of Engineering, Liverpool John Moores University, Byrom Street, Liverpool
L3 3AF, UK
§ Collège Jean-de-Brébeuf, Montréal, Québec, Canada H3T 1C1
Received 22 June 1998
Abstract. Assuming given externally applied magnetic fields and given velocities, we calculate
the currents induced in moving conductors of arbitrary conductivity, and the net magnetic field.
Three of the four conductors are solid, and the fourth is fluid. We find that, in these examples,
the net magnetic field is either unaffected by the moving conductor, or distorted downstream, or
distorted upstream. In all cases the net magnetic field is static.
1. Introduction
Conductors that move in magnetic fields are everywhere. For example, the solar plasma
convects in its own magnetic field. Certain features, such as sunspots, are self-excited dynamos
that generate magnetic fields (Lorrain 1995, Lorrain and Koutchmy 1993, 1996, 1998), while
in other regions the solar plasma convects in magnetic fields generated elsewhere. The liquid
outer shell of the earth’s core similarly convects in its own magnetic field (Lorrain 1993).
There exists an abundant literature on the interaction between magnetic fields and moving
conductors. But only Lundquist (1952) has calculated the distortion of magnetic field lines by
moving conductors in specific cases. He found that the net magnetic field is static. Shercliff
(1965) gives one example of a magnetic field that is not affected by a moving conductor, and
two examples where the moving conductor shifts the magnetic field lines upstream, but his
discussion is brief and qualitative. His net magnetic fields are also static.
Teachers of courses on electromagnetic theory, geophysics, or astrophysics will find
thought-provoking exercises here. Readers who are pressed for time should read our
conclusions in section 4 now.
A more extensive discussion, including other examples, is available from one of the authors
(PL).
We calculate here the induced currents and the net magnetic field in simple cases. The
conductors are solid, save for the sphere of Example 4, but our general conclusions apply
equally well to fluids, e.g. to space plasmas (Peratt 1991). The conductivity σ can have any
arbitrary value, finite or infinite.
0143-0807/98/050451+07$19.50
© 1998 IOP Publishing Ltd
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We assume steady state conditions: with respect to a stationary reference frame S, the field
variables E and B , the electric current density J , and the electrostatic volume and surface
e and e
charge densities Q
S, are all independent of the time. All transient phenomena have
become negligible, charges have moved to their equilibrium positions, and the conductors
have reached their asymptotic velocities. We therefore set ∂/∂t = 0. If the conductor moves
in a straight line, the velocity v of a given point inside is constant. If the conductor rotates,
then v is a function of the time, but the angular velocity vector is constant. Then the Maxwell
equations apply only in the neighbourhood of a given point (Møller 1974). The conductors
move in a vacuum, and there are no externally applied electric charges, fields or currents.
We assume non-magnetic conductors and non-relativistic speeds: µr = 1, v 2 c2 ,
where c is the speed of light. Then Maxwell’s equations reduce to (Lorrain et al 1988)
e r 0 )
∇ · E = Q/(
∇·B =0
∇×E =0
∇ × B = µ0 J
(1)
(2)
(3)
(4)
e is the net electric volume charge density, and 0 and µ0 are, respectively, the
where Q
permittivity and the permeability of free space. The relative permittivity r of a conductor
is not measurable, but it is presumably of the order of 3, as for dielectrics. The electric current
density J results from the motion of free charges; we neglect polarization currents.
Recall that
E = −∇V − ∂ A/∂t
(5)
where V is the electric potential of the volume and surface charges, and A is the magnetic
vector potential, defined by B = ∇ × A. Here,
∂ A/∂t = 0 and E = −∇V .
(6)
Recall also that, if a body of conductivity σ moves in a field E , B , then the conduction
current density is
J = σ [E + (v × B )]
(7)
where all five variables are measured with respect to a stationary reference frame S. This is
Ohm’s law for moving conductors.
Note that B is the net field:
B = Bext + Bind
(8)
where Bext is the externally applied field, and Bind is the magnetic field of the induced currents.
So
J = σ {E + [v ×(Bext + Bind )]}
(9)
and Bind is a function of J through equation (4).
The above equation is complex because, except for the imposed Bext and the imposed v ,
all the variables are functions of all the others.
Call S 0 a reference frame that follows the conductor. Then (Lorrain et al 1988) B 0 = B
as long as v 2 c2 : observers in the fixed and moving frames S and in S 0 , measure identical
magnetic fields. In the examples below, the induced electric currents and the net magnetic
fields are static in both frames.
Magnetic fields in moving conductors
453
2. The electrostatic volume charge density inside moving conductors
It is well known that the relaxation time of electrostatic volume charges in conductors is
normally exceedingly short (Lorrain et al 1988). It is less well known that, if a conductor
moves in a magnetic field, then it may carry a volume charge of density (van Bladel 1984,
Lorrain 1990)
e = −r 0 ∇·(v × B ).
Q
(10)
The proof is simple. Take the divergence of equation (7), assuming a uniform conductor:
∇ · J = σ [∇ · E + ∇·(v × B )].
(11)
e
From the law of conservation of charge ∇ · J = −∂ Q/∂t and, under steady conditions, the
left-hand side is zero and
∇ · E = −∇·(v × B ).
(12)
e
Applying now equation (1) leads to equation (10). Observe that Q is independent of σ .
The electrostatic volume charges result from the fact that the v × B field sweeps free
electrons inside. In a region where ∇·(v × B ) is positive, conduction electrons move in, and
the region becomes negative. But if ∇·(v × B ) is negative, then some electrons move out and
e and ∇·(v × B ) have opposite signs, as above.
the region becomes positive. So Q
Electrostatic volume and surface charges play an essential role whenever conductors move
in magnetic fields because, as we shall see below, their field E cancels at least part of the v × B
field in equation (7).
e is a function of B , which depends on the other variables of equation
From equation (10), Q
e and hence on B .
(9). Of course, the potential V , and hence E in equation (9), depends on Q,
So equations (9) and (10) are both complex, and it seems hopeless to attempt a general
analysis. We therefore investigate only specific cases where either Bind = 0, or v × B ind = 0.
Then we can disregard the Bind terms in equations (9) and (10), but we can still calculate Bind
from J .
3. Examples
Example 1. Rigid object, no rotation, J = 0,
Bind = 0
A rigid conducting object of arbitrary conductivity, shape, and size moves at a constant velocity
v and without rotating, in a vacuum, in a uniform magnetic field Bext . Figure 1 shows a copper
plate as an example. There is no external circuit.
Initially, the plate is stationary in the field Bext . Next, an externally applied force F pulls
the plate to the right. The plate accelerates and acquires a velocity v . The v × B ext field drives
an induced electric current of density J in the downward direction, and electrostatic charges
build up at the top and bottom of the plate. Figure 1 shows the braking force, of density J × B ,
that opposes F , as well as the orientation of the induced magnetic field. What happens once
a steady state has been reached?
In the rest frame S 0 of the conductor, B 0 is constant. Using the more general form of
equation (3), with the time derivative of B on the right,
∇0 × J 0 = σ ∇0 × E 0 = −σ ∂ B 0 /∂t 0 = 0.
(13)
The conductivity σ is the same in both frames if v c .
Since there is no external circuit, the induced currents, if any, can only flow in closed loops.
Call C 0 an arbitrary closed loop inside the plate, and A0 an arbitrary surface bounded by C 0 .
Then, from Stokes’s theorem,
I
Z
J 0 · d l0 ≡
(∇0 × J 0 ) · dA0 = 0.
(14)
2
C0
A0
2
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Figure 1. Example 1. A rectangular copper plate moves to the right without rotating
at the constant velocity v in a uniform magnetic field B that points out of the paper.
(a) The copper plate is first stationary in the field B . (b) An externally applied force F
pulls the plate to the right at the constant velocity v . The v × B field causes an induced
current, of density J , to flow downward, charging the top and bottom surfaces until
the resulting E cancels v × B . We have not shown the fringing electric field of those
charges. The squares show the orientation of the magnetic field of the induced current.
There is a magnetic braking force, of density J × B , that opposes part of F . (c) Soon, the
electrostatic charges have built up to their equilibrium value, and E = −v × B , J = 0.
There is no braking force, F = 0, and the copper plate continues moving to the right in
the absence of any force. From then on, the moving copper plate has no effect on the
magnetic field because there are no induced currents. There are no electrostatic volume
charges inside the plate, and there is no magnetic braking force.
Since C 0 is arbitrary, then J 0 = 0 and, once a steady state has been reached, there is no induced
current and no induced magnetic field. From equation (7), E + (v × B ext ) = 0: the E of the
electrostatic charges cancels v × B exactly at every point inside the plate, whatever the value
of its conductivity σ . Both E and v × B are independent of σ . Since the moving plate does
not carry a current, it does not disturb the magnetic field lines.
e within the plate is zero because v × B is independent of the
From equation (10), Q
coordinates. So there is a net charge density only on the surface.
If the copper plate moved in a conducting fluid, then E would not completely cancel v × B ,
current would flow both through the plate and the fluid, and there would be a magnetic braking
force. Then the applied force F would have to be adjusted accordingly to maintain the constant
velocity v .
Example 2. Rotating sphere, J = 0,
Bind = 0
A homogeneous conducting sphere rotates at a constant angular velocity in a uniform magnetic
field. The axis of rotation is parallel to the field. What effect does the sphere have on the field?
As in example 1, B 0 is constant. Then there is no induced current in the sphere and no
induced magnetic field. The magnetic field is unaffected by the rotating conducting sphere.
This was known to J J Thomson (1893) over a century ago.
This fact has baffled many an author: how can there be no induced current when there is
a v × B field? The explanation is that the E of the electrostatic charges cancels v × B at
every point. van Bladel (1984) and Lorrain (1990) showed that the volume and surface charge
densities are, respectively,
e = −2r 0 ωB
e
Q
S = 0 ωBR 23 + r sin2 θ − 1
(15)
where R is the radius of the sphere and θ is the angle between the position vector r and B .
The magnetic field need not be uniform. As long as the applied magnetic field is
axisymmetric, ∂ B 0 /∂t = 0, and there is no induced current and no induced magnetic field.
Magnetic fields in moving conductors
455
The rotating body need not even be spherical or homogeneous! Inside a conducting body of
any shape, size, or conductivity that rotates at a constant angular velocity in an axisymmetric
magnetic field, a rotating observer sees a constant field. There again, there is no induced
magnetic field: the magnetic field is unaffected by the rotating body.
Example 3. Faraday disc, J 6= 0,
v × B ind = 0
Figure 2 shows a disc dynamo, or Faraday disc, or homopolar generator (Lorrain et al 1988,
Lorrain 1990, 1995). The v × B field in the disc drives a current I in the direction shown.
Disregard the field of the external circuit. The magnetic fields of the current along the axle
and in the disc add azimuthal fields to the applied axial field B . So v × B ind = 0.
Figure 2. Example 3. Cross section through a Faraday
disc. The device is symmetrical about the vertical axis. A
motor M rotates the disc D in the axial magnetic field B of
the solenoid S. With the polarities shown, the v × B field in
the disc drives a current I in the direction shown through
the switch and the resistance R. There are contacts all
around the axle and the rim.
It is easy to show that the net magnetic field is static, despite the rotation: stop the rotation,
open the circuit at the switch, and apply at its terminals a voltage that restores the current I .
The current distribution and the magnetic field are unchanged. The magnetic field is static,
whether the disc rotates or not.
Example 4. Fluid within a sphere, J 6= 0,
v × B ind = 0
The conducting fluid inside a sphere of radius R has an azimuthal velocity v that is a function
of ρ and z:
v = ωρ[1 − K(z2 /R 2 )]φ̂.
(16)
If K is positive, the equator rotates faster than the poles. The sphere rotates in a uniform axial
magnetic field Bext ẑ .
Inside the sphere, the field v × B ext points everywhere in the ρ̂ direction, while E
results from the presence of both volume and surface electrostatic charges. Because of the
axisymmetry, E has only ρ̂ and ẑ components. Then, from equation (7), J also has only ρ̂
and ẑ components, and a current sheet has the shape of a toroid. So the induced magnetic field
is azimuthal, like the velocity v = ωρ φ̂, and v × B ind = 0.
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P Lorrain et al
Under steady conditions, if σ is uniform,
v × B = ωρ φ̂×(Bext ẑ + Bind φ̂) = ωρBext [1 − K(z2 /R 2 )]ρ̂
e = −r 0 ∇ · (v × B ) = −2r 0 ωBext [1 − K(z /R )]
Q
e and E , are all independent of σ .
where v × B , Q
Also,
2
2
∇ × J = σ ∇×[−∇V + (v × B ext )] = σ ∇×(v × B ext )
= − 2σ ωBext K(ρz/R 2 )φ̂.
(17)
(18)
(19)
(20)
If ω, Bext , K, z are all positive, ∇ × J points in the −φ̂ direction, and current flows in
closed loops in the direction that gives a magnetic field pointing also in the −φ̂ direction. The
magnetic field is that of a toroidal coil, and ∇ × J and Bind point in the same direction.
With ω, Bext , K positive, Bind points in the −φ̂ direction in the upper hemisphere, and
in the +φ̂ direction in the lower hemisphere: magnetic field lines are distorted downstream,
but they are stationary.
Now suppose that ω and Bext are again positive, but that K lies between −1 and 0. This
changes the signs of ∇ × J and of Bind φ̂, and the magnetic field lines are distorted upstream.
For a given angular velocity, they are stationary.
Equation (20) shows that ∇ × J , and hence J , are proportional to σ , which is uniform by
hypothesis. Then equation (7) shows that E + v × B is independent of σ .
Now consider a closed curve C that follows the conductor, and the magnetic flux 8
linking C. The net magnetic flux density is the sum of Bext , which is axial, plus Bind , which
is azimuthal. The velocity of a given point on C is azimuthal.
For C, choose a circle that lies originally in a radial plane through the axis of symmetry.
The center of C is at ρ = R/2, z = R/2. Since the velocity of a point on C is purely
azimuthal, the flux of the azimuthal Bind field through C is constant.
Whether K > 0 or K < 0, the azimuthal velocity of a point on C changes with z, as
in equation (16). Seen from above, the area enclosed by C changes with time, and 8 also
changes with time; it is not constant.
The magnetic flux 8 linked by a closed curve C that moves with the fluid is in general not
constant, whatever the value of σ .
4. Conclusions
These four examples are instructive, even though either Bind = 0 or v × B ind = 0, and even
though they concern only specific, steady state situations. Many readers will think of other
specific cases, possibly more general.
For these examples, and for the Lundquist (1952) and Shercliff (1965) cases, we have the
following results that apply whatever the value of the conductivity σ .
(1) Under steady conditions, in the presence of a moving conductor, magnetic field lines can
be either unaffected, or distorted downstream, or distorted upstream. The net magnetic field is
static. Magnetic field lines are not dragged by the moving conductor (Alfven and Falthammar
1963).
(2) Either E cancels v × B exactly at every point inside the conductor, and the moving
conductor then has no effect on the magnetic field lines, or E cancels only part of v × B ,
and there are induced currents, an induced magnetic field, the moving conductor distorts the
magnetic field lines upstream or downstream, and there is a magnetic braking force. The net
magnetic field is static.
(3) Both E and v × B are independent of σ ; increasing σ increases J proportionately.
Magnetic fields in moving conductors
457
Acknowledgments
PL thanks Luce Gauthier for her comments, and Professors Ernesto Martin and José Margineda
of the Departamento de Fisica of the Universidad de Murcia, Espinardo, España, Jean-Louis
Le Mouël of the Institut de Physique du Globe de Paris, and Serge Koutchmy of the Institut
d’Astrophysique de Paris for their hospitality. JMcT thanks Professor C Michael of the
University of Liverpool for making available the facilities of the DAMTP. This work was
subsidised in part by the NATO Cooperative Research Grant CRG 940291.
References
Alfven H and Falthammar C G 1963 Cosmical Electrodynamics (Oxford: Clarendon) p 189
Lorrain P 1990 Eur. J. Phys. 11 94
—— 1993 Phys. Scr. 47 461
—— 1995 Phys. Scr. 52 349
Lorrain P, Corson D R and Lorrain F 1988 Electromagnetic Fields and Waves 3rd edn (New York: Freeman) pp 75,
287, 303, 314, 321, 402 and 492
Lorrain P and Koutchmy O 1998 The sunspot as a self-excited dynamo Astron. Astrophys. submitted
Lorrain P and Koutchmy S 1993 Astron. Astrophys. 269 518
—— 1996 Solar Phys. 165 115
Lundquist S 1952 Ark. Fys. 5 297
Møller C 1974 The Theory of Relativity (Oxford: Oxford University Press) ch 8 and 9
Peratt A L 1991 Physics of the Plasma Universe (Berlin: Springer)
Shercliff J A 1965 A Textbook on Magnetohydrodynamics (Oxford: Pergamon) p 49
Thomson J J 1893 Notes on Recent Researches on Electricity and Magnetism (Oxford: Clarendon) p 534
van Bladel J 1984 Relativity and Engineering (Berlin: Springer) p 287