Magnetic fields in moving conductors: four simple examples

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 451 452 P Lorrain et al 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 ǫ0 ) ∇ · E = Q/(ǫ ∇·B =0 ∇×E =0 ∇ × B = µ0 J (1) (2) (3) (4) e is the net electric volume charge density, and ǫ0 and µ0 are, respectively, the where Q permittivity and the permeability of free space. The relative permittivity ǫr 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 ′ a reference frame that follows the conductor. Then (Lorrain et al 1988) B ′ = B as long as v 2 ≪ c2 : observers in the fixed and moving frames S and in S ′ , 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 = −ǫr ǫ0 ∇·(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 ′ of the conductor, B ′ is constant. Using the more general form of equation (3), with the time derivative of B on the right, ∇′ × J ′ = σ ∇′ × E ′ = −σ ∂ B ′ /∂t ′ = 0. 2 2 (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 ′ an arbitrary closed loop inside the plate, and A′ an arbitrary surface bounded by C ′ . Then, from Stokes’s theorem, I Z (∇′ × J ′ ) · dA′ = 0. (14) J ′ · d l′ ≡ C′ A′ 454 P Lorrain et al 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 ′ is arbitrary, then J ′ = 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 ′ 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 = −2ǫr ǫ0 ωB e Q (15) S = ǫ0 ωBR 23 + ǫr sin2 θ − 1 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 ′ /∂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. 456 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 = −ǫr ǫ0 ∇ · (v × B ) = −2ǫr ǫ0 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. 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