Faradayís "Paradox" and Spin-Induced Electric Fields
Abstract. Evidently Faraday was puzzled when he found that current might flow in a wire when the wire and the current-inducing magnet are at rest relative to one another. The "paradox" is shown to be attributable to a conservative electric field, induced by a spinning, disc-shaped permanent magnet.
Note: The magnets in this article are ceramic, non-conducting magnets
1. Faraday's "Paradox".
Fig. 1_1 depicts a Faraday disc variation. A conducting disc and a non-conducting, disc-shaped magnet can be independently spun around the y-axis. A closing wire electrically connects the conducting discís periphery with its (conducting) drive shaft via sliding contacts. In series with the closing wire is a galvanometer.
Faraday Disc Variation
Faraday investigated 3 cases: (1) magnet at rest, disc spins; (2) disc at rest, magnet spins; and (3) magnet and disc spin in tandem.
In Case 1 the galvanometer indicates current flow Ö an expected result in view of the magnetic (Lorentz) force experienced by the disc's conduction charge. In Case 2 no current is detected, again an expected result since the conduction charge is at rest and experiences no magnetic force. In case 3 current is detected.
Faraday found Case 3 to be somewhat paradoxical, since he usually found an induced emf only when the magnetic field source and the conducting circuit move relative to one another. For example, one finds a nonzero emf when the magnetic flux, threading the area spanned by a circuit, varies in time. Noteworthy in this regard is that, in Case 3, the magnetic flux through any area is constant in time.
The other type of emf occurs when a conductor moves relative to a magnet (Faraday's experiment, Case 1). But in Case 3 the magnet and conductor are at rest relative to one another. Whence the emf?
Faradayís explanation for the nonzero emf in Case 3 was that the magnetís field lines do not rotate with the magnet, but remain at rest in inertial space. Thus in both Cases 1 and 3 the discís conduction charge cuts across such lines and experiences a magnetic force.
2. Translation-Induced Electric Fields.
Fig. 2_1 depicts an uncharged current loop, consisting of a circulating rectangular positive line charge overlaid on an equal non-circulating negative line charge.
Uncharged Current Loop
Since the negative charge in Fig. 2_1 is at rest, B at the loopís center points out of the page, and E=0 everywhere. If the loop translates in the positive x-direction, then the Lorentz transformation indicates a net positive charge density in the bottom leg, and a net negative charge density in the top leg.
Owing to the nonzero charge densities when the loop translates, the translating loop has a nonzero electric dipole moment. Among other things, the translating loop has a nonzero electric field whereas the non-translating loop has none. Noteworthy is the fact that the loopís net charge is zero in both cases.
3. A Model for a Permanent, Non-conducting Magnet.
Uncharged, resting permanent magnets have nonzero B fields but no E fields. A convenient model for the magnet is thus an array of microscopic, uncharged current loops a la Fig. 2_1. Of course when such a magnet translates, then dB/dt at points in space will be nonzero and there will be a nonzero E field with nonzero curl. But what if the magnet does not translate, but does spin?
In the disc-shaped permanent magnet case, the hypothetical microscopic current loops do translate when the magnet is spun, despite the fact that the magnet as a whole does not. In this case a nonzero conservative E field with radial components can be expected in, above, and below the magnet Ö a possibility that Faraday seems not to have been aware of. For even if he did suspect that there might be microscopic current loops, he had no knowledge of the Lorentz transformation and the result that such tiny loops are electrically polarized when they translate in or close to their planes.
4. A Simple Experiment.
Fig. 4_1 depicts a positive test charge, kept more or less at rest relative to a disc-shaped magnet. We shall suppose that the supporting column is rubber, so that the test charge can move toward or away from the magnet's axle when subjected to a net radial force.
Let us suppose that the magnet's B field points upward at points just above the magnet's upper surface.
Test Charge And Magnet
When the uncharged magnet and the test charge are at rest, then the magnet has only a B field. The test charge experiences no electric force. And, since it is at rest, it experiences zero centrifugal and zero magnetic force. The supporting column remains straight.
When the magnet/charge is spun, Faraday (now wiser from Case 3 of his dual disc experiment) would perhaps suggest that the test charge experiences a mysterious radial magnetic force, in addition to an outward-pointing centrifugal force. Since the magnet's B field always points upward regardless of which way it spins, the magnetic force on the test charge should either point in the same direction as the centrifugal force, or opposed to it. To the extent the charge's support is flexible, it should bend either more or less than it would if the spinning disc were unmagnetized. Indeed Faraday might suggest that, for a given spin direction, one could tell from the bending of the rubber stand, (a) whether or not the disc was in fact a magnet, and (b) if so, which surface was north.
However, if our myriad microscopic current loop model is feasible then the spinning, permanent, non-conducting magnet presumably has a radial electric field that, above the magnet, points either radially toward or away from the spin axle. And the test charge should experience an electric force in addition to the magnetic and centrifugal forces.
Now here is an interesting point: when the magnet and the test charge spin around the magnet's axle, the magnetic force and the electric force are always in opposite directions. If they always sum to zero, then the only force on the test charge would be the centrifugal force. The rubber stand should bend outward the same amount as it would if the spinning disc were unmagnetized. We can test whether the charge feels an electric force by conducting this simple experiment. Let us suppose that this is actually found to be the case. We shall dub it "Faraday's surprise."
5. Faraday Revisited.
We now reconsider the 3 cases investigated by Faraday, but this time taking into account the spinning magnetís hypothetical nonzero, radial E field.
Case 1. Disc Spins, Magnet is at Rest.
No change here. The moving conduction charge in the spinning conducting disc experiences a strictly magnetic, radial force. The (resting) closing wire conduction charge experiences no such force, and hence there is a net emf around the circuit. Current is detected.
Case 2. Magnet Spins, Disc is at Rest.
The net E field from the magnet's microscopic electric dipoles is conservative, and thus there is no current around the closing wire-conducting axle-conducting disc circuit. The galvanometer reads zero.
Case 3. Magnet and Disc Spin in Tandem.
The radial magnetic force on disc conduction charge is hypothetically canceled by the radial electric force. There is zero emf in the disc portion of the circuit. But the conduction charge in the (resting) closing wire experiences an electric force. There is consequently a net emf around the circuit. Current flows through the galvanometer.
6. Testing for the Hypothetical E Field.
Figure 6_1 depicts a resting disc-shaped magnet with a test charge (say a charged pith ball) hanging from a silk thread. When the magnet is spun there is hypothetically a radial E field at the test charge's position. The test charge should swing toward or away from the magnet's center, depending on the direction of the spin. The possibility that air currents from the spinning magnet contribute to the test charge's motion can be determined by initially doing the experiment with an uncharged pith ball.
Test For E Field