Faradayís "Paradox" and Spin-Induced Electric Fields
1. Faradayís "Paradox"
Fig. 1_1 depicts a Faraday disc variation. A conducting disc and a non-conducting (e.g. a ceramic), disc-shaped magnet can be independently spun around the y-axis. A closing wire electrically connects the 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) forces experienced by disc conduction charges. In Case 2 no current is detected. 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 moved relative to one another. That is, he usually found a nonzero emf only when the magnetic flux, threading the area spanned by the circuit, varied in time. Noteworthy in this regard is that, in Case 3, the magnetic flux through any area is constant in time.
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 charges cut across such lines and experience magnetic forces.
2. Spin-Induced Electric Fields.
Fig. 2_1 depicts an uncharged current loop, consisting of a circulating rectangular positive line charge and an equal non-circulating negative line charge.
Uncharged Current Loop
If the negative charge in Fig. 2_1 is at rest, then B at the loopís center points out of the page, and E=0 everywhere. If the negative charge 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.
The nonzero E field in the translating case is consistent with the Lorentz field transformations. That is, if Fig. 2_1 is viewed from an inertial frame moving to the left, then from that perspective there will be both nonzero B and E fields.
3. A Model for a Permanent Magnet
Uncharged, permanent magnets have nonzero B fields and no E fields. A convenient model for the magnet is thus an array of microscopic, uncharged current loops. Of course when such a magnet translates, then dB/dt at points in space may be nonzero, and there may be a nonzero E field with nonzero curl.
In the disc-shaped magnet case, the hypothetical microscopic current loops translate when the magnet is spun, despite the fact that the magnet as a whole does not translate. In this case a nonzero E field with radial components can be expected Ö a result that Faraday seems not to have been cognizant of.
4. Quasi-Rotational Relativity
Fig. 4_1 depicts a test charge, kept permanently at rest relative to a disc-shaped magnet.
When the magnet/charge is at rest in the lab frame, then the magnet has only a B field. The test charge experiences no electric force. And, since it is at rest, it experiences zero magnetic force.
When the magnet/charge is spun, Faraday (or his ghost) would perhaps suggest that the test charge experiences a radial magnetic force. However, the spinning magnet presumably also has an electric field, and the test charge also experiences an electric force. Furthermore, the radial component of this electric force points opposite to the magnetic force. Experiment indicates that the two forces may sum to zero. In both the non-spinning and spinning cases, then, the force on the test charge is zero! Hence the "quasi-relativistic" description.
5. Faraday Revisited
Let us consider the 3 cases investigated by Faraday once again, but this time taking into account the spinning magnetís nonzero E field.
Case 1. Disc Spins, Magnet is at Rest.
No change here. The moving conduction charges in the spinning disc experience strictly magnetic, radial forces. The (resting) closing wire conduction charges experience 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 spinning magnetís electric field is conservative (curl E=0). Thus the net emf around any circuit is zero. No current is detected.
Case 3. Magnet and Disc Spin in Tandem.
The magnetic forces on disc conduction charges are canceled by electric forces. There is zero emf in this part of the circuit. The emf in the closing wire portion of the circuit is thus unbalanced; there is a net emf around the circuit, and current is detected.
6. Detecting the Spin-Induced Electric Field
Fig. 6_1 depicts a resting test charge, suspended above the disc-shaped magnet. The test charge is attached to a sensitive dynamometer.
In Fig. 6_1 we expect no magnetic force on the test charge, regardless of whether or not the magnet spins. And since the non-spinning magnetís E field is zero, the dynamometer should read zero in the non-spinning case.
But the test charge should experience a radial electric force component when the magnet spins. Thus the dynamometer should indicate a radial force. Note that this positive reading should be obtained even though the original Faraday apparatus registers zero current in Case 2.
The author is uncertain whether or not this experiment has been performed to date. The dynamometer should of course be such as to not itself be affected by the spinning magnetís E field.