Michael Faraday
Note: The magnet in this article is ceramic and non-conducting.
1. Faraday’s “Paradox”.
Fig. 1_1 depicts a Faraday disk variation. A conducting disk and a non-conducting, disk-shaped magnet can be independently spun around the y-axis. A permanently resting, closing wire electrically connects the conducting disk’s periphery with its (conducting) drive shaft via sliding contacts. In series with the closing wire is a galvanometer.
Figure 1_1
Faraday Disk Variation
Faraday investigated 3 cases: (1) magnet is at rest, disk spins; (2) disk is at rest, magnet spins; and (3) magnet and disk spin in tandem.
In Case 1 the galvanometer indicates current flow … an expected result in view of the magnetic (Lorentz) force experienced by the disk’s conduction charge. In Case 2 no current is detected, again an expected result since the conduction charge in the disk 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 looped wire 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?
Faraday’s explanation for the nonzero emf in Case 3 was that the magnet’s field lines do not rotate with the magnet, but rather remain at rest in inertial space. Thus in both Cases 1 and 3 the disc’s conduction charge cuts across such lines and experiences a radial magnetic force. (Evidently Faraday was never happy with this explanation.)
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.
Figure 2_1
Uncharged Current Loop
Since the negative charge in Fig. 2_1 is at rest, B at the loop’s center points out of the page. 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 loop’s 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 inertial 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 disk-shaped permanent magnet case, the hypothetical microscopic current loops do translate when the magnet is spun, despite the fact that the magnet’s center does not. In this case a nonzero conservative E field with radial components can be expected in, above, and below the magnet … 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, hanging on a non-conducting thread and initially at rest along with a non-spinning disk-shaped magnet. Let us suppose that the magnet’s B field points upward at points just above the magnet’s upper surface.
Figure 4_1
Test Charge And Magnet
When the uncharged magnet and the test charge are at rest, the magnet has only a B field. The test charge experiences no force other than gravity.
When the magnet is spun, Faraday (now wiser from Case 3 of his dual disk experiment) would perhaps suggest that the test charge experiences a radial electric force in addition to the force of gravity. The charge does not hang verically downward.
5. Faraday Revisited.
We now reconsider the 3 cases investigated by Faraday, but this time taking into account the spinning magnet’s hypothetical nonzero, radial E field.
Case 1. Disk Spins, Magnet is at Rest.
No change here. The moving conduction charge in the spinning conducting disk experiences a strictly magnetic, radial force. Current is detected.
Case 2. Magnet Spins, Disk js at Rest.
The net E field from the magnet’s microscopic electric dipoles is conservative, and thus there is no net emf around the closing wire-conducting axle-conducting disk circuit. The galvanometer reads zero.
Case 3. Magnet and Disk Spin in Tandem.The radial magnetic force on disk conduction charge is hypothetically canceled by the radial electric force. There is zero emf in the disk 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. But note that this current is in the opposite direction than that in case
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