Electromagnetism and Rotational Relativity
In The Feynman Lectures on Physics, V2, Sect. 14-4, R.P.Feynman discusses the following apparatus: an arbitrarily long cylinder of charge (solenoid) has a small wire soldered to its inner wall as depicted in Fig. 1. Two cases are considered: (1) the cylinder is at rest in an inertial frame, and (2) the cylinder rotates.
Since the electric field is zero inside the cylinder, there is no electric polarization in the wire when the cylinder is at rest. When the cylinder rotates, there is a uniform B field within. And since the wire cuts across these B field lines, each charge in the wire experiences a magnetic force:
Thus there is electric polarization in the wire.
Feynman’s point is that, in electromagnetism, "There is no relativity of rotation … We must be sure to use equations of electromagnetism only with respect to inertial coordinate systems." For example, let us dub the inertial frame in which the cylinder’s axis is at rest "frame K." And let us dub the noninertial frame, in which the cylinder is at rest when rotating in frame K, "frame K’." The wire’s polarization when it and the cylinder rotate in K (case 2) is readily understood (Eq. 1). And its lack of polarization when it and the cylinder are at rest in K (case 1) is also explainable.
Viewed from frame K’, case 1 becomes case 2’; the cylinder and wire rotate. Yet in K’ there is no polarization of the wire (and for that matter no B’ field). Similarly case 2 becomes case 1’. The cylinder and wire are at rest in K’; yet there is polarization (and more generally a B’ field). Feynman concludes that Maxwell’s equations and the Lorentz force law apply only in inertial frames of reference.
This idea stressed by Feynman … the principle that the laws of electromagnetism do not work in rotating frames of reference … was seemingly challenged in experiments performed recently by Guala-Valverde et al. Using a rotating/stationary disc-shaped permanent magnet and an independently rotatable wire probe, they found that the probe is polarized both (a) when the magnet is at rest and the probe rotates, and (b) when the probe is at rest and the magnet rotates. Fig. 2 depicts relevant portions of their apparatus.
Schematic of the Guala-Valverde Apparatus
The probe polarization is readily understood in case a. Here again the charges in the probe experience a magnetic force (Eq. 1) in the disc magnet’s B field. The surprising result is in case b; there is evidently polarization when the probe is at rest and the magnet spins. Since the magnet’s B field is presumably constant in time in this case (as in case a), the theoretical basis for this polarization was not immediately obvious.
We should include the additional cases where (c) both probe and magnet are at rest, and (d) both probe and magnet spin together (at rest relative to each other). In these two cases it is found that (c) there is no polarization, and (d) there is polarization, but not what might be expected solely from magnetic forces on the probe charges (a la Feynman). It should be noted that, viewed from the rotating frame K’, the polarization in case d’ (magnet and probe mutually at rest) and the lack of polarization in case c’ (magnet and probe rotate together) are difficult to explain.
The crux of the problem seemed to be the observed probe polarization in case b, where the magnet spins and the probe is at rest. Since there is (according to the Lorentz force law) no magnetic force acting on probe charges, the observed polarization must be ascribable to an electric field/force. The question is, to what might such an electric field be attributed? The magnet is uncharged, and its magnetic field in every case is presumably constant in time.
The theoretical basis of this observed phenomenon … the probe polarization when it is at rest and the magnet spins … is found in the electric polarization of moving current loops. This relativity of current loop electric polarization is discussed in several articles on this website. The disc-shaped, uncharged, permanent magnet can be modeled as an array of microscopic current loops. When the magnet spins relative to the lab (inertial) frame, each of these tiny current loops translates and consequently becomes electrically polarized. The direction of the electric dipole moment is, in each tiny loop’s case, radially toward or away from the magnet’s spin axis (depending upon whether the magnet’s spin is CW or CCW, viewed from above). Thus there is a net radial electric field component above and below the magnet when it spins. This radial E field component is the theoretical basis for the observed probe polarization in case b (probe at rest, magnet spinning). It also explains why the force experienced by probe charges in case d (probe and magnet spin together) are not explained solely by magnetic forces. In this case the force on a probe charge is a sum of electric and magnetic forces: