A Suggested Theoretical Basis for Effects Measured
by Guala-Valverde et al.
Fig. 1 depicts an uncharged current loop, at rest in the xz-plane of inertial frame K. The loop’s magnetic field on the y-axis has only a positive y-component. The loop has no E field. The small test charge is also at rest in K, and hence experiences no Lorentz force.
An Uncharged Current Loop
Viewed from frame K’, moving in the positive x-direction of K at speed v, there are both an electric and a magnetic field. At time t=0 and at points on the momentarily coincident y and y’ axes, Ez’ is positive and proportional to |v X B|. The explanation for this E’ field lies in the fact that the loop is electrically polarized in K’. That is, Leg A has a negative charge density in K’, and Leg B has a positive density.
Since F, the force on the small test charge, is zero in K, F’ must also be zero in K’. The E’ field in K’ must therefore be such that the K’ electric force on qtest is equal and oppositely directed to the K’ magnetic force. Indeed this requirement forms a basis for the electromagnetic field transformations from K to K’.
Fig. 2 depicts the top view of a disc-shaped, permanent, uncharged magnet. The "upper" surface is north. The magnet can be rotated clockwise (looking down from positive y) around the y-axis. The axle is conducting, as is a conducting ring around the magnet’s periphery. A wire labeled "A" has one end soldered to the axle, and the other to the peripheral ring.
Disc-Shaped Magnet and Wire "A"
If the magnet is at rest in the (inertial) lab frame, then the conduction electrons in wire A experience no force along the wire. But what if the magnet/Wire A/Ring is rotated around the y-axis? A conduction electron in Wire A will experience a Lorentz force:
The magnetic force points from the axle to the ring (radially outward). What might eE be?
Let us model the permanent magnet as an array of uncharged, microscopic current loops such as the loop portrayed in Fig. 1. Evidently each tiny loop is electrically polarized when the magnet rotates. For clockwise rotation Leg A would have a positive charge density, and Leg B would have a negative density. Collectively all of the microscopic electric dipoles have an electric field with a radial outward-pointing component above and below the magnet. A conduction electron in Wire A accordingly experiences a radial inward-pointing electric force. Let us again assume that the outward magnetic force and the inward electric force sum to zero. This being the case, there is no emf across Wire A’s ends when the apparatus spins. (And of course there is no emf when the apparatus does not spin.)
Fig. 3 depicts the magnet, shaft and peripheral ring in cross section. The shaft has been electrically connected to the ring by a wire/load/volt meter, which is always at rest in the lab frame. The contacts at the shaft and ring are free to slip when the magnet assembly rotates.
Magnet and Load Across Shaft and Periphery
Now assuming Wire A has zero emf induced across its ends (shaft and ring) when the assembly rotates, we might at first glance conclude that there will never be any current through the load. However, experiments by J. Guala-Valverde et al do in fact measure a current. For clockwise rotation the current flows radially outward through the load. Whence the emf?
Since the connecting wire/load assembly is at rest in the lab frame, there can be no magnetic force on any of its conduction electrons. However, the rotation-induced, outward-pointing electric field exists above and below the magnet. And it is this E field that generates the emf across the load when the magnet spins.
Fig. 4 depicts a variation of the disc-shaped magnet. Guala-Valverde et al removed a small wedge from the magnet and soldered a second wire (Wire B) to shaft and peripheral ring, said wire running down the center of the removed wedge region.
Magnet with Wedge Removed
Now in the removed wedge section the magnetic field points in the negative y-direction. Its magnitude is greatest down the wedge’s centerline (Wire B), and is least just inside the wedge edges. When the magnet does not spin there is no E field, anddBB/dt is zero. However, when the magnet spins dBB/dt is nonzero at points inside the wedge region. A radial E field is induced, said field pointing inward. In brief there is B field inversion inside the wedge, and also E field inversion when the magnet spins. For CW rotation a conduction electron in Wire B experiences an inward magnetic force and an outward electric force. We again assume that these forces cancel; there is no rotation-induced emf across the ends of Wire B.
The B and E field reversals, found inside the wedge, are for most practical purposes confined to the wedge region. Above (y>0) and below (y<0) the wedge E still points outward (when the magnet spins) and B still points upward (positive y-direction). Consequently the same connecting wire/load again experiences an outward emf when connected across the ring and axle. This is precisely what Guala-Valverde et al observed. Indeed they configured two rings, one in electrical contact with the shaft via Wire A (Fig. 2) and the other in contact with the shaft via Wire B (Fig. 4). Since neither Wire A nor Wire B have rotation-induced emf’s across their ends, the two rings are at the same potential when the apparatus spins. Guala-Valverde et al showed that this was the case by connecting the load across the two peripheral rings. Virtually no current was measured. That is, in Fig. 5 the rotation-induced, outward pointing E field integrates to zero along the connecting wire.
Rings "A" and "B" at Common Potential