On the Difference Between Spinning Electromagnets
And Spinning Permanent Magnets
1. An "Ideal" Electromagnet
Fig. 1-1 depicts a positive, circular line charge of uniform densityl++. Superimposed on this charge is a negative line charge of uniform density l- - = -l++. The net charge is zero, and indeed the net charge of any increment, of length R dq, is zero.
Superimposed Negative and Positive Circular Line Charges
Let us stipulate that the rest frame of an earthbound laboratory is an inertial frame. If the positive charge is at rest in the lab frame, and if the negative charge rotates at an angular rate ofw, then we have an uncharged current loop. Such a current loop has an associated magnetic field. For example, if the negative charge’s sense of rotation is clockwise, then at points in the loop’s plane and within its perimeter B points out of the page.
The net charge of this current loop is zero, as is the net charge of any increment of length R dq. Since charge is invariant, this also applies if the positive charge rotates clockwise at a rate of w’ and the negative charge’s rotation rate is (w + w’). The net charge density is still everywhere zero, and there is no electric field. Also, B at any given point is the same in both cases.
2. When the Current Loop Translates
It has been quantitatively demonstrated that the net charge density is not everywhere zero when the current loop of Sect. 1 translates along a line in its plane. Indeed a simple proof qualitatively indicates that uncharged current loops are electrically polarized when they translate. At any given moment the translating loop has an excess of positive or negative charge on one "side," and an excess of negative or positive charge on the other. In brief, the translating loop has an electric dipole moment (as well as a magnetic moment), and consequently there is a nonzero electric field.
This effect is only present when the loop translates. It is not present when the loop merely spins.
3. An Uncharged, Disc-shaped Permanent Magnet
Let us replace the plane interior of the current loop in Fig. 1-1 with an uncharged permanent magnet whose north pole points out of the figure. Again E will be zero everywhere, and B will point out of the page at points within the magnet’s periphery. Such a magnet certainly has a nonzero E field when it translates, sincedB/dt is nonzero at practically any given point. On the other hand, dB/dt is zero if the magnet simply spins, and we might therefore conclude that E = 0 both when the magnet is at rest and when it spins. Experiments by Guala-Valverde et al, however, suggest that this is not in fact the case.
4. A Model for the Permanent Magnet
Let us model the uncharged, disc-shaped, permanent magnet as an array of uncharged, microscopic current loops a la Fig. 1-1. The charge density is everywhere zero when the magnet is at rest, and consequently E = 0 everywhere. What happens if the magnet spins?
At any given moment one of the microscopic current loops translates relative to the lab frame. Although that current loop is not permanently at rest in any inertial frame, it is not unreasonable to suppose that it is electrically polarized (as viewed from the lab frame). That is, when the magnet spins the charge density is not everywhere zero. Consequently there is a nonzero electric field.
The electric field above and below the plane of any given microscopic current loop has a component that points radially inward or outward (depending on the loop’s translation-induced electric dipole moment). Collectively the E fields of the individual current loops sum to produce a radial component of E above and below the spinning magnet.
This, then, constitutes an important difference between a spinning, uncharged electromagnet and a spinning permanent magnet. The spinning electromagnet does not translate, and accordingly there is no spin-associated electric field. But, although the spinning permanent magnet does not translate, the microscopic current loops that generate its magnetic field do translate. Consequently the spinning permanent magnet has a nonzero E field, even thoughdB/dt = 0 for most practical purposes.
5. Stokes’ Theorem
Let us reconfigure the circular, uncharged, microscopic current loops in our permanent magnet model with microscopic trapezoids, whose sides are slightly curved. The (straight) "front" and "back" of each tiny trapezoid, which lie along radial lines, overlap with the backs and fronts of adjacent trapezoids. And the slightly curved side of one overlaps part of the side of one slightly closer to the magnet’s axis of rotation. Let us say that the magnet’s spin is such as to result in a trapezoidal current loop’s inner side being positively charged, and its outer side negatively charged.
By Stokes’ theorem the charges of the overlaid sides collectively cancel everywhere except at the magnet’s center hole (where it attaches to a shaft) and around the outer periphery. (The fronts and backs are uncharged both when the loop is at rest and when it translates.) In these two places there are excess charges of opposite sign. It is not difficult to appreciate that there will be an associated electric field with a nonzero radial component. This spin-induced electric field is electrostatic and thus conservative. The integral of E around any closed loop is zero.
6. The emf In a Radial Wire, Rotated in the Magnetic Field
Fig. 6-1 depicts a wire, soldered to the spin axis and rotating in an electromagnet’s magnetic field.
A Wire Rotating in a B Field
Any given conduction electron in the wire experiences a magnetic force away from the spin axis (shaft). There is a nonzero emf across the wire’s ends. This emf exists both when the electromagnet source of B is at rest and when it spins (e.g. in tandem with the wire).
The situation is quite otherwise if the source of B is a permanent magnet. Certainly when the magnet is at rest and the wire rotates, there is a nonzero emf. But when the magnet spins with the wire, there is a radial E field along the wire. And, as demonstrated in a previous article, the resulting electric force on each conduction electron is equal but oppositely directed to the magnetic force. The net force on each conduction electron is zero, and there is no emf!
7. The Guala-Valverde Experiment
Fig. 7-1 depicts a spinning, uncharged, permanent disc-shaped magnet. A conducting wire is soldered to the (conducting) drive shaft and to a peripheral conducting ring. At rest in the lab frame is a load, electrically connected via sliding contacts to the shaft and the peripheral ring.
Guala-Valverde has measured current through the load when the magnet spins. If the net Lorentz force, experienced by conduction electrons in the wire that is at rest on the magnet, is zero, whence the emf across the load?
The answer lies in the magnet’s spin-induced radial E field. Since the load is at rest, the electric force on its conduction electrons is not canceled by a magnetic force. Current flows. But (as Guala-Valverde has found), if the load is rotated along with the magnet, then its conduction electrons experience the same equal but oppositely directed electric and magnetic forces as the electrons in the magnet-mounted wire do. No current through the load is measured!
8. Rotational Relativity
The results obtained by Guala-Valverde and others suggest a sort of rotational relativity. That is, it seems that the emf across the load depends only upon the motion of magnet and load relative to each other. (Obviously there is a nonzero emf when the magnet is at rest and the load rotates.)
The problem with this conclusion is that Maxwell’s equations (like Newton’s laws) apply only in inertial frames of reference. The contention of this article is that the experimental results are consistent with the electric polarization of translating current loops. In brief, spinning permanent magnets have spin-induced electric fields with radial components. Such radial electric fields can produce emfs across resting loads, quite as the magnetic forces produce emfs when the load moves through the B field of a resting magnet.
It is worth noting another explanation suggested for the zero emf observed when magnet and load rotate in tandem. It has been theorized that the spinning magnet "drags" its B field along, so that the (also moving) load is not cutting across any B field lines. However, the emf is nonzero when the load spins within an enclosing electromagnet, regardless of whether the electromagnet spins or remains at rest. And the emf is zero when the load is at rest and the electromagnet spins! It seems somewhat ad hoc to suppose that a spinning permanent magnet drags its magnetic field along, whereas a spinning electromagnet does not.
The key to this difference between spinning electromagnets and spinning permanent magnets lies in the fact that the spinning electromagnet is not translating, and hence has no electric dipole moment. But when the permanent magnet spins, so that each of its microscopic current loops translates, there is electric polarization and an associated nonzero electric field, even though the magnet itself does not translate.