5. Stable Distributions of Charge.
Although static (constant shape) distributions of charge (e.g. spherical shells of charge) are routinely discussed in electromagnetic texts, the simple truth is that Maxwell’s equations rule out the prospect of any charge increment being in equilibrium in an electrostatic field. (See for example Sands’ discussion in chapter 5 of The Feynman Lectures on Physics, V2.) According to Maxwellian theory alone, the very distributions so routinely discussed in the literature would quickly disperse in the real world.
A legitimate question is whether some dynamic arrangement would maintain its shape in time. In the case of moving charge there are also magnetic fields and forces. Is there, for example, some Gordian knot of circulating line charge that would not change its contour with time? Even with the advent of super computers, no such systems have been found. Evidently a given distribution of charge/current can be held constant only by introducing something extra-Maxwellian.
One of the intriguing extra-Maxwellian candidates has been magnetic charge. Rigorously speaking the equation "div B = 0" rules out magnetic charge. And to be sure none has been found. But experimentally speaking this implies that if magnetic charges (of both signs) do exist, they are invariably so tightly bound that a positive or negative one can never be isolated.
In any case, Fig. 5_1 depicts a circular, negative, spinning line charge situated between two magnetic point charges, q+’ and q-‘. The circular line charge’s spin vector is in the negative y-direction, so that its magnetic dipole moment points toward positive y.
Figure 5_1
A Stable System
The stability of the system depicted in Fig. 5_1 might be qualitatively described as follows. The opposite sign magnetic charges attract one another, but each also experiences a force in the opposite direction in the spinning line charge’s magnetic field. The right combination of charge magnitudes, magnetic charge separation and spinning line charge angular rate should hypothetically result in a net zero force on either magnetic charge.
Similarly every charge increment in the negative current loop experiences an electric force away from the origin, in the electrostatic field of the overall loop. But the magnetic field of the magnetic charges points in the negative y-direction at all points on the spinning line charge, and hence every charge increment also experiences a magnetic force toward the origin. Again the idea is that the net incremental force on every loop charge increment is zero.
Of historic note is Dirac’s demonstration that the quantization of electric charge theoretically implies the existence of magnetic charge. And to be sure, charge is quantized in increments of 1.6E-19 coulombs (or 1/3 this amount in the case of quarks). Equally certain is the fact that if magnetic charges exist in electrons, etc., then they are separated by very small distances (and the radii of current loops are very small). The unanswered question is, how do we open an electron up (if that is possible) and see what, if anything, is going on inside?