A Simple Proof that Moving Current Loops are Electrically Polarized

G.R.Dixon

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When E is zero everywhere in inertial frame K and B is nonzero, then in frame K’ (moving in the positive x-direction of K at speed v)

Ey’ = -gvBz, (1a)

Ez’ = gvBy. (1b)

Or,

E’ = gv x B. (2)

Imagine a positive, circular line charge of radius R, at rest in the xz-plane of K and centered on the origin. Superimposed is a negative line charge that circulates clockwise, looking down from positive y. Together the two charges constitute an uncharged current loop with E=0 everywhere and with a nonzero dipolar B field.

Construct a pillbox that encases part of the yz-plane, with all internal z<0. Let the periphery coincide with a magnetic field line. It is clear from Eq. 2 that E’ (a) points outward everywhere on the periphery, and (b) is parallel to the pillbox faces. Thus there is a net outward electric field flux. By Gauss’ law the enclosed increment of current loop has excess positive charge.

Repeating this exercise with all pillbox internal points at z>0, there is an inward flux and excess negative charge.

If a similar pillbox is constructed in the xy-plane with x>0, then E’ is parallel to the peripheral surface and, by symmetry, equal at opposing points in the two faces. Thus the E’ flux is zero in this case. Similarly for x<0. Again by Gauss’ law, the loop contains no net charge of either sign at x’=+R and z’=0.

Summarizing, a current loop which is uncharged in its rest frame is electrically polarized in other frames (at least when the loop’s velocity lies in its plane). As suggested in another article, this result is consistent with the computation of each positive and negative line charge increment’s position in K’ at the single instant t’=0 (or at any other instant in K’). Similar remarks would apply to a current loop of pure positive or negative charge, although in this case E would not be zero in frame K. Current loop electric polarization is relative, varying from frame to frame.

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