Current Loops That Move Parallel to Their Spin Axes

G.R.Dixon, 7/6/2005

1. Ground Rules for the Field Transformations.

The general field transformations provide answers to the following question: Given __E__(x,y,z,t) and __B__(x,y,z,t) in inertial frame K, what will __E’__(x’,y’,z’,t’) and __B’__(x’,y’,z’,t’) be in frame K’? Here x’ is the Lorentz transformation of x, etc. In general if __E__ and __B__ are specified throughout space __at a particular instant__ in K, then the formulas for __E’__ and __B’__ will produce field vectors __over a range of t’__ and at values of x’ that map to values of x via the Lorentz transformation.

A notable exception to this rule is if K and K’ measure the fields in their yz/y’z’ planes at t’=t=0 (where t’ and t are the readings of the origin clocks). By convention the yz and y’z’ planes coincide at this moment. That is,

, (1_1a)

, (1_1b)

, (1_1c)

. (1_1d)

At t’=t=0 and at points in the coincident yz/y’z’ planes, therefore,

, (1_2a)

, (1_2b)

, (1_2c)

, (1_2d)

, (1_2e)

. (1_2f)

2. The Fields in the Plane of an Uncharged Current Loop.

In its rest frame, an uncharged current loop can be modeled as a spinning, circular, positive line charge q_{+} superimposed on a resting, negative line charge q_{-}=-q_{+}. The loop can conveniently be located in the yz plane of inertial frame K and centered on the origin. __E__ is everywhere zero, and __B__ is specified by Biot-Savart. At points in the loop’s plane, B_{y}=B_{z}=0. At all points __B__ is constant in time.

According to Sect. 1, Eqs. 1_2a through f can be applied to find __E’__ and __B’__ at points in the
y'z' plane at the single instant t’=0. (Relative to K’ the loop moves in the negative x’-direction at a constant velocity.) Thus at points (0,y’=y,z’=z,0) we have

, (2_1a)

, (2_1b)

, (2_1c)

, (2_1d)

, (2_1e)

. (2_1f)

Since t=0 is arbitrary, the following is generally true: *At any given instant the fields in the plane of an uncharged current loop, that moves at a constant velocity perpendicular to its plane, are identical to the fields measured when the loop is at rest.*