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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/yz planes at t=t=0 (where t and t are the readings of the origin clocks). By convention the yz and yz 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/yz 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 loops plane, By=Bz=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.