Current Loops That Move Parallel to Their Spin Axes
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,
At t’=t=0 and at points in the coincident yz/y’z’ planes, therefore,
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, 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
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.