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The Fields of an Infinitely Long, Uncharged Solenoid that Translates Parallel to its Longitudinal Axis

G.R.Dixon, 7/7/2005

Let an infinitely long, uncharged solenoid be at rest in inertial frame K, with its longitudinal axis coincident with the x-axis. E=0 everywhere, and B=0 outside the solenoid. Within, B is single-valued with By=Bz=0.

For reasons previously discussed, the general field transformations can be used to determine the fields in the yz plane of inertial frame K at time t=0. Inside the solenoid

, (1a)

, (1b)

, (1c)

, (1d)

, (1e)

. (1f)

At all points outside the solenoid,

, (2a)

. (2b)

In brief, the fields in the yz plane of K are, at time t=0, the same as those in K.

Since the election of where (in the +x-direction) to locate the yz plane of K is arbitrary, the fields in K are everywhere the same as in K. This is true at all times.