The Fields of an Infinitely Long, Uncharged Solenoid that Translates Parallel to its Longitudinal Axis
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 y’z’ plane of inertial frame K’ at time t’=0. Inside the solenoid
At all points outside the solenoid,
In brief, the fields in the y’z’ 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.