= Longitudinal solenoid field under an axial boost
An ideal long <solenoid> has $B=\mu_0nI$, with axial turn <density> n. Under an axial boost, turn <density> becomes $n'=\gamma n$. The transverse circulation path is unchanged while a turn's charge-circulation rate, measured relative to its moving material, becomes $I'=I/\gamma$. Thus $n'I'=nI$ and the axial <magnetic field> remains unchanged. Equivalently, the ideal transverse current-sheet <density> is invariant under this boost. End effects are excluded.
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