 Longitudinal solenoid field under an axial boost

An ideal long solenoid has $B={\mu }_{0}nI$, with axial turn density n. Under an axial boost, turn density becomes $n^{\prime }=\gamma n$. The transverse circulation path is unchanged while a turn's charge-circulation rate, measured relative to its moving material, becomes $I^{\prime }=I/\gamma$. Thus $n^{\prime }{I}^{\prime }=nI$ and the axial magnetic field remains unchanged. Equivalently, the ideal transverse current-sheet density is invariant under this boost. End effects are excluded.