 Solution

For a long solenoid at rest, $B={\mu }_{0}nI$, where n is turns per unit axial length and I is the current around a turn. Under a boost along its axis, longitudinal contraction gives $n^{\prime }=\gamma n$. The circumference of a turn is transverse to the boost and does not contract; time dilation reduces its charge-circulation rate, measured relative to the moving material of that turn, to $I^{\prime }=I/\gamma$. HenceEquivalently, the transverse volume-current density of an ideal continuous current sheet is unchanged by an axial boost. The increase in turn density must not be included while leaving the per-turn current unchanged. End effects and the microscopic winding geometry are outside this ideal-solenoid argument.