 Oscillation of a uniformly wound coil

For a light nonmagnetic rod of length $l$ wound uniformly with $N$ thin turns of mean radius $a$, the magnetic dipole moment is $m_B=NI\pi a^2$. Wire of cross-sectional area $A_w$ and density $\rho$ has mass approximately $M=2\pi aN\rho A_w$. The transverse central moment of inertia is $J=M(l^2/12+a^2/2)$ in the thin-winding approximation. Consequently the oscillation period in a transverse magnetic field does not depend on the number of turns: wire inertia and magnetic moment both scale with $N$.