What's the magnetic moment induced by an external magnetic field on a ferromagnetic material?

First you need to find the magnetization $M$, this is equal to $chi_mH$ where $H=frac{B}{mu}$. Then the magnetic moment is just $iiint{MdV}$, over the volume of the rod.

In Gaussian units, ${bf B=H}+4pi{bf M}$. If it is long and thin, $bf H$ is negligible, and ${bf M =B}/4pi$ for most of the length. The magnetic moment is $bf M$ times the volume. When $bf B$ is removed, $bf M$ will reduce to smaller value, depending on the material.