 Uniform magnetic field in an off-axis cylindrical cavity

Subtract a cylinder of current density $\mathbf{J}$ occupying the cavity from a complete current-carrying cylinder. Inside each cylinder, $\mathbf{B}={\mu }_0\mathbf{J}\times \mathbf{r}/2$. The difference is independent of observation point and equals ${\mu }_0\mathbf{J}\times \mathbf{b}/2$, where $\mathbf{b}$ is the offset of the cavity axis. This assumes parallel axes, uniform current density in the remaining metal, and negligible end effects.