= Magnetic field of a rotating charged solid cylinder
An infinite solid cylinder with radius $R$, prescribed uniform volume <charge density> $\rho$, and <angular velocity> $\omega\hat{\mathbf z}$ has <current density> $\mathbf J=\rho\omega r\hat{\boldsymbol\phi}$. Treat a shell at radius $a$ and thickness $da$ as a <solenoid> carrying surface current $\rho\omega a\,da$. With no applied background <magnetic field> and permeability $\mu$,
$$
\mathbf B(r)=\begin{cases}\dfrac{\mu\rho\omega}{2}(R^2-r^2)\hat{\mathbf z},&r<R,\\0,&r>R.\end{cases}
$$
The same result follows from <Ampère's circuital law>, since $-dB_z/dr=\mu\rho\omega r$ inside.
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