Question #42841

A non-uniform, spherically symmetric charge distribution has a volume charge density $\rho(r) = \frac{\alpha}{r}$ for $r \le a$, where $r$ is the spherical radial distance from the origin and $\alpha$ is a positive constant. In the same region, there exists a magnetic field given in cylindrical coordinates $(s, \phi, z)$ as $\vec{B} = \frac{B_0 R}{s} \hat{s}$, where $B_0$ and $R$ are positive constants. A thin, straight conducting rod of length $L$ is placed parallel to the $z$-axis at a cylindrical radial distance $s = R$. The ends of the rod are at $z = 0$ and $z = L$. The rod rotates about the $z$-axis with a constant angular velocity $\vec{\omega} = \omega \hat{k}$. Assuming the rod is entirely within the charge distribution ($R^2 + L^2 < a^2$), what is the steady-state potential difference $V(L) - V(0)$ between the ends of the rod?