Question #43844

A thin n-type semiconductor ring of radius $R$, radial width $w$, and axial thickness $t$ ($w, t \ll R$) lies in the $xy$-plane, centered at the origin. A magnetic dipole of moment $\vec{m} = m \hat{i}$ is fixed at the origin. A total steady current $I$ is driven around the ring in the $+\hat{\phi}$ direction. The charge carriers have mobility $\mu$ and charge magnitude $e$.

Due to the spatially varying magnetic field, a Hall electric field $E_z$ develops in the axial direction. To satisfy steady-state boundary conditions ($\nabla \times \vec{E} = 0$), the azimuthal current density $J_\phi$ must vary across the thickness of the ring.

Assuming the variation in drift velocity is small compared to its average value, find the magnitude of the difference in azimuthal current density, $\Delta J = |J_\phi(z=t/2) - J_\phi(z=-t/2)|$, at the angular position $\phi = \pi/2$ (on the $y$-axis).