Question #43015

A long, straight, thick cylindrical conductor of radius $R$ carries a steady DC current. The current density inside this main conductor varies radially as $J(r) = J_0 \left(1 - \frac{r}{R}\right)$, where $r$ is the distance from the central axis and $J_0$ is a positive constant. At a certain junction, this main conductor splits into $4$ identical, long, straight cylindrical branches, each of radius $\frac{R}{2}$. After a sufficient distance from the junction, the current in each branch redistributes such that its current density becomes uniform across its cross-section. Let $B_1$ be the magnitude of the magnetic field inside the main conductor at a radial distance $r = \frac{R}{2}$ from its axis. Let $B_2$ be the magnitude of the magnetic field inside one of the branches at a radial distance $r' = \frac{R}{4}$ from its own central axis. Find the ratio $\frac{B_1}{B_2}$.