Question

A large cylindrical solid conductor of radius $R$ is carrying current along the axis such that magnetic field inside conductor varies $B = B_0r^2$ where $r$ is radial distance from axis of cylinder. Determine the current density at radial distance of $r$ (given $r < R$ and $B_0$ is a constant)

Answer 1

$J(r) = \frac{3 B_0 r}{\mu_0}$

Answer 2

1. Apply Ampere's Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$.
2. For a loop of radius $r$: $(B_0r^2)(2\pi r) = \mu_0 I_{enc} \implies I_{enc} = \frac{2\pi B_0r^3}{\mu_0}$.
3. Current density $J(r)$ relates to $I_{enc}$ by $\frac{dI_{enc}}{dr} = J(r) 2\pi r$.
4. Differentiate $I_{enc}$: $\frac{dI_{enc}}{dr} = \frac{6\pi B_0r^2}{\mu_0}$.
5. Equate and solve: $J(r) 2\pi r = \frac{6\pi B_0r^2}{\mu_0} \implies J(r) = \frac{3 B_0 r}{\mu_0}$.