2006-50.md

course	course_year	question_number	tags	title	year
Electromagnetism	IB	50	IB 2006 Electromagnetism	2.I.6G	2006

Given that the electric field $\mathbf{E}$ and the current density $\mathbf{j}$ within a conducting medium of uniform conductivity $\sigma$ are related by $j=\sigma \mathbf{E}$, use Maxwell's equations to show that the charge density $\rho$ in the medium obeys the equation

$$\frac{\partial \rho}{\partial t}=-\frac{\sigma}{\epsilon_{0}} \rho .$$

An infinitely long conducting cylinder of uniform conductivity $\sigma$ is set up with a uniform electric charge density $\rho_{0}$ throughout its interior. The region outside the cylinder is a vacuum. Obtain $\rho$ within the cylinder at subsequent times and hence obtain $\mathbf{E}$ and $\mathbf{j}$ within the cylinder as functions of time and radius. Calculate the value of $\mathbf{E}$ outside the cylinder.