Question

A uniform current of density $j$ flows inside an infinite plate of thickness $2d$ parallel to its surface. Find the magnetic induction induced by this current as a function of the distance $x$ from the median plane of the plate. The magnetic permeability is assumed to be equal to unity both inside and outside the plate.

Answer 1

We assume that the current flows perpendicular to the plane of the paper, by circulation theorem,
$2Bdl={\mu }_0\left(2xdl\right)j$
or, $B={\mu }_{0}xj$, $|x|\le d$
Outside, $2Bdl={\mu }_0(2d\ast dl)j$
or, $B={\mu }_{0}dj$, $|x|\ge d$