Question

Find the magnitude and direction of the magnetic induction vector $B$.
(a) of an infinite plane carrying a current of linear density $i$; the vector $i$ is the same at all points of the plane;
(b) of two parallel infinite planes carrying currents of linear densities $i$ and $-i$; the vectors $i$ and $-i$ are constant at all points of the corresponding planes.

Answer 1

(a) We apply circulation theorem as shown. The current is vertically upwards in the plane and the magnetic field is horizontal and parallel to the plane.

$\oint \overrightarrow{B}\cdot d\overrightarrow{l}=2Bl={\mu }_{0}il$ or $B=\frac{{\mu }_{0}i}{2}$

(b) Each plane contributes ${\mu }_0\frac{i}{2}$ between the planes and outside the plane that cancel.

Thus

$B=\{\begin{array}{c}{\mu }_{0}i\left(betweentheplane\right)\\ 0\left(outside\right).\end{array}$