Question

A direct current $I$ flows along a lengthy straight wire. From the point $O$ (Fig.) the current spreads radially all over an infinite conducting plane perpendicular to the wire. Find the magnetic induction at all points of space.

Answer 1

It is easy to convince oneself that both in the regions. $1$ and $2$ there can only be a circuital magnetic field (i.e. the component $B_{\phi })$. Any radial field in region $1$ or any $B_Z$ away from the current plane will imply a violation of Gauss' law of magnetostatics, $B_{\phi }$ must obviously by symmetrical about the straight wire. Then in $1$,
$B_{\phi }2\pi r={\mu }_{0}I$
or, $B_{\phi }=\frac{{\mu }_0}{2\pi }\frac{I}{r}$
$In2,B_{\phi }\cdot 2\pi r=0$, or $B_{\phi }=0$.