Question

The wire loop $PQRSP$ formed by joining two semicircular wires of radii $R_1$ and $R_2$ carries a current $I$ as shown in figure. If the magnitude of the magnetic induction at centre $C$ is $\frac{{\mu }_{0}I}{x}\left[\frac{1}{R_1}\frac{I}{R_2}\right]$. find x.

Answer 1

The effect of current in $PQ$ and $RS$ for producing magnetic field at centre is zero.
Magnetic field due to current in semicircular arc $QAR=\frac{1}{2}\left[\frac{{\mu }_0}{2}\frac{I}{R_1}\right]$
exited towards the reader perpendicular to the plane of paper.
Magnetic field due to current in semicircular arc $SBP=\frac{1}{2}\left[\frac{{\mu }_0}{2}\frac{I}{R_2}\right]$
directed away from the reader perpendicular to the plane of paper.
$\therefore$ Net magnetic field $=\frac{1}{2}\left[\frac{{\mu }_0}{2}\frac{I}{R_1}\right]-\frac{1}{2}\left[\frac{{\mu }_0}{2}\frac{I}{R_2}\right]$
(directed towards the reader perpendicular to plane of paper)
$=\frac{{\mu }_{0}I}{4}\left[\frac{1}{R_1}\frac{I}{R_2}\right]$