Question

A wire loop $PQRSP$ is constructed by joining two semi-circular coils of radii $r_1$ and $r_2,$ respectively as shown in the figure. If the current flowing in the loop is $i$, then the magnetic induction at the point $O$

• A. $\frac{{\mu }_{0}i}{4}\left[\frac{1}{r_1}-\frac{1}{r_2}\right]$
• B. $\frac{{\mu }_{0}i}{4}\left[\frac{1}{r_1}+\frac{1}{r_2}\right]$
• C. $\frac{{\mu }_{0}i}{2}\left[\frac{1}{r_1}-\frac{1}{r_2}\right]$
• D. $\frac{{\mu }_{0}i}{2}\left[\frac{1}{r_1}+\frac{1}{r_2}\right]$

Answer 1

Answer: (A)

$B_{PQ}=B_{RA}=0$
$B_{QR}=\frac{{\mu }_0}{4}\cdot \frac{i}{r_1}$ (vertically upward)
$B_{SP}=\frac{{\mu }_0}{4}\cdot \frac{i}{r_2}$ (vertically downward)
So, $B_{net}=B_{QR}-B_{SP}$
$=\frac{{\mu }_0}{4}\left(\frac{1}{r_1}-\frac{1}{r_2}\right)i$