Question

The current flowing in the loop is shown in the figure, then the magnetic field at point $O($centre$)$ is :

• A. $\frac{3{\mu }_{0}i}{8r}$
• B. $\frac{5{\mu }_{0}i}{8r}$
• C. $\frac{{\mu }_{0}i}{8r}$
• D. none of these

Answer 1

Answer: (C)

$\frac{{\mu }_{0}i}{8r}$

Magnetic field due to semicircular outer loop at center of the loop,
$B_{outer}=\frac{1}{2}\frac{{\mu }_0\frac{i}{2}}{2r}$
$=\frac{{\mu }_{0}i}{8r}$
where $2r$ is the radius of the outer loop. A factor of $\frac{1}{2}$ is multiplied since $\frac{{\mu }_0\frac{i}{2}}{2r}$ is the field due to circular outer loop.
Magnetic field due to the inner circular loop has two halves carrying current in the opposite direction
$B_{inner}=0$ since magnetic field due to one half cancels that due to other half.