Question

A current carrying loop consists of 3 identical quarter circles of radius R, lying in the positive quadrants of the x-y, y-z and z-x planes with their centres at the origin, joined together. Find the direction and magnitude of B at the origin.

Answer 1

From Biot-Savart law we find the relation of magnetic field at centre of the current carrying coil which subtends an angle θ, B = ${\displaystyle \frac{{\mu }_0}{4\pi }\frac{I\theta }{R}}$.

Magnetic field at origin due to the quarter circle lying in x-y plane:

${\displaystyle {\overrightarrow{B}}_1=\frac{{\mu }_0}{4\pi }\frac{I\left(\frac{\pi }{2}\right)}{R}\hat{k}=\frac{{\mu }_0}{4}\frac{I}{2R}\hat{k}}$

Similarly, magnetic field at origin due to the quarter circle lying in the y-z plane:

${\displaystyle {\overrightarrow{B}}_2=\left(\frac{{\mu }_0}{4}\right)\frac{I}{2R}\hat{i}}$

Similarly, magnetic field at origin due to the quarter circle lying in the z-x plane:

${\displaystyle {\overrightarrow{B}}_3=\left(\frac{{\mu }_0}{4}\right)\frac{I}{2R}\hat{j}}$

Now, vector sum of magnetic field at origin due to each quarter is given by

${\displaystyle {\overrightarrow{B}}_{\text{net}}=\frac{1}{4}\left(\frac{{\mu }_{0}I}{2R}\right)(\hat{i}+\hat{j}+\hat{k})}$.