Question

A circular loop of radius $r$ is bent along a diameter and given a shape as shown in the figure. One of the semicircles lies in $xz-$plane and the other one in $yz-$plane, with their centres at origin. The same current $I$ flows through each of the semicircles as shown. The net magnetic field at the origin is,

• A. $\frac{{\mu }_{o}I}{4r}(\hat{i}+\hat{j})$
• B. $\frac{{\mu }_{o}I}{4r}(-\hat{i}-\hat{j})$
• C. $\frac{{\mu }_{o}I}{4r}(-\hat{i}+\hat{j})$
• D. $\frac{{\mu }_{o}I}{4r}(-\hat{i}-\hat{j})$

Answer 1

The correct option is C $\frac{{\mu }_{o}I}{4r}(-\hat{i}+\hat{j})$

The magnetic field due to current carrying circular arc at its centre is given by,
$$B=\frac{{\mu }_{o}I}{4\pi r^2}l$$
For both the given loops, length of the arc, $l=\pi r$

$\Rightarrow B=\frac{{\mu }_{o}I}{4\pi r^2}\pi r=\frac{{\mu }_{o}I}{4r}$

For the loop in $xz-$plane,

${\overrightarrow{B}}_{xz}=\frac{{\mu }_{o}I}{4r}\left(\hat{j}\right)$

For the loop in $yz-$plane,

${\overrightarrow{B}}_{yz}=\frac{{\mu }_{o}I}{4r}(-\hat{i})$

So, the net magnetic field at the origin is,

${\overrightarrow{B}}_{net}={\overrightarrow{B}}_{xz}+{\overrightarrow{B}}_{yz}=\frac{{\mu }_{o}I}{4r}(-\hat{i}+\hat{j})$

Hence, option $\left(c\right)$ is the correct answer.