Question

A wire carrying current $I$ has the shape as shown in the figure.


Linear parts of the wire are very long and parallel to $X$-axis while semi-circular portion of radius $R$ is lying in $Y-Z$ plane. Magnetic field at point $O$ is

• A. $\overrightarrow{B}=-\frac{{\mu }_0}{4\pi }\frac{I}{R}\left(\pi \hat{i}+2\hat{k}\right)$
• B. $\overrightarrow{B}=\frac{{\mu }_0}{4\pi }\frac{I}{R}\left(\pi \hat{i}-2\hat{k}\right)$
• C. $\overrightarrow{B}=\frac{{\mu }_0}{4\pi }\frac{I}{R}\left(\pi \hat{i}+2\hat{k}\right)$
• D. $\overrightarrow{B}=-\frac{{\mu }_0}{4\pi }\frac{I}{R}\left(\pi \hat{i}-2\hat{k}\right)$

Answer 1

The correct option is A $\overrightarrow{B}=-\frac{{\mu }_0}{4\pi }\frac{I}{R}\left(\pi \hat{i}+2\hat{k}\right)$

Given situation is shown in the figure.


Parallel wires $1$ and $3$ are semi-infinite, so magnetic field at $O$ due to them
$\overrightarrow{B_1}=\overrightarrow{B_3}=-\frac{{\mu }_{0}I}{4\pi R}\hat{k}$
Magnetic field ar $O$ due to semi-circular are in $YZ$-plane is given by
$\overrightarrow{B_2}=-\frac{{\mu }_{0}I}{4R}\hat{i}$
Net magnetic field at point $O$ is given by
$\overrightarrow{B}=\overrightarrow{B_1}+\overrightarrow{B_2}+\overrightarrow{B_3}=-\frac{{\mu }_{0}I}{4\pi R}\hat{k}-\frac{{\mu }_{0}I}{4R}\hat{i}-\frac{{\mu }_{0}I}{4\pi R}\hat{k}=-\frac{{\mu }_{0}I}{4\pi R}(\pi \hat{i}+2\hat{k})$