Question

A wire carrying current $I $ has the shape as shown in adjoining figure.
Linear parts of the wire are very long and parallel to . $X$-axis while semicircular portion of radius $R$ is lying in $Y-Z$ plane. Magnetic field at point $O$ is

• A. $\vec{B}=-\frac{\mu_{0} I}{4 \pi} \frac{I}{R}(\pi \hat{i}+2 \hat{k})$
• B. $\vec{B}=\frac{\mu_{0}}{4 \pi} \frac{I}{R}(\pi \hat{i}-2 \hat{k})$
• C. $\vec{B}=\frac{\mu_{0}}{4 \pi} \frac{I}{R}(\pi \hat{i}+2 \hat{k})$
• D. $\vec{B}=-\frac{\mu_{0}}{4 \pi} \frac{I}{R}(\pi \hat{i}-2 \hat{k})$

Answer 1

Answer: (A)

Given situation is shown in the figure.

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