Question

The magnetic induction at the point $O$, if the wire carries a current $i$, is

• A. $\frac{\mu_{0} i}{2 R}$
• B. $\frac{\mu_{0} i}{2 \pi R}$
• C. $\frac{\mu_{0} i\left(\pi^{2}+4\right)^{1 / 2}}{4 \pi R}$
• D. $\frac{\mu_{0} i\left(\pi^{2}+4\right)}{4 \pi R}$

Answer 1

Answer: (C)

Due to straight wires $\vec{B_{1}}$ at $O$
$B_{1} =\frac{\mu_{0} i}{4 \pi R}(-\hat{j})+\frac{\mu_{0} i}{4 \pi R}(-\hat{j}) $
$=-\frac{\mu_{0} i}{2 \pi R} \hat{j}$
Due to semicircle
$\vec{B_{2}}=-\frac{\mu_{0} i}{4 R} \hat{k}$
Net $\vec{B}=\vec{B_{1}}+\vec{B_{2}}$
$=\sqrt{\left(\frac{\mu_{0} i}{2 \pi R}\right)^{2}+\left(\frac{\mu_{0} i}{4 R}\right)^{2}}$
$=\frac{\mu_{0} i}{2 R} \sqrt{\frac{1}{\pi^{2}}+\frac{1}{4}}$
$=\frac{\mu_{0} i}{2 R(2 \pi)} \sqrt{\pi^{2}+4}$