Question

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

• A. $\frac{{\mu }_{0}i}{2R}$
• 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 $\overrightarrow{B_1}$ 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
$\overrightarrow{B_2}=-\frac{{\mu }_{0}i}{4R}\hat{k}$
Net $\vec{B}=\overrightarrow{B_1}+\overrightarrow{B_2}$
$=\sqrt{{\left(\frac{{\mu }_{0}i}{2\pi R}\right)}^2+{\left(\frac{{\mu }_{0}i}{4R}\right)}^2}$
$=\frac{{\mu }_{0}i}{2R}\sqrt{\frac{1}{{\pi }^2}+\frac{1}{4}}$
$=\frac{{\mu }_{0}i}{2R(2\pi )}\sqrt{{\pi }^2+4}$