Question

A long wire is bent into the shape $PQRST$ as shown in the following Figure with $QRS$ being a semicircle with centre $O$ and radius r metre. A current of $I$ ampere flows through it in the direction $P\to Q\to R\to S\to T$. Then, the magnetic induction at the point $O$ of the figure in vacuum is

• A. ${\mu }_{0}i[\frac{1}{2\pi r}+\frac{1}{4r}]$
• B. ${\mu }_{0}i[\frac{1}{2\pi r}-\frac{1}{4r}]$
• C. $\frac{{\mu }_{0}i}{4r}$
• D. $\frac{{\mu }_{0}i}{\pi r}$

Answer 1

The correct option is A ${\mu }_{0}i[\frac{1}{2\pi r}+\frac{1}{4r}]$

Magnetic field at $O$ due to the semi-infinite segment $PQ$ is half of that due to an infinite wire.
$\overrightarrow{B_{PQ}}=\frac{1}{2}\frac{{\mu }_{0}i}{2\pi r}(-\hat{k})$
Similarly,
$\overrightarrow{B_{ST}}=\frac{1}{2}\frac{{\mu }_{0}i}{2\pi r}(-\hat{k})$
$\overrightarrow{B_{QRS}}=\frac{1}{2}\frac{{\mu }_{0}i}{4r}$ where $\frac{{\mu }_{0}i}{2r}(\hat{k}$ is the magnetic field due to a circular wire at the center of the circle.
The magnetic field due to semi circular portion is half of that due to a circular wire.
$\therefore \overrightarrow{B_{total}}=\overrightarrow{B_{PQ}}+\overrightarrow{B_{ST}}+\overrightarrow{B_{QRS}}={\mu }_{0}i[\frac{1}{2\pi r}+\frac{1}{4r}]$