Question

A long wire is bent into the shape PQRST as shown in the figure with QRS being a semicircle with centre O and radius r. A current I flows through it in the direction $P\to Q\to R\to S\to T$. The net magnetic field at point O is

• A.
  ${\mu }_0[\frac{I}{2\pi r}+\frac{I}{4r}]$
• B.
  ${\mu }_0[\frac{I}{2\pi r}+\frac{I}{2r}]$
• C.
  $\frac{{\mu }_{0}I}{4r}$
• D.
  $\frac{{\mu }_{0}I}{4\pi }$

Answer 1

The correct option is A

${\mu }_0[\frac{I}{2\pi r}+\frac{I}{4r}]$

The magnetic field at point O due to the wires PQ, QRS and ST is perpendicularly inwards $(⨂)$.

Magnetic field at O due to (semi-infinite) wire $PQ,B_{PQ}=\frac{{\mu }_{0}I}{4\pi r}⨂$

Magnetic field at O due to (semi-infinite) wire ST,$B_{ST}=\frac{{\mu }_{0}I}{4\pi r}⨂$

Magnetic field at O due to semi-circular wire QRS, $B_{QRS}=\frac{{\mu }_{0}I}{4r}⨂$

Net magnetic field at the centre O, $B=\frac{{\mu }_{0}I}{4\pi r}+\frac{{\mu }_{0}I}{4\pi r}+\frac{{\mu }_{0}I}{4r}={\mu }_0[\frac{I}{2\pi r}+\frac{I}{4r}]$

Hence, the correct answer is option (a).