Question

A square wire of each side $l$ carries a current $I .$ What is the magnetic field at the midpoint of the square?

• A. $4 \sqrt{2} \frac{\mu_{0}}{4 \pi} \frac{I}{l}$
• B. $8 \sqrt{2} \frac{\mu_{0}}{4 \pi} \frac{I}{l}$
• C. $16 \sqrt{2} \frac{\mu_{0}}{4 \pi} \frac{I}{l}$
• D. $32 \sqrt{2} \frac{\mu_{0}}{4 \pi} \frac{I}{l}$

Answer 1

Answer: (B)

$B=B_{1}+B_{2}+B_{3}+B_{4} $
$B=4\left[\frac{\mu_{0}}{4 \pi} \cdot \frac{I}{a}\left(\sin \phi_{1}+\sin \phi_{2}\right)\right]$
$=4 \times \frac{\mu_{0}}{4 \pi} \frac{I}{(l / 2)}\left[\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\right]$
Here, $a=\frac{l}{2}, \phi_{1}=\phi_{2}=45^{\circ}$
$\therefore B=4 \times \frac{16}{\sqrt{2}}\left[\frac{\mu_{0}}{4 \pi} \frac{I}{l}\right]=8 \sqrt{2}\left(\frac{\mu_{0}}{4 \pi} \frac{I}{l}\right)$