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)

$B=B_1+B_2+B_3+B_4$
$B=4\left[\frac{{\mu }_0}{4\pi }\cdot \frac{I}{a}\left(\sin {\varphi }_1+\sin {\varphi }_2\right)\right]$
$=4\times \frac{{\mu }_0}{4\pi }\frac{I}{(l/2)}\left[\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\right]$
Here, $a=\frac{l}{2},{\varphi }_1={\varphi }_2=45^{\circ }$
$\therefore B=4\times \frac{16}{\sqrt{2}}\left[\frac{{\mu }_0}{4\pi }\frac{I}{l}\right]=8\sqrt{2}\left(\frac{{\mu }_0}{4\pi }\frac{I}{l}\right)$