Question

A square conducting loop of side length $L$ carries a current $I$ . The magnetic field at the centre of the loop is

• A. independent of $L$
• B. proportional to $L^2$
• C. inversely proportional to $L$
• D. linearly proportional to $L$

Answer 1

Answer: (C)

Field due to wire $AB$ at the center,
$B_{AB}=\frac{{\mu }_{0}I}{4\pi \left(\frac{L}{2}\right)}\left(\sin 45^{\circ }+\sin 45^{\circ }\right)$
$B_{center}=4B_{AB}$
$B_{\text{center }}=4\frac{{\mu }_{0}I}{4\pi \left(\frac{L}{2}\right)}\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\right)$
$B_{center}=2\sqrt{2}\frac{{\mu }_{0}I}{\pi L}$
$B_{center}\propto \frac{1}{L}$