JEE Advanced · 2012 · 08 Apr 2012 · Paper 2 · Magnetic Effects of Current

The magnetic moment is given by $\vec{M}=I\vec{A}$, where $A$ is the area enclosed by the loop. The loop consists of a square of side $a$ (area $a^2$) and a quarter circle of radius $a$ (area $\frac{\pi a^2}{4}$). The total area is $a^2+\frac{\pi a^2}{4}=\left(1+\frac{\pi }{4}\right)a^2$. However, examining the figure more carefully, the loop encloses a square plus a semicircle, giving area $a^2+\frac{\pi a^2}{2}=\left(\frac{\pi }{2}+1\right)a^2$. Therefore, $\vec{M}=I\left(\frac{\pi }{2}+1\right)a^2\hat{k}$.