Question

A loop carrying current $I$ lies in the $x-y$ plane as shown in figure. The unit vector $\hat{k}$ is coming out of the plane of the paper. The magnetic moment of the current loop is

• A. $a^{2}I\hat{k}$
• B. $(\frac{\pi }{2}+1)a^{2}I\hat{k}$
• C. $-(\frac{\pi }{2}+1)a^{2}I\hat{k}$
• D. $(2\pi +1)a^{2}I\hat{k}$

Answer 1

The correct option is B $(\frac{\pi }{2}+1)a^{2}I\hat{k}$

Area of the loop: $A=a^2+4\times \frac{\pi (\frac{a}{2}{)}^2}{2}=a^2+\frac{\pi a^2}{2}$

$\overrightarrow{\mu }=IA\hat{k}=I(a^2+\frac{\pi a^2}{2})\hat{k}=I{a}^2(1+\frac{\pi }{2})\hat{k}$