# A metallic wire is folded to form a square loop of side $a$. It carries a current $i$ and is kept $\perp$ to a uniform magnetic field. If the shape of the loop is changed from a square to a circle without changing its length or current, the amount of work done is

$\begin {array} {1 1} (a)\;i Ba^2 \bigg( \large\frac{\pi}{2} + 1\bigg) & \quad (b)\;i Ba^2 \bigg( \large\frac{\pi}{2} - 1\bigg) \\ (c)\;i Ba^2 \bigg(1+ \large\frac{4}{\pi}\bigg) & \quad (d)\;i Ba^2 \bigg( 1-\large\frac{4}{\pi}\bigg) \end {array}$

work done in changing the shape of the wire is
$W = U_f – U_i = i B(A_i – A_f)$........(1)
the radius of the circular loop is given by
4a = 2$\pi$ r
=> r= 2a/$\pi$
area = $4 a^2/\ / \pi$
substituting in (1) we get the answer
Ans : (d)

edited Sep 16, 2014