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# A wire of length $\ell$ is bent to form a circular coil of some turns. A current $i$ flows through the coil and is placed in a uniform magnetic field $B$. The maximum torque on the coil can be

$\begin {array} {1 1} (a)\;\large\frac{iB\ell^2}{4 \pi} & \quad (b)\;\large\frac{iB\ell^2}{2 \pi} \\ (c)\;\large\frac{iB\ell^2}{ \pi} & \quad (d)\;\large\frac{2iB\ell^2}{4 \pi} \end {array}$

Let $n$ be the number of turns and $R$ be the radius of the coil
$\ell = 2\pi Rn$
$R = \large\frac{\ell}{2 \pi n}$
$M = niA = \large\frac{i \ell^ 2} {4 \pi n}$
$M$ is maximum when $n = 1$ so is the torque.
Torque can be found by calculating its cross product with $B$
Ans : (a)