A metal wire of length ' $\mathrm{L}$ ' is bent to form a circular coil of number of turns 'n'. The coil is…
A metal wire of length ' $\mathrm{L}$ ' is bent to form a circular coil of number of turns 'n'. The coil is placed in magnetic field 'B' and current is passed through the coil. The maximum torque acting on the coil is
$\frac{\mathrm{BIL}^{2}}{4 \pi}$
$\frac{\mathrm{BIL}^{2}}{2 \pi}$
$\frac{\mathrm{B}^{2} \mathrm{IL}}{2 \pi}$
$\frac{\mathrm{B}^{2} \mathrm{IL}}{4 \pi}$
Solution
Let \(r\) be the radius of the coil and \(n\) be the number of turns formed. Then
\(\begin{aligned}
& L=2 \pi r n \text { or } r=\frac{L}{2 \pi n} \ldots . .(\mathrm{i}) \\
& \text { Maximum torque, } \tau_{\max }=B n I A=B n I \pi r^2 \\
& =B n I \pi \times \frac{L^2}{4 \pi^2 n^2}=\frac{B I L^2}{4 \pi n}
\end{aligned}\)
Torque will be maximum if \(n=1\)
\(\therefore \quad \tau_{\max }=\frac{B I L^2}{4 \pi}\)