A wire of length $L$ is bent in the form of a circular coil and current '$i$' is passed through it. This coil is kept in magnetic field. The torque acting on the coil will be maximum, when the number of turns is ___

The torque acting on a circular coil of $N$ turns in a magnetic field is given by $\tau = N i A B \sin \theta$.

For maximum torque, $\sin \theta = 1$, giving $\tau = N i A B$.

Let $r$ be the radius of the coil. The total length of the wire is $L = N(2 \pi r)$, which gives $r = \dfrac{L}{2 \pi N}$.

The area of the coil is $A = \pi r^2 = \pi \left( \dfrac{L}{2 \pi N} \right)^2 = \dfrac{L^2}{4 \pi N^2}$.

Substituting the area into the torque expression, we get $\tau = N i \left( \dfrac{L^2}{4 \pi N^2} \right) B = \dfrac{i L^2 B}{4 \pi N}$.

For the torque to be maximum, the number of turns $N$ must be minimum.

Since $N$ is a positive integer, the minimum possible value is $N = 1$.

Answer: $1$