Question

A small circular coil with a mass of m consists of N turns of fine wire and carries a current of I. The coil is located in a uniform magnetic field of B with the coil axis originally parallel to the field direction.
What is the period of small angle oscillations of the coil axis around its equilibrium position, assuming no friction and no mechanical force?

Answer 1

Magnetic moment of coil:
$M=NIA=NI\pi R^2$
Let the coil displaced by small angle $\theta$, the torque on the coil: $\tau =MBsin\theta$
$\Rightarrow -I\alpha =MB\theta$ (for small angle $sin\theta \approx \theta )$
$\Rightarrow -\frac{1}{2}m{R}^2\alpha =NI\pi R^{2}B\theta$
$\Rightarrow \alpha =-\left(\frac{2NI\pi B}{m}\right)\theta$
Comparing with $\alpha ={\omega }^2\theta$ we get $\omega =\sqrt{\frac{2NI\pi B}{m}}$
Time period:
$T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{m}{2NI\pi B}}$