Q.Which of the following relations is correct for earth's magnetic field (B_H) and frequency of magnetic needle (n)? (A) n^2 ∝ B_H
(B) n ∝ B_H
(C) n ∝ B_H^2
(D) n^2 ∝ 1/B_H

A magnetic dipole disturbed slightly from its stable equilibrium orientation ($\vec m$ parallel to $\vec B$) and released performs angular simple harmonic motion about that equilibrium, with angular frequency $\omega=\sqrt{mB/I}$ and time period $T=2\pi\sqrt{\dfrac{I}{mB}}$, where I is the magnet's moment of inertia about the axis of oscillation. This result follows from combining the restoring torque $\tau=-mB\sin\theta\approx-mB\theta$ (for small $\theta$) with the rotational equation of motion $I\ddot\theta=\tau$, giving exactly the form of linear SHM's equation, $\ddot\theta=-\omega^2\theta$. …