Q.Derive the formula for the time period of simple harmonic motion of a magnetic dipole placed in a uniform magnetic field.

A magnetic dipole displaced from equilibrium in a uniform field oscillates like a torsional pendulum, with time period T = 2π√(I/mB).

Consider a bar magnet of magnetic moment $m$ and moment of inertia $I$, free to rotate about its centre, placed in a uniform field $B$. Let it be displaced by a small angle $\theta$ from alignment with B.

Restoring torque: $\tau = -mB\sin\theta$ (negative sign since the torque acts to restore alignment).

Equation of motion: By Newton's law for rotation, $\tau = I\alpha = I\dfrac{d^2\theta}{dt^2}$

$I\dfrac{d^2\theta}{dt^2} = -mB\sin\theta$

For small angles, $\sin\theta \approx \theta$:

$I\dfrac{d^2\theta}{dt^2} = -mB\theta$

$\dfrac{d^2\theta}{dt^2} = -\dfrac{mB}{I}\theta$

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