**Question:**

A magnetic dipole is under the influence of two magnetic fields. The angle between the field directions is 60º, and one of the fields has a magnitude of 1.2 × 10−2 T. If the dipole comes to stable equilibrium at an angle of 15º with this field, what is the magnitude of the other field?

**Solution:**

Magnitude of one of the magnetic fields, *B*1 = 1.2 × 10−2 T

Magnitude of the other magnetic field = *B*2

Angle between the two fields, *θ* = 60°

At stable equilibrium, the angle between the dipole and field *B*1, *θ*1 = 15°

Angle between the dipole and field *B*2, *θ*2 = *θ* − *θ*1 = 60° − 15° = 45°

At rotational equilibrium, the torques between both the fields must balance each other.

∴Torque due to field *B*1 = Torque due to field *B*2

*MB*1 sin*θ*1 = *MB*2 sin*θ*2

Where,

*M* = Magnetic moment of the dipole

$\therefore B_{2}=\frac{B_{1} \sin \theta_{1}}{\sin \theta_{2}}$

$=\frac{1.2 \times 10^{-2} \times \sin 15^{\circ}}{\sin 45^{\circ}}=4.39 \times 10^{-3} \mathrm{~T}$

Hence, the magnitude of the other magnetic field is 4.39 × 10−3 T.