# A magnetic dipole is under the influence of two magnetic fields.

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, B1 = 1.2 × 10−2 T

Magnitude of the other magnetic field = B2

Angle between the two fields, θ = 60°

At stable equilibrium, the angle between the dipole and field B1θ1 = 15°

Angle between the dipole and field B2θ2 = θ − θ= 60° − 15° = 45°

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

Torque due to field B1 = Torque due to field B2

MB1 sinθ1 = MB2 sinθ2

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.