Question

A magnetic dipole is under the influence of two magnetic fields. Theangle between the field directions is 60°, and one of the fields has amagnitude of 1.2 × 10–2 T. If the dipole comes to stable equilibrium atan angle of 15° with this field, what is the magnitude of the otherfield?

Answer 1

Given, the magnitude of the first magnetic field is 1.2× 10 −2  T, the angle between the two fields is 0° and the angle between the dipole and the first magnetic field is 15°.

The angle between the dipole and the other field B 2 is given by the equation,

θ 2 =θ− θ 1

Substituting the values in the above equation, we get:

θ 2 =60°−15° =45°

The formula to calculate the torque due to the first magnetic field is,

T 1 =m B 1 sin θ 1

Here, the magnitude of the first magnetic field is B 1 , the magnetic moment of the dipole is m and the angle of dip is θ 1 .

The formula to calculate the torque due to the second magnetic field is,

T 2 =m B 2 sin θ 2

Here, the magnitude of the second magnetic field is B 2 , the magnetic moment of the dipole is m and the angle of dip is θ 2 .

At rotational equilibrium, the torques between both the fields must be equal. Therefore,

m B 1 sin θ 1 =m B 2 sin θ 2 B 2 = B 1 sin θ 1 sin θ 2

Substituting the values in the above equation, we get:

B 2 = ( 1.2× 10 −2 )sin15° sin45° =4.39× 10 −3  T ≈4.4× 10 −3  T

Thus, the magnitude of the second magnetic field is 4.4× 10 −3  T.