Question

A short bar magnet of magnetic moment \( 10^4 \, \text{J T}^{-1} \) is free to rotate in a horizontal plane. The work done in rotating the magnet slowly from the direction parallel to a horizontal magnetic field of \( 4 \times 10^{-5} \, \text{T} \) to a direction \( 60^\circ \) to the direction of the field is

• A. 0.2 J
• B. 2.6 J
• C. 0.4 J
• D. 6.2 J

Hint:

- Potential energy of a magnetic dipole (moment \(M\)) in a uniform magnetic field \(B\) is \( U = -\vec{M} \cdot \vec{B} = -MB\cos\theta \). - Work done in rotating the dipole slowly from angle \( \theta_1 \) to \( \theta_2 \) is \( W = \Delta U = U_2 - U_1 = -MB(\cos\theta_2 - \cos\theta_1) \). - Parallel to field: \( \theta = 0^\circ \), \( \cos 0^\circ = 1 \). - Perpendicular to field: \( \theta = 90^\circ \), \( \cos 90^\circ = 0 \). - Anti-parallel to field: \( \theta = 180^\circ \), \( \cos 180^\circ = -1 \). - \( \cos 60^\circ = 1/2 \).

Answer 1

The Correct Option is A

Magnetic moment \( M = 10^4 \, \text{J T}^{-1} \).
Magnetic field strength \( B = 4 \times 10^{-5} \, \text{T} \).
The potential energy of a bar magnet in a magnetic field is \( U = -MB\cos\theta \), where \( \theta \) is the angle between the magnetic moment \( \vec{M} \) and the magnetic field \( \vec{B} \).
The magnet is rotated from a direction parallel to the field to a direction \( 60^\circ \) to the field.
Initial angle \( \theta_1 = 0^\circ \) (parallel to the field).
Final angle \( \theta_2 = 60^\circ \).
Initial potential energy \( U_1 = -MB\cos(0^\circ) = -MB(1) = -MB \).
Final potential energy \( U_2 = -MB\cos(60^\circ) = -MB\left(\frac{1}{2}\right) = -\frac{1}{2}MB \).
The work done in rotating the magnet slowly is equal to the change in its potential energy: \( W = U_2 - U_1 = \left(-\frac{1}{2}MB\right) - (-MB) = -\frac{1}{2}MB + MB = \frac{1}{2}MB \).
Substitute the values of M and B: \[ W = \frac{1}{2} (10^4 \, \text{J T}^{-1}) (4 \times 10^{-5} \, \text{T}) \] \[ W = \frac{1}{2} \times 10^4 \times 4 \times 10^{-5} \, \text{J} \] \[ W = \frac{1}{2} \times 4 \times 10^{4-5} \, \text{J} = 2 \times 10^{-1} \, \text{J} \] \[ W = 0.
2 \, \text{J} \] This matches option (1).