Arrange the following cases in the ascending order of magnitude of potential energies of a magnetic dipole in different uniform magnetic fields. Where \( \theta \) is the angle between magnetic moment \( (\vec{m}) \) and magnetic field \( (\vec{B}) \).
(A) \( B = 6\,\mathrm{T},\ \theta = 180^\circ \)
(B) \( B = 4\,\mathrm{T},\ \theta = 0^\circ \)
(C) \( B = 8\,\mathrm{T},\ \theta = 90^\circ \)
(D) \( B = 8\,\mathrm{T},\ \theta = 0^\circ \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (C), (B), (A), (D)
The potential energy of a magnetic dipole in a uniform field is \( U = -\vec{m}\cdot\vec{B} = -mB\cos\theta \). The same dipole (the same m) is used in all four cases.
(A) \( B = 6\,\mathrm{T},\ \theta = 180^\circ \) — \( U = -m(6)\cos180^\circ = +6m, \quad |U| = 6m \)
(B) \( B = 4\,\mathrm{T},\ \theta = 0^\circ \) — \( U = -m(4)\cos0^\circ = -4m, \quad |U| = 4m \)
(C) \( B = 8\,\mathrm{T},\ \theta = 90^\circ \) — \( U = -m(8)\cos90^\circ = 0, \quad |U| = 0 \)
(D) \( B = 8\,\mathrm{T},\ \theta = 0^\circ \) — \( U = -m(8)\cos0^\circ = -8m, \quad |U| = 8m \)
Ascending order of magnitude: \( 0 \lt 4m \lt 6m \lt 8m \), that is (C), (B), (A), (D).