When is the potential energy of a magnetic dipole in a uniform magnetic field maximum? (\( \theta \) is the angle between the magnetic moment \( \vec{m} \) and the field \( \vec{B} \).)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \theta=180^\circ \)
A magnetic dipole of moment \( m \) in a uniform field \( B \) has potential energy
\( U_m=-mB\cos\theta \)
Since \( \cos\theta \) lies between \( -1 \) and \( +1 \), \( U_m \) lies between \( -mB \) and \( +mB \). It is maximum when \( \cos\theta \) has its least value:
\( \cos\theta=-1 \)
\( \Rightarrow\; \theta=180^\circ \)
\( \begin{aligned} U_{\max} &= -mB\cos180^\circ \\ &= +mB \end{aligned} \)
Therefore, the energy is maximum when \( \vec{m} \) points opposite to \( \vec{B} \). This is the most unstable position of the dipole.
NCERT (2026-27):
The Dipole in a Uniform Magnetic Field – “… Equation (5.3) shows that potential energy is minimum (= –mB) at θ = 0° (most stable position) and maximum (= +mB) at θ = 180° (most unstable position).”
\( U_m=-mB\cos\theta=-\mathbf{m}\cdot\mathbf{B} \qquad (5.3) \)