Magnetic Potential Energy of a Dipole

Section 10.8 established a direct structural analogy between a magnetic dipole's torque, $\vec{\tau}=\vec{P}\times\vec{B}$, and an electric dipole's torque, $\vec{\tau}=\vec{p}\times\vec{E}$ (from Class XI). That same analogy extends naturally to POTENTIAL ENERGY. A freely-suspended magnetic dipole, like a freely-suspended electric dipole, possesses potential energy purely on account of its ORIENTATION within the external field -- work must be done to rotate it away from its preferred (lowest-energy) alignment, and that work is stored as potential energy which can, in principle, be released again as the dipole is allowed to swing back.

Recall that, for an electric dipole $\vec{p}$ in an external field $\vec{E}$, the potential energy due to orientation is

(Class XII, Chapter 8, Electrostatics). By direct analogy, the magnetic potential energy of a magnetic dipole $\vec{P}$ in an external magnetic field $\vec{B}$ is

where $\theta$ is the angle between $\vec{P}$ and $\vec{B}$. …