Potential Energy of Magnetic Dipole in B

Minimum energy (stable equilibrium) at θ = 0; maximum (unstable) at θ = π. Change in PE between orientations equals work done against the torque.

Derivation

Work done by external agent rotating the dipole from angle $\theta_1$ to $\theta_2$:

$$dW = \tau_{\text{ext}}\,d\theta = mB\sin\theta\,d\theta$$ $$W = \int_{\theta_1}^{\theta_2} mB\sin\theta\,d\theta = mB[-\cos\theta]_{\theta_1}^{\theta_2}$$

Taking the reference position as $\theta = 90°$ (zero PE):

$$U(\theta) = -mB\cos\theta$$ $$\boxed{U = -\vec{m} \cdot \vec{B}}$$

Equilibria

Work Done in Rotation

To rotate dipole from $\theta_1$ to $\theta_2$:

$$W = U(\theta_2) - U(\theta_1) = mB(\cos\theta_1 - \cos\theta_2)$$