Potential Energy of a Magnetic Dipole
As τ = MB sinθ
If the dipole is rotated against the action of this torque, work has to be done. This work is stored as potential energy of the dipole.
The work done in turning the dipole through a small angle dθ is
dW = τdθ = MB sinθdθ
If the dipole is rotated from an initial position θ = θ1 to the final position θ = θ2, then the total work done will be
W = (\int d W=\int{\theta{1}}^{\theta{2}} M B \sin \theta d \theta=-M B[-\cos \theta]{\theta{1}}^{\theta{2}})
= -MB(cosθ2 – cosθ1)
This work done is stored as the potential energy U of the dipole.
∴ U = -MB(cosθ2 – cosθ1)
The potential energy of the dipole is zero when
(\vec{M} \perp \vec{B}). So the potential energy of the dipole in any orientation θ can be obtained by putting θ1 = 90° and θ2 = θ in the above equation.
∴ U = -MB(cosθ – cos 90°)
or U = -MBcosθ = -M. B
