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Potential Energy of Magnetic Dipole

Potential Energy of Magnetic Dipole :- The energy stored in a magnetic dipole due to its special position in a magnetic field is called the potential energy of magnetic dipole or the work done in bringing a magnetic dipole from a position perpendicular to the magnetic field to any other angular position is called the potential energy of magnetic dipole.

When a bar magnet of magnetic dipole moment M is placed in a uniform magnetic field B at an angle θ (between M and B), then a restoring torque τ = MB sinθ acts on it which tries to bring the bar magnet in stable equilibrium position ( to bring M and B in same direction). If an attempt is made to rotate the bar magnet from the stable equilibrium position to any other position, then work has to be done against this torque. This work gets stored in the magnetic dipole in the form of its potential energy.

Now the work done in rotating the magnetic dipole through a small angle dθ against the restoring torque τ is given by :-

$\displaystyle dW=\tau d\theta =MB\sin \text{ }\!\!\theta\!\!\text{ }d\theta$

The total work done in rotating the dipole from θ = θ₁ to θ = θ₂,

$\displaystyle W=\int\limits_{{{\theta }_{1}}}^{{{\theta }_{2}}}{MB\sin \text{ }\!\!\theta\!\!\text{ }d\theta =MB\left[ -\cos \text{ }\!\!\theta\!\!\text{ } \right]}_{{{\theta }_{1}}}^{{{\theta }_{2}}}$