Potential Energy of magnetic dipole in a magnetic field is defined as the amount of work done in rotating the dipole from zero potential energy position to any desired position.

A current loop does not experience a net force in a magnetic field. It, however, experiences a torque. This is very similar to the behavior of an electric dipole in an electric field. A current loop, therefore, behaves like a magnetic dipole.

Potential Energy of a Bar Magnet in Uniform Magnetic Field

When a bar magnet of dipole moment M is kept in a uniform magnetic field B it experiences a torque $\tau=M B \sin \theta$ which tries to align it parallel to the direction of the field.

If the magnet is to be rotated against this torque work has to be done.

The work done in rotating dipole by small-angle d$\theta $ is $d W =\tau d \theta$

Total work done in rotating it from angle $\theta_{1}$ to $\theta_{2}$ is

$\mathrm{W}=\int \mathrm{dW}=\int_{\theta_{1}}^{\theta_{2}} \tau \mathrm{d} \theta=\mathrm{MB} \int_{\theta_{1}}^{\theta_{2}} \sin \theta \mathrm{d} \theta$

$=\operatorname{MB}\left(\cos \theta_{1}-\cos \theta_{2}\right)$

This work done in rotating the magnet is stored inside the magnet as its potential energy.

So U = MB $\left(\cos \theta_{1}-\cos \theta_{2}\right)$

The potential energy of a bar magnet in a magnetic field is defined as work done in rotating it from a direction perpendicular to the field to any given direction.

$U = W _{ \theta }- W _{\frac{\pi}{2}}=- MB \cos \theta=-\overrightarrow{ M } \cdot \overrightarrow{ B }$



Also Read:

Biot Savart's Law    

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