Potential Energy of Magnetic Dipole in Magnetic Field || Magnetism Class 12 Physics Notes

Potential Energy of magnetic dipole in 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 direction of field.

If 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

$W =\int d W =\int_{\theta_{1}}^{\theta_{2}} \tau d \theta= MB \int_{\theta_{1}}^{\theta_{2}} \sin \theta d \theta= MB \left(\cos \theta_{1}-\cos \theta_{2}\right)$

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 field to any given direction.

$U = W _{ \theta }- W _{\frac{\pi}{2}}=- MB \cos \theta=-\overrightarrow{ M } \cdot \overrightarrow{ B }$ Biot Savart’s Law