## Potential Energy of a Bar Magnet in Uniform Magnetic Field **
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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.
$U = W _{ \theta }- W _{\frac{\pi}{2}}=- MB \cos \theta=-\overrightarrow{ M } \cdot \overrightarrow{ B }$
**Also Read:**Biot Savart’s Law

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Where thita 1 taken

Is angle take from equlibrium Or perpendicular I cont

It is very comfortable for students to learn

Expression for period of a magnetic dipole kept in a uniform magnetic field