In this article, we are going to derive an expression for the potential energy of a magnetic dipole in a uniform magnetic field. So let’s get started…
Derivation of the potential energy of a magnetic dipole in a uniform magnetic field class 12
Look at the following figure.
When a magnetic dipole is placed in a uniform magnetic field at angle with it, then it experiences a torque given as:
This torque tends the magnetic dipole to align with the direction of magnetic field .
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This torque acts on the magnetic dipole in the direction of decreasing . If the dipole is rotated against the action of this torque then some work has to be done. This work is stored as the potential energy of the dipole. The work done in turning the magnetic dipole through small-angle is-
If the dipole is rotated from initial position to final position , then the total work done will be –
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This work done is stored as the potential energy of the dipole.
The potential energy of a dipole is zero when magnetic moment is perpendicular to the magnetic field i.e. . So the potential energy of the dipole for any orientation can be obtained by putting in the above formula.
- When , , Thus, the potential energy of a dipole is minimum when is parallel to . In this state, magnetic dipole is said to be in stable equilibrium.
- When ,
- When , , Thus, the potential enegry of a dipole is maximum when is antiparallel to . In this state, the magnetic dipole is said to be in unstable equilibrium.
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