Potential energy of a bar magnet in a uniform magnetic field
When a bar magnet (magnetic dipole) of dipole moment m is held at an angle θ with the direction of a uniform magnetic field , as shown in Figure 3.20 the magnitude of the torque acting on the dipole is
If the dipole is rotated through a very small angular displacement dθ against the torque τB at constant angular velocity, then the work done by external torque ( ) for this small angular displacement is given by
The bar magnet has to be moved at constant angular velocity, which implies that
Total work done in rotating the dipole from θʹ to θ is
This work done is stored as potential energy in bar magnet at an angle θ when it is rotated from θʹ to θ and it can be written as
In fact, the equation (3.26) gives the difference in potential energy between the angular positions θʹ and θ. We can choose the reference point θʹ = 90o, so that second term in the equation becomes zero and the equation (3.26) can be written as
The potential energy stored in a bar magnet in a uniform magnetic field is given by
(i) If θ = 0º, then
U = − pm B (cos0 º ) = − pm B
(ii) If θ = 180 º, then
U= − pm B (cos180 º) = pm B
We can infer from the above two results, the potential energy of the bar magnet is minimum when it is aligned along the external magnetic field and maximum when the bar magnet is aligned anti-parallel to external magnetic field.
Consider a magnetic dipole which on switching ON external magnetic field orient only in two possible ways i.e., one along the direction of the magnetic field (parallel to the field) and another anti-parallel to magnetic field. Compute the energy for the possible orientation. Sketch the graph.
Let m be the dipole and before switching ON the external magnetic field, there is no orientation. Therefore, the energy U = 0.
As soon as external magnetic field is switched ON, the magnetic dipole orient parallel (θ = 0º) to the magnetic field with energy,
Uparallel = Uminimum = − pm B cos0
Uparallel = − pm B
since cos 0º = 1
Otherwise, the magnetic dipole orients anti-parallel (θ = 180º) to the magnetic field with energy,
Uanti− parallel = U maximum =− pmB cos180
⇒U anti−parallel = pmB
since cos 180º = -1