Question

In analogy with an electric dipole, state an expression for the magnetic potential energy of an magnetic dipole in a uniform magnetic field. Discuss the orientations of the dipole moment for the maximum and minimum of potential energy.

Answer 1

Consider an electric dipole of dipole moment  \(\vec P\) placed in a uniform electric field  \(\vec P\) making an angle Φ with \(\vec P\).The torque \(\vec{\tau}\) = \(\vec P\) x \(\vec E\) tends to rotate the dipole and align it with \(\vec E\).

If the dipole was initially parallel to \(\vec E\), its potential energy is minimum. We arbitrarily assign U₀ = 0 to the minimum potential energy for this position. Then, at a position where \(\vec P\) makes an angle θ with \(\vec E\), the potential energy of the dipole is

U_θ = -pE cos θ = - \(\vec P\). \(\vec E\).

A current-carrying coil placed in a magnetic field \(\vec B\) experiences a torque,

\(\vec{\tau}\) \(\vec \mu\) x \(\vec B\)…………. (1)

where \(\vec \mu\) is the magnetic dipole moment of the coil. In analogy with an electric dipole, the potential energy of a magnetic dipole is

U_θ = – μB cos θ = – \(\vec \mu\). \(\vec B\) ……….. (2)

U_θ is also known as orientation energy. A magnetic dipole has its lowest energy ( = – μB cos 0 = – μB) when its dipole moment is lined up with the magnetic field, and has its highest energy ( = – μBcos 180° = + μB) when is directed opposite the field.

Orientations of highest and lowest energy of a magnetic dipole moment in an external magnetic field \(\vec B\)