Figure shows a small magnetized needle P placed at a point O. The arrow shows the direction of magnetic moment. The other arrows show different positions (and orientations of the magnetic moment) of another identical magnetized needle Q.

(a) In which configuration is the system not in equilibrium?
(b) In which configuration is the system in (i) stable and (ii) unstable equilibrium?
(c) Which configuration corresponds to the lowest potential energy among all the configurations shown?

Explanation

(a) Magnetic field due to magnetic needle P at a point on its axis, `B_P=(mu_0)/(4pi)(2M)/(d^3)` along the axis and
Magnetic field due to magnetic needle P on the normal bisector, `B_P^'=(mu_0)/(4pi)(M)/(d^3)`, along the axis, and antiparallel to the axis.
(a) System is not in equilibrium, when magnetic moment of Q is perpendicular to `B_P^'` i.e. in configuration `PQ_1` and `PQ_2`.
(b) Equilibrium is stable, when magnetic moment of `Q` is parallel to `B_P`, and unstable, when magnetic moment of Q is antiparallel to `B_P`.
`:.` configuration `PQ_5` and `PQ_6` represent stable equilibrium and cofigurations `PQ_3` and `PQ_4` represent unstable equilibrium.
(c) Potential energy of configuration arises due to potential energy of one dipole (say Q) in the magnetic field due to the other (P).
Lowest potential energy is the maximum negative energy, which is for configuration `PQ_6`. (Field due to P is more on axial line, than on the right bisector).