Figure shows a small magnetised needle P placed at a point O. the arrow the direction of its magnetic moment. The other arrows show different positions (and orientations of the magnetic moment) of another identical magnetised needle Q.
a. In which configuration the system is 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 configuration shown ?

Explanation

Potential energy of the configuration arises due to the potential energy of one dipole (say, Q) in the magnetic field due to other (P). Use the result that the field due to P is given by the expression [Eqs. (5.7) and (5,8)]
`B_(P)=-(mu_(0)m_(p))/(4pi r^(3))` (on the normal bisector)
`B_(p)=(mu_(0)2)/(4pi) (m_(p))/(r^(3))` (on the axis)
where `m_(p)` is the magnetic moment of the dipole P.
Equilibrium is stable when `m_Q` is parallel to `B_p` and unstable when it is anti parallel to `B_P`.
For instance for the configuration `Q_(3)`, for which Q is along the perpendicular bisector of the dipole P, the magnetic moment of Q is parallel to the magnetic field at the position 3. Hence Q, is stable. Thus
(a) `PQ_(1) and PQ_(2)`
(b) (i) `PQ_(3), PQ_(6) ("stable") (ii) PQ_(5), PQ_(4) ("unstable")`
(c) `PQ_(6)`