Example 5.4 What is the magnitude of the equatorial and axial fields due to a bar magnet of length 5.0 cm at a distance of 50 cm from its mid-point? The magnetic moment of the bar magnet is 0.40 A m2, the same as in Example 5.2. 

Solution From Eq. (5.7)

From Eq. (5.8),

Example 5.5 Figure 5.5 shows a small magnetised needle P placed at a point O. The arrow shows 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 configurations shown?

Figure 5.5

Solution

 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)]:

(on the normal bisector)

(on the axis)

where mP 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 Q₃ 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 Q₃ is stable.

Thus,

(a) PQ₁ and PQ₂

(b) (i) PQ₃, PQ₆ (stable); (ii) PQ₅, PQ₄ (unstable)

(c)   PQ₆