(b) A magnetised needle in a uniform magnetic field experiences a torque but no net force. An iron nail near a bar magnet, however, experiences a force of attraction in addition to a torque. Why ?

Explanation

(b) (i) A magnetised needle placed in uniform magnetic field :

Consider a magnetic needle with magnetic dipole moment `overset(to) (m) = q_m (2 overset(to) (l) )` placed in uniform magnetic field `overset(to) (B)` at an angle `theta`. Here resultant force exerted on the magnetic needle is,
`overset(to) (F) - overset(to) (F_N) + overset(to) (F_S)`
`= q_m B hat(i) + q_m B (- hat(i) )`
`therefore overset(to) (F) = overset(to) (0)" "...(1)`
Amount of resultant torque (or moment of couple) exerted on the magnetic needle is,
`tau= (F_N)` (Perpendicular distance ND)
`therefore tau = (q_m B) (2l sin theta)`
`= (q_m) (2l) (B) (sin theta)`
`therefore tau m B sin theta [ because m=q_m (2l)]`
Considering directions,
`overset(to) (tau) = overset(to) (m) xx overset(to) (B)" "...(2)`
In above situation, direction of `overset(to) (tau)` is perpendicularly inside the plane of figure.
From equations (1) and (2), we can see that in above situation `overset(to) (F) = overset(to) (0)` but `overset(to) (tau) ne overset(to) (0)`.
(iii) An iron nail is a ferromagnetic substance which gets magnetised in the presence of external magnetic field. Its one end acts as induced north pole whereas another end acts as induced south pole. Here, because of non-uniform magnetic field, strengths of magnetic field at the positions of induced north and induced south pole are different and so magnetic forces exerted on them have different magnitudes giving rise to some resultant magnetic force.
Here induced south pole in a nail is closer to north pole of bar magnet as compared to induced north pole. Hence, above resultant force is attractive in nature.
When iron nail is placed in non-uniform magnetic field, magnetic forces exerted on its two induced poles are not colinear and not equal in magnitude. Hence, such a pair of forces produce some torque also.