Consider a bar magnet NS as shown in Figure.
Let N be the North Pole and S be the south pole of the bar magnet, each of pole strength `q_(m)` and separated by a distance of 2l. The magnetic field at a point C ( lies along the axis of the magnet ) at a distance from the geometrical center O of the bar magnet can be computed by keeping unit north pole `(q_(mc) = 1A m ) ` at C. The force experienced by the unit north pole at C due to pole strength can be computed using Coulomb's law of magnetism as follows :
The force of repulsion between north pole of the bar magnet and unit north pole at point C ( in free space ) is
`vecF_(N) = (mu_(0))/(4 pi) (q_(m))/((r - l)^(2)) hat i ` ....(1)
where r - l is the distance between north pole of the bar magnet and unit north pole at C .
The force of attraction between South Pole of the bar magnet and unit North Pole at point C ( in free space ) is
`vecF_(S) = (mu_(0))/(4 pi) (q_(m))/((r + l)^(2)) hati ` .....(2)
where `r + l ` is the distance between south pole of the bar magnet and unit north pole at C . From equation (1) and (2) , the net force at point C is `vecF = vecF_(N) + vecF_(S) ` . From our definition, this net force is the magnetic field due to magnetic dipole at a point `C (vecF = vecB) `
`vecB = (mu_(0))/(4 pi) (q_(m))/((r - l)^(2)) hati + (- (mu_(0))/(4 pi) (q_(m))/((r + l )^(2))hati)`
`vecB = (mu_(0)q_(m))/(4 pi) (1/((r - l)^(2))- 1/((r + l)^(2)))hati`
`vecB = (mu_(0)2r)/(4 pi) ((q_(m)*(2l))/(r^(2) - l^(2))) hati `
Since, magnitude of magnetic dipole moment is `|vecp_(m)| = p_(m) = q_(m). 2l` the magnetic field at a point C equation (3) can be written as
`vecB _("axial") = mu_(0)/ (4pi) ((2rp_(m))/((r^(2) - l^(2))^(2))) hati` ...... (4)
If the distance between two poles in a bar magnet are small ( looks like short magnet ) compared to the distance between geometrical centre O of bar magnet and the location of point C i.e., r `gt gt ` l then ,
`(r^(2) - l^(2))^(2) approx r^(4) ` ....... (5)
Therefore , using equation (5) in equation (4) , we get
`vecB_("axial") = mu_(0)/(4 pi) ((2 p_(m))/(r^(3))) hati = mu_(0)/ (4 pi ) 2/r^(3) vecp_(m)` ......(6)
Where `vecp_(m) = p_(m) hati`.
