Question

Derive an expression for magnetic field intensity due to a magnetic dipole at a point lies on its equatorial line.

Answer 1

Consider a point P on equatorial position (or broad side on position) of short bar magnet of length 2l, having north pole (N) and south pole (S) of strength +qm and – qm respectively. The distance of point P from the mid-point (O) of magnet is r. Let B₁ and B₂ be the magnetic field intensities due to north and south poles respectively. NP = SP

= \(\sqrt{r^2+l^2}\) .

Clearly, magnitudes of and are equal

i.e., |\(\vec{B_1}\)| = \(\vec{B_2}\)| or B₁ = B₂

To find the resultant of \(\vec{B_1}\) and \(\vec{B_2}\) we resolve them along and perpendicular to magnetic axis SN. Components \(\vec{B_1}\) of along and perpendicular to magnetic axis are B₁ cos θ and B₂ sin θ respectively.

Components of \(\vec{B_2}\) along and perpendicular to magnetic axis are B₂ cos θ and B₂ sin θ respectively. Clearly, components of and perpendicular to axis SN. B₁ sin θ and B₂ sin θ are equal in magnitude and opposite in direction and hence, cancel each other; while the components of \(\vec{B_1}\) and \(\vec{B_2}\) along the axis are in the same direction and hence, add up to give to resultant magnetic field parallel to the direction \(\vec{Ns}\) .

∴ Resultant magnetic field intensity at P.

B = B₁ cosθ + B₂ cos θ

If the magnet is very short and point P is far away, we have l<<r; so I² may be neglected as compared to r² and so equation (3) takes the from

B = \(B=\frac{\mu_0}{4\pi}\frac{m}{r^3}\) ...(4)

This is expression for magnetic field intensity at equatorial position of the magnet. 

It is clear from equations (2) and (4) that the magnetic field strength due to a short magnetic dipole is inversely proportional to the cube of its distance from the centre of the dipole and the magnetic field intensity at axial position is twice that at equatorial position for the same distance.