Magnetic Field Intensity at a Point on Equatorial Line of Bar Magnet :
Let (P) be a point on the equatorial line of a bar magnet such that its distance from centre (O) is (r).
Magnetic field intensity at ( P ) due to N-pole of the bar magnet:
$\vec{B_1} = \frac{\mu_0}{4\pi} \cdot \frac{q_m}{(\sqrt{r^2 + l^2})^2} \quad \text{along PC}$
$= \frac{\mu_0}{4\pi} \cdot \frac{qm}{(r^2 + l^2)} \quad \text{along PC} \quad ...(i)$
Magnetic field intensity at ( P ) due to S-pole of the bar magnet:
$\vec{B_2} = \frac{\mu_0}{4\pi} \cdot \frac{q_m}{(r^2 + l^2)} \quad \text{along PS} \quad ...(ii)$
Resultant Magnetic Field on Equatorial Line
$ ( \vec{B_1}$ ) and $( \vec{B_2} )$ are inclined at an angle $(2\theta)$.
Therefore, the resultant of these two field intensities is:
$B_e = \sqrt{B1^2 + B2^2 + 2B1B_2 \cos 2\theta}$
Since $( |\vec{B1}| = |\vec{B2}|$:
$B_e = \sqrt{2B1^2 + 2B_1^2 \cos 2\theta}$
$= \sqrt{2B_1^2 (1 + \cos 2\theta)}$
$(\because 1 + \cos 2\theta = 2 \cos^2 \theta)$
$= \sqrt{2B_1^2 \cdot 2 \cos^2 \theta}$
$= 2B_1 \cos \theta$
Using eqn. (i):
$B_e = 2 \cdot \frac{\mu0}{4\pi} \cdot \frac{q_m}{(r^2 + l^2)} \cos \theta$
From geometry:
$\cos \theta = \frac{l}{\sqrt{r^2 + l^2}}$
$B_e = \frac{\mu0}{4\pi} \cdot \frac{q_m \cdot 2l}{(r^2 + l^2)^{3/2}}$
Since ( $q_m \cdot 2l = m $) (dipole moment):
$B_e = \frac{\mu0}{4\pi} \cdot \frac{m}{(r^2 + l^2)^{3/2}} \quad ...(iii)$
For a very small magnet $(( l^2 \ll r^2)$:
$Be = \frac{\mu0}{4\pi} \cdot \frac{m}{r^3} \quad ...(iv)$
Magnetic field due to a short bar magnet at a distance (r) from the centre of the bar magnet on the axial line:
$B_a = \left( \frac{\mu0}{4\pi} \right) \frac{2m}{r^3} \quad \text{...(1)}$
Magnetic field due to a short bar magnet at a distance (r) from the centre of the bar magnet on the equatorial line:
$B_e = \left( \frac{\mu0}{4\pi} \right) \frac{m}{r^3} \quad \text{...(2)}$
Dividing eqn. (1) by eqn. (2), we get:
$\frac{B_a}{B_e} = 2$or
$B_a = 2B_e \quad \text{...(3)}$
Thus, the magnetic field due to a short magnet at a distance (r) from the centre of the bar magnet on the axial line is two times the magnetic field of the short magnet at the same distance (r) on the equatorial line.