Magnetic Field of a Bar Magnet on Its Axis

Consider a short bar magnet of dipole moment $m = q_m(2l)$, with its S pole at point $S$ and N pole at point $N$, a distance $2l$ apart, centred at $O$. Let $P$ be a point on the extension of the magnetic axis beyond the N pole, at distance $d$ from the centre $O$ (so $P$ is at distance $d-l$ from the N pole and $d+l$ from the S pole).

Field due to each pole. Treating each pole as a point source of magnetic field, in exact analogy with a point charge in electrostatics (with $\mu_0/4\pi$ replacing $1/4\pi\epsilon_0$ and pole strength $q_m$ replacing electric charge $q$), the field at $P$ due to the N pole points AWAY from the magnet along the axis, with magnitude

and the field at $P$ due to the S pole points TOWARD the magnet along the axis, with magnitude

Combining the two fields. Both fields lie exactly along the axis, and since $P$ is nearer the N pole, $B_N>B_S$, so the resultant points away from the magnet (in the direction of $\vec{m}$):

using $m=q_m(2l)$ from Section 1.4.

The short-magnet limit. For a point far from the magnet, $d\gg l$, so $l^2$ may be dropped compared with $d^2$ in the denominator, giving $(d^2-l^2)^2\approx d^4$:

$\boxed{B_{\text{axial}} \approx \frac{\mu_0}{4\pi}\cdot\frac{2m}{d^3}}$ …