Q.State the expression for the magnetic field intensity on the axial line of a short bar magnet of dipole moment $m$ at distance $d$ from its centre, and briefly outline how it is obtained by treating the magnet as two point poles $\pm q_m$ a distance $2l$ apart.

Model the magnet as two poles $+q_m$ (N) and $-q_m$ (S), separated by $2l$, centred at $O$. At an axial point $P$ beyond the N pole, at distance $d$ from $O$: the N pole (distance $d-l$) gives field $B_N=(\mu_0/4\pi)q_m/(d-l)^2$ pointing away from the magnet, and the S pole (distance $d+l$) gives $B_S=(\mu_0/4\pi)q_m/(d+l)^2$ pointing toward it. Both lie along the axis, and since $P$ is nearer $N$, the resultant points away from the magnet:

For a short magnet ($d\gg l$), $(d^2-l^2)^2\approx d^4$, giving

Add the two poles' fields along the axis as scalars (they're collinear), then take the $d\gg l$ short-magnet limit.

• Dropping $l^2$ too early, before combining the fractions, gives the wrong intermediate expression.
• Forgetting the $2$ in $2m/d^3$ (it comes from the difference of two nearly-equal large numbers, not a single pole's field).