Q.Two small and similar bar magnets have magnetic dipole moment of 1.0 A m$^2$ each. They are kept in a plane in such a way that their axes are perpendicular to each other. A line drawn through the axis of one magnet passes through the centre of the other magnet. If the distance between their centres is 2 m, find the magnitude of the magnetic field at the mid-point of the line joining their centres.

[!TLDR] The mid-point is an axial point of one magnet ($B_a=2\times10^{-7}$ T) and an equatorial point of the other ($B_{eq}=1\times10^{-7}$ T); since the two fields are mutually perpendicular, $B_{net}=\sqrt{B_a^2+B_{eq}^2}=\ …

Call the two magnets A and B, each of moment $m=1.0$ A m$^2$, with their axes perpendicular and the axis of A passing through the centre of B. The mid-point of the line joining their centres is at $r=1$ m from each centre (half of the given $2$ m separation). Since this mid-point lies ON the axis of magnet A, the field due to A there is the AXIAL field, $B_a=\dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3}=10^{-7}\times\dfrac{2\times1}{1^3}=2\times10^{-7}$ T, directed along A's axis (i.e. along the line joining the centres). Since magnet B's axis is perpendicular to this same line, the mid-point lies on B's EQUATORIAL line, so the field due to B there is the equatorial field, $B_{eq}=\dfrac{\mu_0}{4\pi}\dfrac{m}{r^3}=10^{-7}\times\dfrac{1}{1^3}=1\times10^{-7}$ T, directed perpendicular to the line joining the centres (anti-parallel to B's own moment). Because $B_a$ (along the connecting line) and $B_{eq}$ (per …

Recognise that the mid-point is simultaneously an axial point of one magnet and an equatorial point of the other (since the axes are perpendicular and one axis passes through the other's ce …

Simply adding the two field magnitudes ($2\times10^{-7}+1\times10^{-7}=3\times10^{-7}$ T) instead of combining them …