Figure shows two identical magnetic dipoles `a and b` of magnetic moments `M` each, placed at a separation `d`, with their axes perpendicular to each other. Find the magnetic field at the point `P` midway between the dipoles.

Explanation

The point p is in end-on position for the dipole a and in broadside-on position for the dipole b’. The magnetic field at P due to a is `B_(a)=(mu_(0))/(4pi)(2M)/((d//2)^(3))` along the axis of a, and that due to b is `B_(b)=(mu_(0))/(4pi)(M)/((d//2)^(3))` parallel to the axis of b as shown in figure. The resultant field at P is, therefore
`B=sqrt(B_(a)^(2)+B_(b)^(2))`
`=(mu_(0)M)/(4pi(d//2)^(3))sqrt(1^(2)+2^(2))`
`=(2sqrt(5)mu_(0)M)/(pid^(2))`
The direction of this field makes an angle `alpha` with `B_(a)` such that `tanalpha=B_(b)//B_(a)=1//2`.