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 2M)/(2pi (d/2)^3)` along the axis of `a`, and that due to `b` is `B_b = (mu_0 M)/(4pi (d/2)^3)` parallel to the axis of `b` as shown in figure. The resultant field at `P` is, therefore,
`B = sqrt(B^2_a + B^2_b)`
`= (mu_0 M)/(4pi (d/2)^3) sqrt(1^2 + 2^2)`
`(2 sqrt(5) mu_0 M)/(pid^3)`
The direction of this field makes an angle `alpha with B_a` such that `tanalpha = B_b/B_a = 1/2`.