Magnetic Dipole and Permeable Medium

A point magnetic dipole m in vacuum (medium 1) is pointing toward the plane surface of a medium with permeability μ (medium 2). The distance between the dipole and surface is d (see Figure 1.1).

Figure 1.1

a) Solve for the magnetic field B within the medium.

b) What is the force acting on the dipole?

SOLUTION

a) Use the method of images. We place another dipole m' = α m at the point O', at the same distance d on the other side of the plane separating the vacuum from the permeable medium (see Figure 1.2). Compute the field in medium 1 as a superposition of the dipole fields from m and m'. To find the field in medium 2, we put yet another dipole m'' = β m at point O. We can write this in the form

(1)

(2)

where n, n' are unit vectors in the directions r, r' respectively, and indices 1 and 2 correspond to media 1 and 2. As usual, we write the boundary

Figure 1.2

conditions at some point P on the plane where |r| = |r'|

(3)

(4)

Substituting (1) and (2) into (3) and (4), we obtain

(5)

(6)

From (5) and (6), we have

(7)

which yield

So the field in medium 2 is

b) The force acting on a dipole m is determined only by the field B^(1)' of the image dipole m' in medium 2

(8)

The dipole is attracted to the medium. The result for the equivalent problem of an electric dipole near a half space filled with an ideal conductor can be obtained from (8) by letting μ → ∞

where p is the electric dipole moment.