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Date: 2-8-2016
1530
Date: 22-8-2016
1075
Date: 10-9-2016
1371
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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.
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