Question 3.7.3: Determine the force on a cylindrical bar magnet a distance z...
Determine the force on a cylindrical bar magnet a distance z above an infinite ferromagnetic plate that has an infinite permeability μ ≈ ∞ (Fig. 3.46a). The magnet has a radius R and length L and is magnetized to a level M_{s} along its axis. Assume R << L.
We replace the plate by an image magnet as shown in Fig. 3.46b. The image magnet is identical to the source magnet, and it is easy to check that the field due to the two magnets will satisfy the boundary conditions (3.301). The force between the source and image magnet was determined in Example 3.4.3. It is given by Eq. (3.126) with d replaced by 2z,
Eq. (3.126): F(d) = \frac{μ_{0}Q_{m}^{2}}{4π} \{ -\frac{1}{(d)^{2}} + \frac{2}{(L+d)^{2}} – \frac{1}{(2L+d)^{2}}\}.
F(z) = \frac{μ_{0}Q_{m}^{2}}{4π} \{ -\frac{1}{(2z)^{2}} + \frac{2}{(L+2z)^{2}} – \frac{1}{4(L+z)^{2}}\} , (3.304)
where Q_{m}=M_{s}πR^{2}.