Question 3.7.2: Compute the field due to a cylindrical magnet of radius R an...
Compute the field due to a cylindrical magnet of radius R and length L that is polarized along its axis with uniform magnetization
M = M_{s}\hat{z},
The magnet is resting on an infinite plate. Assume that the plate has an infinite permeability μ ≈ ∞ (Fig. 3.45a).
We replace the plate by an image magnet (Fig. 3.45b). The image magnet is identical to the source magnet, and positioned as shown. From symmetry, we know that the field due to the two magnets will satisfy the boundary conditions (3.301) at the interface. The resulting magnet structure is equivalent to a single magnet of length 2L (Fig. 3.45c). Thus, the field is given by Eq. (3.98) with L replaced by 2L:
Eq. (3.98): B_{z}(z)=\frac{μ_{0}M_{s}}{2} [\frac{z+(L)}{\sqrt{(z+L)^{2}+R^{2}}} – \frac{z}{\sqrt{z^{2}+R^{2}}}].
B_{z}(z)=\frac{μ_{0}M_{s}}{2} [\frac{z+(2L)}{\sqrt{(z+2L)^{2}+R^{2}}} – \frac{z}{\sqrt{z^{2}+R^{2}}}]. (3.303)