Question 3.4.3: Consider two thin cylindrical magnets oriented along the x-a...
Consider two thin cylindrical magnets oriented along the x-axis as shown in Fig. 3.19a. Both magnets have a radius R and length L and are polarized to a level Ms. Assume R << L and determine the force between the two magnets as a function of their separation d.
We compute the force on the magnet to the right. Because the magnets are thin (R <<L), we can treat the magnets as dipoles to first order (Fig. 3.19b). The charge at each pole face is given by Eq. (3.110), which reduces to
Eq. (3.110): Q_{m}(x^{\prime}) = σ_{m}(x^{\prime})Δ A (surface charge),
Q_{m} = ± M_{s}πR^{2},
where the ± sign refers to the north and south poles, respectively. The force is obtained by summing the forces between the charges as given by Eq. (3.125). We obtain
Eq. (3.125): F_{1 2} = \frac{μ_{o}}{4π} \frac{Q_{m} (x_{1}) Q_{m} (x_{2}) }{ (x_{1} – x_{2})^{2}}.
F(d) = \frac{μ_{o}Q_{m}^{2}}{4π} \{ -\frac{1}{d^{2}} + \frac{2}{(L + d)^{2}} – \frac{1}{(2L + d)^{2}} \}. (3.126)
The force on the magnet to the left is of the same magnitude but opposite direction as shown.