Imagine two charged magnetic dipoles (charge q, dipole moment m), constrained to move on the z axis (same as Problem 6.23(a), but without gravity). Electrically they repel, but magnetically (if both m’s point in the z direction) they attract.
(a) Find the equilibrium separation distance.
(b) What is the equilibrium separation for two electrons in this orientation.
\text { [Answer: } 4.72 \times 10^{-13} m \text {.] }(c) Does there exist, then, a stable bound state of two electrons?
(a) Forces on the upper charge:
F _{q}=\frac{1}{4 \pi \epsilon_{0}} \frac{q^{2}}{z^{2}} \hat{ z }, \quad F _{m}=\nabla( m \cdot B )=\nabla\left(m \frac{2 \mu_{0} m}{4 \pi z^{3}}\right)=\frac{\mu_{0} m^{2}}{2 \pi}\left(\frac{-3}{z^{4}}\right) \hat{ z } .
At equilibrium,
\frac{1}{4 \pi \epsilon_{0}} \frac{q^{2}}{z^{2}}=\frac{3 \mu_{0} m^{2}}{2 \pi z^{4}} \quad \Rightarrow z^{2}=\frac{6 \mu_{0} \epsilon_{0} m^{2}}{q^{2}} \Rightarrow z=\sqrt{6} \frac{m}{q c} ,
where 1 / \sqrt{\epsilon_{0} \mu_{0}}=c , the speed of light.
(b) For electrons, q=1.6 \times 10^{-19} C (actually, it’s the magnitude of the charge we want in the expression above), and m=9.22 \times 10^{-24} Am ^{2} (the Bohr magneton—see Problem 5.58), so
z=\sqrt{6} \frac{9.22 \times 10^{-24}}{\left(1.6 \times 10^{-19}\right)\left(3 \times 10^{8}\right)}=4.72 \times 10^{-13} m .
(For comparison, the Bohr radius is 0.5 \times 10^{-10} m , so the equilibrium separation is about 1% of the size of a hydrogen atom.)
(c) Good question! Certainly the answer is no. Presumably this is an unstable equilibrium, so unless you could find a way to maintain the orientation of the dipoles, and keep them on the z axis, the structure would fall apart.