Magnetic Dipole Moment of a Revolving Electron

Section 1.2 showed that ANY current loop carries a magnetic dipole moment. An electron revolving around the nucleus of an atom, in the simple Bohr picture, is exactly such a loop -- a single negative charge $-e$ circulating repeatedly around a fixed circular path -- and this section works out the size of the dipole moment that motion produces.

Setting up the equivalent current. Consider an electron of charge magnitude $e$ moving with speed $v$ in a circular orbit of radius $r$. The time taken for one complete revolution (period) is $T = 2\pi r/v$, so the electron passes any fixed point on the orbit with frequency $f = 1/T = v/2\pi r$. Since a charge $e$ passing a point $f$ times per second is, on average, equivalent to a steady current

the orbiting electron behaves exactly like a tiny current loop of this current $I$ enclosing the orbit's area $A=\pi r^2$.

Orbital magnetic moment. Using $\mu_l = IA$ from Section 1.2 (the subscript $l$ stands for "orbital", and $\mu_l$ is used here instead of $m$ purely to avoid clashing with the electron's mass $m_e$, which appears next):

Because the electron's charge is NEGATIVE, applying the right-hand rule to the conventional (positive) current direction and then reversing it shows that $\vec{\mu}_l$ points OPPOSITE to the electron's orbital angular momentum vector $\vec{L}$, unlike the case of a positive orbiting charge, where the two would point the same way.

Relation to angular momentum -- the gyromagnetic ratio. The electron's orbital angular momentum has magnitude $L = m_evr$ (mass $m_e$, speed $v$, radius $r$). Comparing this with $\mu_l = evr/2$ found above,

The constant of proportionality, $e/2m_e$, is called the gyromagnetic ratio, and its value, $\approx 8.8\times10^{10}\ \text{C/kg}$, is the same for every electron in every atom regardless of which orbit it occupies -- a direct consequence of $\mu_l$ and $L$ both being proportional to $vr$. …