Magnetic Dipole Moment of Revolving Electron

An electron circling a nucleus in a stationary orbit of radius $R$ with speed $v$ constitutes a tiny current loop (moving charge = current), with $I = e/T$ (magnitude; the electron's charge is $-e$) and period $T=2\pi R/v$. Its orbital magnetic dipole moment, from §3.8.5's $\mu_L=IA$, is

Meanwhile its orbital angular momentum about the same centre has magnitude $L = m_evR$ (with $m_e$ the electron mass). Dividing the two,

a constant of proportionality called the gyro-magnetic ratio, connecting the electron's orbital magnetic moment to its orbital angular momentum, independent of the orbit's actual size or speed. (The minus sign that would appear in a fully vector treatment simply says $\vec\mu_L$ and $\vec L$ point opposite ways, since the electron's charge is negative.)

Bohr's quantization rule restricts $L$ to integer multiples of $h/2\pi$: $L_n = nh/2\pi$ ($n=1,2,3,\dots$). Substituting,

$\mu_L = \frac{e}{2m_e}\cdot\frac{nh}{2\pi} = n\,\frac{eh}{4\pi m_e}$ …