3.1 Orbital Magnetic Dipole Moment: the Bohr Magneton
An electron in orbit is a current loop. If the orbital period is \(T\), the
orbital current is
\begin{equation}
i = -\frac{e}{T} = -\frac{e v}{2\pi r} ,
\label{eq:orbital-current}
\end{equation}
and if \(A = \pi r^{2}\) is the area enclosed, the orbital magnetic moment is
\begin{equation}
\mu_{\ell} = i A = -\frac{e v r}{2} .
\label{eq:mu-orbital}
\end{equation}
The orbital angular momentum is \(L = m v r\), so
\begin{equation}
\boxed{\;\frac{\mu_{\ell}}{L} = -\frac{e}{2m}\;}
\label{eq:gyromag-ratio}
\end{equation}
The ratio of magnetic moment to angular momentum is independent of the
parameters of the orbit — \(v\) and \(r\) have cancelled. The quantity \(e/2m\) is
the gyromagnetic ratio of the electron, exactly half the charge-to-mass
ratio \(e/m = 1.76\times 10^{11}\,\mathrm{C}\,\mathrm{kg}^{-1}\).
Note
The electron is negatively charged, so its magnetic dipole moment points
opposite to its angular momentum. Every sign in this chapter follows from
that one fact.
Written as a vector relation,
\begin{equation}
\vec\mu_{\ell} = -\,g_{\ell}\,\frac{\mu_{\mathrm B}}{\hbar}\,\vec L ,
\qquad g_{\ell} = 1 ,
\label{eq:mu-vec-orbital}
\end{equation}
where the Bohr magneton is
\begin{equation}
\mu_{\mathrm B} = \frac{e\hbar}{2m}
= 9.274\times 10^{-24}\,\mathrm{J}\,\mathrm{T}^{-1}
= 5.788\times 10^{-5}\,\mathrm{eV}\,\mathrm{T}^{-1} .
\label{eq:bohr-magneton}
\end{equation}
The orbital \(g\) factor \(g_{\ell}=1\) is introduced only to preserve
symmetry with the spin and Landé \(g\) factors that follow; it is not equal to
\(g_{s}\).
From quantum mechanics \(\lvert\vec L\rvert = \sqrt{\ell(\ell+1)}\,\hbar\), so the
magnitude of the orbital moment is
\begin{equation}
\lvert\vec\mu_{\ell}\rvert
= g_{\ell}\,\mu_{\mathrm B}\sqrt{\ell(\ell+1)} ,
\qquad
\mu_{\ell,z} = -g_{\ell}\,\mu_{\mathrm B}\, m_{\ell} .
\label{eq:mu-magnitude}
\end{equation}
