Equation of Motion

The torque $\mbox{\bf l}$ is given by the rate of change of angular momentum $\mbox{\bf g}$ with time

$$\frac{d\mbox{\bf g}}{dt} = \mbox{\bf l} \quad , \quad \vert\mbox{\bf g}\vert=\hbar \; .$$

Moreover the torque acting on a magnetic moment $\mbox{\bf m}$ in a magnetic field $\mbox{\bf H}$ is given by

$$\mbox{\bf l}=\mbox{\bf m} \times \mbox{\bf H} \; .$$

The gyromagnetic ratio $\gamma_0$ is defined by the ratio of magnetic moment to angular momentum, which are antiparallel due to the negative charge of electrons.

$$\gamma_0=\frac{m}{\vert\mbox{\bf g}\vert} \quad , \quad \mbox{\bf g}=-\frac{\mbox{\bf m}}{\vert\gamma_0\vert}$$

Therefore we obtain

$$\frac{d\mbox{\bf m}}{dt}=-\vert\gamma_0\vert\mbox{\bf m} \times \mbox{\bf H} \; .$$

An orbiting electron with charge e=-e₀ gives rise to a current I in a loop covering an area F. Its magnetic moment m is given by (SI units)

$$m=I \; F= \frac{e v}{2r\pi} r^2 \pi= \frac{e vr}{2}= \fra... ...derbrace{m_e vr}_{=\hbar}= -\frac{e_0 \hbar}{2 m_e}= -\mu_B$$

In cgs units $m=\frac{I}{c} \; F \Rightarrow \mu_B=\frac{e_0 \hbar}{2 m_e c}$.

However, the magnetic moment caused by the spin of an electron is larger. The appropriate gyromagnetic factor $g_s \approx -2$ can be calculated by solving Dirac's equation. However, experiments give $g_s=-2 \cdot 1.001\,159\,652$ , which can be verified using perturbation theory in quantumelectrodynamics.

Finally we can calculate the gyromagnetic ratio

$$\gamma_0=\frac{g_s \mu_B}{\hbar} \approx 1.759\cdot 10^{11} \ \frac{\mbox{rad}}{\mbox{Ts}}$$