Angular Momenta and Magnetic Dipole Moments — Question 1

Question 1

The sign of an orbital magnetic moment follows the electron charge

A classical circular electron orbit has radius r and speed v. Use charge $-e$, electron mass m, current magnitude $I=e/T$, and magnetic-moment magnitude $I\pi r^2$.

1. Derive the vector relation between orbital magnetic moment and angular momentum.

2. For $l=2,m_l=-1$, give the magnetic-moment magnitude and z projection in units of $\mu_B=e\hbar/(2m)$.

Question 1 - Solution

The sign of an orbital magnetic moment follows the electron charge

(a) Method and calculation

$$T=2\pi r/v,\quad |\mu_L|=\frac{ev}{2\pi r}\pi r^2=\frac{evr}{2},\quad |L|=mvr.$$$$\boxed{\boldsymbol\mu_L=-\frac{e}{2m}\boldsymbol L=-\frac{\mu_B}{\hbar}\boldsymbol L}.$$

(b) Interpretation and further calculation

$$|\mu_L|=\mu_B\sqrt{l(l+1)}=\boxed{\sqrt6\mu_B}.$$$$\mu_{Lz}=-m_l\mu_B=\boxed{+\mu_B}.$$

Physical check

The conventional current circulates opposite to electron motion. The quantum relation uses the same charge sign even though a stationary atomic orbital is not a literal planetary trajectory.