Angular Momenta and Magnetic Dipole Moments — Question 4

Question 4

Magnetic torque produces precession without changing angular momentum magnitude

An angular momentum $\boldsymbol J$ has moment $\boldsymbol\mu=\gamma\boldsymbol J$ in uniform $\boldsymbol B=B\hat z$. Its dynamics obey $d\boldsymbol J/dt=\boldsymbol\mu\times\boldsymbol B$. The initial transverse component points along x.

1. Derive the signed azimuthal angular velocity.

2. For an electron orbital moment, state the direction and period using $\gamma=-e/(2m)$.

Question 4 - Solution

Magnetic torque produces precession without changing angular momentum magnitude

(a) Method and calculation

$$\dot J_x=\gamma BJ_y,\quad\dot J_y=-\gamma BJ_x,\quad\dot J_z=0.$$$$J_x=J_\perp\cos\omega t,\ J_y=J_\perp\sin\omega t\Rightarrow\boxed{\omega=-\gamma B}.$$

(b) Interpretation and further calculation

$$\gamma=-e/(2m)\Rightarrow\boxed{\omega=eB/(2m)>0}\quad(x\text{ toward }y).$$$$\boxed{T=2\pi/|\omega|=4\pi m/(eB)}.$$

Physical check

The torque is perpendicular to J, so its magnitude remains fixed. Positive azimuth means counterclockwise viewed from positive z toward the origin.