Magnetism and Electrons — Question 1

Question 1

A classical electron orbit has opposite moment and angular momentum

In a classical circular-orbit model an electron of charge -e and mass me moves counterclockwise, viewed from +z, at speed v and radius r. Use the conventional-current definition of magnetic moment.

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

2. If radius doubles while angular momentum is conserved, find the new speed, current magnitude, and moment relative to their initial values.

Question 1 - Solution

A classical electron orbit has opposite moment and angular momentum

(a) Method and calculation

$$I=\frac{ev}{2\pi r},\quad\boldsymbol\mu=-I\pi r^2\hat z=-\frac{evr}{2}\hat z.$$$$\mathbf L=m_evr\hat z\Rightarrow\boxed{\boldsymbol\mu=-\frac{e}{2m_e}\mathbf L}.$$

(b) Interpretation and further calculation

$$r_2=2r_1,\ L_2=L_1\Rightarrow v_2=v_1/2,\quad I_2=I_1/4.$$$$\boxed{\boldsymbol\mu_2=\boldsymbol\mu_1}.$$

Physical check

The classical orbit is a model for deriving the gyromagnetic ratio, not a literal quantum electron trajectory.