The Magnetic Dipole Moment — Question 9

Question 9

The orbital moment of a single circulating charge is linked to its angular momentum

A nonrelativistic charge q of mass m moves in a circular orbit of radius r and speed v. Define the orbit normal by the direction of angular momentum L. The signed equivalent current is q/T, where T is the orbital period.

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

2. If the orbit is maintained solely by a uniform magnetic field perpendicular to the motion, show that the orbital moment points opposite to the applied field and has magnitude K/B.

Question 9 - Solution

The orbital moment of a single circulating charge is linked to its angular momentum

(a) Method and calculation

$$I=q/T=qv/(2\pi r),\quad\vec\mu=I\pi r^2\hat L=\tfrac12qvr\hat L.$$$$\vec L=mvr\hat L\Rightarrow\vec\mu=\frac q{2m}\vec L.$$

(b) Interpretation and further calculation

$$r=\frac{mv}{|q|B}\Rightarrow|\mu|=\frac{|q|vr}{2}=\frac{mv^2}{2B}=K/B.$$$$q>0:\ \vec L\text{ opposes }\vec B;\quad q<0:\ \vec L\text{ follows }\vec B.$$In both cases, the factor q in the moment makes $\vec\mu$ oppose $\vec B$.

Physical check

This is an orbital classical relation; intrinsic spin moments require their own gyromagnetic ratios.