Induced Electric Fields — Question 5

Question 5

Changing enclosed flux transfers angular momentum to a constrained charge

A particle of charge q and mass m is constrained to a frictionless circular track of radius r outside an axially symmetric changing-flux region. The enclosed magnetic flux changes by DeltaPhi. The track force is radial and the particle starts at rest.

1. Find the change in angular momentum about the axis.

2. Find final kinetic energy, neglecting radiation and the particle back-reaction on the source.

Question 5 - Solution

Changing enclosed flux transfers angular momentum to a constrained charge

(a) Method and calculation

$$2\pi rE_\phi=-\dot\Phi,\quad\dot L_z=rqE_\phi=-q\dot\Phi/(2\pi).$$$$\boxed{\Delta L_z=-q\Delta\Phi/(2\pi)}.$$

(b) Interpretation and further calculation

$$K_f=\frac{L_z^2}{2mr^2}.$$$$\boxed{K_f=\frac{q^2(\Delta\Phi)^2}{8\pi^2mr^2}}.$$

Physical check

A magnetic force, if present at the track, is radial and does no work. The induced electric field transfers the energy.