A particle with charge q and velocity v = z enters a perfectly conducting radio-frequency resonant cavity

A particle with charge q and velocity v = νẑ enters a perfectly conducting radio-frequency resonant cavity (length L) through a tiny entrance hole and then exits the cavity through an equally tiny exit hole. If ν is large, we can use the impulse approximation to compute the momentum “kick” transverse to z.

(a) For a cavity with oscillation frequency ω, show that

A particle with velocity v = ż experiences a change in vector potential dA/dt = ∂A/∂t + (v · ∇)A.

(b) The formula derived in part (a) gives Δp_⊥ = 0 for TE modes. Reconcile this with the fact that both TE and TM modes produce Lorentz forces in the transverse direction.

Transcribed Image Text:

L 9 ΔΡΙ Į dz [E+vx B]₁ V 0 L 9 4% dzv 0 dz V₁ E₂.