Classical Electrodynamics · Problem 13.15

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · transition-radiation-photon-number

Problem

(a) Find the number

N_\gamma

of transition radiation quanta with frequencies greater than

\omega_p

emitted per interface, starting from the energy spectrum (13.85). Show that for

\gamma \gg 1

N_\gamma = \frac{z^2 e^2}{\pi \hbar c}\left[(\ln\gamma - 1)^2 + \frac{\pi^2}{12}\right]

where terms of order

1/\gamma^2

have been neglected. (b) Using the result from part a for the number of photons and the value

\hbar\omega_p = 20

eV, find the mean energy of the photons (in keV) for

\gamma = 10^3

10^4

10^5