Classical Electrodynamics · Problem 13.1

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · coulomb-scattering-energy-transfer

Problem

If the light particle (electron) in the Coulomb scattering of Section 13.1 is treated classically, scattering through an angle

\theta

is correlated uniquely to an incident trajectory of impact parameter

b

according to

b = \frac{ze^2}{pv} \cot\frac{\theta}{2}

where

p = \gamma mv

and the differential scattering cross section is

\dfrac{d\sigma}{d\Omega} = \dfrac{b}{\sin\theta}\left|\dfrac{db}{d\theta}\right|

. (a) Express the invariant momentum transfer squared in terms of impact parameter and show that the energy transfer

T(b)

T(b) = \frac{2z^2e^4}{mv^2} \cdot \frac{1}{b^2 + b_{\min}^{(c)2}}

where

b_{\min}^{(c)} = ze^2/pv

and

T(0) = T_{\max} = 2\gamma^2\beta^2mc^2

. (b) Calculate the small transverse impulse

\Delta p

given to the (nearly stationary) light particle by the transverse electric field (11.152) of the heavy particle

q = ze

as it passes by at large impact parameter

b

in a (nearly) straight line path at speed

v

. Find the energy transfer

T \approx (\Delta p)^2/2m

in terms of

b

. Compare with the exact classical result of part a. Comment.