Classical Electrodynamics · Problem 13.5

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · energy-loss-plasma-close-collisions

Problem

Consider the energy loss by close collisions of a fast, but nonrelativistic, heavy particle of charge zeze passing through an electronic plasma. Assume that the screened Coulomb interaction V(r)=ze2exp⁡(−kDr)/rV(r) = ze^2\exp(-k_D r)/r, where kDk_D is the Debye screening parameter, acts between the electrons and the incident particle. (a) Show that the energy transfer in a collision at impact parameter bb is given approximately by ΔE(b)≈2(ze2)2mv2 kD2K12(kDb)\Delta E(b) \approx \frac{2(ze^2)^2}{mv^2}\, k_D^2 K_1^2(k_D b) where mm is the electron mass and vv is the velocity of the incident particle. (b) Determine the energy loss per unit distance traveled for collisions with impact parameter greater than bmin⁡b_{\min}. Assuming kDbmin⁡≪1k_D b_{\min} \ll 1, show that (dEdx)kDb<1≈(ze)2v2 ωp2ln⁡(11.47kDbmin⁡)\left(\frac{dE}{dx}\right)_{k_D b < 1} \approx \frac{(ze)^2}{v^2}\, \omega_p^2 \ln\left(\frac{1}{1.47 k_D b_{\min}}\right) where bmin⁡b_{\min} is given by the larger of the classical and quantum minimum impact parameters [(13.16) and above].

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