Classical Electrodynamics · Problem 13.6

Radiating Systems, Multipole Fields and Radiation — John David Jackson

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

Problem

The energy loss in a plasma from distant collisions can be found with Fermi's method for the density effect. Consider the nonrelativistic limit of (13.36) with the relative dielectric constant of a plasma given by (7.59) augmented by some damping, ϵ(ω)=1−ωp2ω2+iωΓ\epsilon(\omega) = 1 - \frac{\omega_p^2}{\omega^2 + i\omega\Gamma} Assume that the arguments of the Bessel functions are small (corresponding to a speed of the incident particle large compared to thermal speeds in the plasma). (a) Show that the energy loss (13.36) for kDb>1k_D b > 1 becomes (dEdx)kDb>1≈2z2e2πv2∫0∞Re(iωϵ(ω))ln⁡(1.123kDvω)dω\left(\frac{dE}{dx}\right)_{k_D b > 1} \approx \frac{2z^2e^2}{\pi v^2} \int_0^\infty \mathrm{Re}\left(\frac{i\omega}{\epsilon(\omega)}\right) \ln\left(\frac{1.123 k_D v}{\omega}\right) d\omega (b) With the assumption that Γ≪ωp\Gamma \ll \omega_p in ϵ(ω)\epsilon(\omega), show that the formula of part a yields (dEdx)kDb>1≈z2e2v2 ωp2ln⁡(1.123kDvωp)\left(\frac{dE}{dx}\right)_{k_D b > 1} \approx \frac{z^2 e^2}{v^2}\, \omega_p^2 \ln\left(\frac{1.123 k_D v}{\omega_p}\right) Combine with the close-collision result of Problem 13.5 to find the total energy loss of a nonrelativistic particle passing through a plasma, (dEdx)≈z2e2v2 ωp2ln⁡(Λvωpbmin⁡)\left(\frac{dE}{dx}\right) \approx \frac{z^2 e^2}{v^2}\, \omega_p^2 \ln\left(\frac{\Lambda v}{\omega_p b_{\min}}\right) where Λ\Lambda is a number of order unity.

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