A report on tokamak heating argues as follows

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Plasma Physics · Kinetic theory, collisions and transport

Plasma Physics, Ch Coulomb collisions: the Coulomb logarithm, the Fokker-Planck equation and Spitzer resistivity

Problem

A report on tokamak heating argues as follows. The parallel resistivity is η∥=0.51 me/(nee2τe)\eta_\parallel = 0.51\,m_e/(n_ee^2\tau_e). Since the electron collision time τe\tau_e is a fixed material property of the plasma at a given temperature, increasing the electron density nen_e simply increases the number of charge carriers, so η∥\eta_\parallel falls as 1/ne1/n_e. Comparing two plasmas at the same Te=1 keVT_e = 1\,\mathrm{keV}, one with ne=1×1019 m−3n_e = 1\times10^{19}\,\mathrm{m^{-3}} and one with ne=1×1020 m−3n_e = 1\times10^{20}\,\mathrm{m^{-3}}, this predicts a resistivity ten times smaller in the denser plasma. ((a)) Identify the error in the quoted reasoning. ((b)) Using τe=62π3/2ϵ02me1/2Te3/2/(niZ2e4ln⁡Λ)\tau_e = 6\sqrt{2}\pi^{3/2}\epsilon_0^2m_e^{1/2}T_e^{3/2}/(n_iZ^2e^4\ln\Lambda) with Z=1Z=1, find the ratio η∥(1020 m−3)/η∥(1019 m−3)\eta_\parallel(10^{20}\,\mathrm{m^{-3}})/\eta_\parallel(10^{19}\,\mathrm{m^{-3}}) at fixed TeT_e, treating ln⁡Λ\ln\Lambda as unchanged between the two cases.

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