Consider problem 11.10, but taking E = 0 and, instead of the adiabatic case, consider a constant kinetic pressure

(a) Show that the electron distribution function is given by

(b) Evaluate the heat flux vector q and show that it can be written as

where the thermal conductivity K is equal to 5kp/(2mv).

(c) What is the value of the electric current density J in this case?

Data from Problem 11.10.

An electron gas (Lorentz gas), in a background of stationary ions, is acted upon by a weak, externally applied electric field E, under steady state conditions. Using the Boltzmann equation for the electrons, with the relaxation model for the collision term (considering a constant collision frequency ν),

and considering the adiabatic case for which

show that the electron distribution function is given by

Assume that f = f₀ + f₁, where |f₁| ≪ f₀ and where f₀ is the following modified Maxwellian distribution

and neglect all second-order terms in the Boltzmann equation. Consider the term involving ∇f₁ as a second-order quantity.

p = n(r) k T(r) = constant