a$)$ In a DC plasma column, the criteria for self$-$sustained breakdown of the gas can be written as

$[exp(d)-1]1$

Explain the overall meaning of this equation and define the meaning of each of the terms, $,d,$ and exp$(d).$

b$)$ Using a simple model of collisions between electrons and neutral particles in a plasma column show that the mean electron drift speed $V_d$ is given by

$V_d=\frac{Ee}{m_e{v}_{ce}}$

Where $E$ is the electric field, $m_e$ is the electron mass, $e$ is the electronic charge and $v_{ce}$ is the electron$-$neutral particle collision frequency.

c$)$ Write down an expression for electron current density $J_e$ in terms of both $V_d$ and the electron density $n_e$ and then use Ohm's law show that the plasma conductivity $$ is given by

$=\frac{n_e{e}^2}{m_e{v}_{ce}}$

d$)$ In an argon plasma column, the argon neutral density is $4{10}^{22}{m}^{-3},$ the cross$-$section for electron$-$argon collisions is $2{10}^{-20}{m}^2$ and the electron thermal velocity is $4{10}^{6}m{s}^{-1}.$

i$)$ Calculate the mean free path ${}_{ce}$ and collision frequency $v_{ce}$ for electron$-$neutral collsions.

ii$)$ If an engineer wishes to have a plasma conductivity of $25S{m}^{-1}$ what must the electron density be$?$