If an electric field
with magnitude is applied to the plasma, the electrons experience
a Lorentz force
(we neglect the ions due to their mass),
where is the electron charge.
However, collisions slow them down while they travel through the plasma.,
This can be modelled as a drag force ,
where is the electron-ion collision frequency
(we neglect since all electrons are moving together),
is their mass,
and their typical velocity relative to the ions in the background.
Balancing the two forces yields the following relation:
Using that the current density ,
we can rearrange this like so:
This is Ohm’s law, where is the resistivity.
From our derivation of the Coulomb logarithm ,
we estimate to be as follows,
where is the ion density,
is the collision cross-section,
and is the reduced mass
of the electron-ion system:
Where we used that ,
and for some ionization ,
and as a result due to the plasma’s quasi-neutrality.
Beware: authors disagree about the constant factors in ;
recall that it was derived from fairly rough estimates.
This article follows Bellan.
Inserting this expression for into
the so-called Spitzer resistivity then yields:
A reasonable estimate for the typical velocity
at thermal equilibrium is as follows,
where is Boltzmann’s constant,
and is the electron temperature:
Other choices exist,
see e.g. the Maxwell-Boltzmann distribution,
but always .
Inserting this into then gives:
- P.M. Bellan,
Fundamentals of plasma physics,
1st edition, Cambridge.
- M. Salewski, A.H. Nielsen,
Plasma physics: lecture notes,