If a charged object is put in a plasma,
it repels like charges and attracts opposite charges,
leading to a Debye sheath around the object’s surface
with a net opposite charge.
This has the effect of shielding the object’s presence
from the rest of the plasma.
We start from Gauss’ law
for the electric field ,
expressing as the gradient of a potential ,
and splitting the charge density into ions and electrons :
The last term represents a test particle,
which will be shielded.
This particle is a point charge ,
whose density is simply a Dirac delta function ,
and is not included in or .
For a plasma in thermal equilibrium,
we have the Boltzmann relations
for the densities:
We assume that electrical interactions are weak compared to thermal effects,
i.e. in both cases.
Then we Taylor-expand the Boltzmann relations to first order:
Inserting this back into Gauss’ law,
we arrive at the following equation for ,
where we have assumed quasi-neutrality such that :
We now define the ion and electron Debye lengths
and as follows:
And then the total Debye length is defined as the sum of their inverses,
and gives the rough thickness of the Debye sheath:
With this, the equation can be put in the form below,
suggesting exponential decay:
This has the following solution,
known as the Yukawa potential,
which decays exponentially,
representing the plasma’s self-shielding
over a characteristic distance :
Note that is a scalar,
i.e. the potential depends only on the radial distance to .
This treatment only makes sense
if the plasma is sufficiently dense,
such that there is a large number of particles
in a sphere with radius .
This corresponds to a large Coulomb logarithm :
The name Yukawa potential originates from particle physics,
but can in general be used to refer to any potential (electric or energetic)
of the following form:
Where and are scaling constants that depend on the problem at hand.
- P.M. Bellan,
Fundamentals of plasma physics,
1st edition, Cambridge.
- M. Salewski, A.H. Nielsen,
Plasma physics: lecture notes,