# Expression for Conductivity of ionized gas (plasma) at low pressure

At low pressure, we can ignore the collisions between the electrons and the gas molecules, and hence energy losses.

Considering a plane electromagnetic wave (EM wave) propagating, the motion of the charge depends almost entirely on the electric field (

**E**-field).

The conductivity of the gas is given by where

**J**

**t**is the electron current density.

Actually, Jt is the sum if all ions in the gas.

where

**Ni**is the number of electrons or ions per metre cube,

**Qi**is the charge of electrons or ions and

**dxi/dt**(=

**vi**) is the speed of the electron or ion along the x-axis.

Now, we require the velocity

**v**of a free electron or ion of mass m subjected to an alternating electric field.

Note that the time derivative of velocity

**v**here can be written as d

**v**/dt = jw

**v**; assuming for example that

**v**= v0 exp(jwt) where j represents the imaginary component, w the frequency and t the time.

Taking the expressions for the force, we obtain

**v**leads the field by 90

^{0}

Then, replacing this expression of

**v**in Jt , taking the pure imaginary part (the "-90

^{0}") and equating the 2 equations for Jt, we get

As the mass of the ions is much larger than that of the electrons while the charge is of the same order of magnitude as that of the electrons, the contribution of the ions can be neglected.

Thus, the conductivity can be written in terms of the charge and mass of electrons only.

So, this is the simple expression that can be used to calculate the conductivity of the plasma.

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