Electrons and ions

We consider now some physical situations in which the field is determined neither by fixed charges nor by charges on a conducting surfaces, but by a combination of two physical phenomenon.In other words, the field will be governed simultaneously by two set of equations.,the equation from electrostatics relating electric fields to charge distribution, and an equation from another part of physics that determines the positions of motions of the charges in the presence of the field.

The first example is a dynamic one in which motion is governed by newton's laws.This is what is happening in plasma, which is an ionised gas consisting of free electrons distributed over a region in space.The ionosphere the upper layer of the atmosphere is an example of such a plasma. In such a plasma the positive ions are very much heavy than the electrons.So we may neglect the ionic motion in comparison to that of the electrons.

Let $n_{o}$ be the density of electrons in the undisturbed state.Assuming the molecules are singly ionised, this must also be the density of positive ions,since the plasma is electrically neutral when undisturbed. If the density of electrons in one region is increased, they will repel each other and tend to return to their equilibrium positions and hence oscillate back and forth.To simplify the situation, we will worry only about a situation in which the motions are all in one dimension say x.

Let us suppose that the electrons originally at x are at the instant t displaced from their equilibrium position by a small amount s(x,t). Since the electrons have been displaced , their density in general be changed. The density in general is calculated. The electrons contained between the planes a and b are now contained between the planes a' and b' the number of electrons that were between a and b is proportional to $n_{0}\triangle x$

The same number are now contained in the space whose width is $\triangle x+\triangle s$ The density has changed to

$n=\frac{n_{0}\triangle x}{\triangle x+\triangle s}=n_{0}(1-\frac{\triangle s}{\triangle x})$ the change is very small.

We assume that the positive ions do not move appreciably because of much of inertia so their density remains $n_{0}$

Each electron carries a charge -q so the average charge density at any point is given by $\rho=-(n-n_{0})q=n_{0}q\frac{ds}{dx}$

the charge density is related to the electric field by maxwell's equations $\nabla\cdot E=\frac{\rho}{\varepsilon_{0}}$

If the problem is indeed one dimensional and there are no other fields but the one due to the displacements of the electrons ,the electric field has a single component along one direction i.e x direction.

$\frac{dE_{x}}{dx}=\frac{n_{0}q}{\epsilon_{0}}\frac{ds}{dx}$

$E_{x}=\frac{n_{0}q}{\epsilon_{0}}s+K$

E=0 when s=0 ; K=0

Force on an electron in the displaced position is

$F_{x}=-\frac{n_{0}q^{2}}{\epsilon_{0}}s=m\frac{d^{2}s}{dt^{2}}$

frequency of oscillations is $f=\frac{1}{2\pi}\sqrt{\frac{n_{0}q^{2}}{\epsilon_{0}m}}=897812.3614Hz$

The frequency of electron oscillation in plasma is given by: $f_p=\cfrac{1}{2\pi} \sqrt{\cfrac{Ne^2}{\epsilon_0m}}$ where,

• $N = 10^{10}$ , the number density
• $e = 1.602176565\times 10^{-19} C$ , the electron charge
• $\epsilon_0= 8.854187818\times 10^{-12} Fm^{-1}$ , the permittivity of space
• $m = 9.109382910\times 10^{-31}kg$ , the electron mass.

Putting in the numbers, $f_p = \boxed{898442.55} Hz$ .