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2) Plasma oscillations

Imagine having a homogeneous distribution of positive and negative charges (electrons and protons or nuclei), with charges +e and -e and masses M and m respectively. M is at least 2000 times larger than m, so we can safely say that the protons stay fixed in a homogeneous distribution, while the electron gas is free to move.

Suppose we have a rectangular box of plasma with equal density of negative (electrons) and positive (ions) charges. We take all the electrons and move them to the right. The situation now is the following

The positive (left) and the negative (right) regions will attract each other. This will lead to the electrons moving to the left. If one thinks of a harmonic oscillator (although we have not proven that this is one!), one expects the electrons to continue moving to the left, because of their inertia, until the above figure is reversed

This will reverse the situation, with the electrons moving to the right, and so on!

Now that we know what we expect, we can use simple math to actually derive it. Ready to be young plasma physicists?

Let us assume that the charge density of the plasma is “n”, and the dislocation distance is “x”. The electric field (remember the parallel plate capacitor) is.

The force is proportional to the distance, so this IS a harmonic oscillator (our intuition was right)!

By using Newton’s second law for the motion of the electrons, we get , which describes an oscillation with frequency . This is called Plasma Frequency.

Wasn’t hard, right?

An important note: Obviously real plasmas are not rectangular boxes that we can manipulate so easily! In that case, what we will have will be an area of increased electron density. They will repel each other, leading to their separation. Due to their inertia they will leave behind an area of small electron density (net positive charge), so they will be then pulled back together.

Do you know any waves that exhibit such a behavior, producing regions of high and low density?

Sound Waves!!!

Sound waves are longitudinal waves, unlike the transverse electromagnetic waves in vacuum!

WHAT? We have the laws of electromagnetism leading to waves different than the good-old vacuum plane-waves!

We see that plasmas increase the complexity of wave phenomena.

Are you intrigued? Do you want to know what other types of waves can occur in plasmas?

If yes, the next session is for you!