How do ion thrusters create a force against the spacecraft? I recently saw an old thread, How do reaction engines create a force against the rocket?, get bumped up, and it asks a good question: in a chemical rocket, the fact that the rocket exhaust gets propelled away means that Newton's Third Law requires that there be some force acting on the rocket in the other direction, but the Third Law itself does not actually specify what that force is, with the answer being that it's the pressure of the gas in the combustion chamber and on the engine bell that produces an unbalanced force on the engine.
I'd like to ask exactly the same question, but for an ion thruster instead. As in the chemical rocket, the fact that there's an ion stream going away at high velocity implies that there needs to be a point at which the outgoing ions exert some form of electric force on the thruster. So: what is the nature of this force, and how does it work?
 A: The principle is very simple, though of course actually constructing the things is a lot more complicated.

A propellant gas is ionised between two charged plates. The cations are attracted to the negative plate and repelled by the positive plate and acquire an energy $E = qV$ and a momentum $p = \sqrt{2mE}$. 
The plates, due to the potential they're being held to, also carry an electric charge (which is what attracts and repels the ions), and these charges also feel a (small) unbalanced force coming from the ions, so the plates (and the spaceship they are attached to) acquire an equal and opposite momentum $-p$. So the electrostatic force pushes the cations one way and the spaceship the other.
The electrons acquire an equal kinetic energy but since they are much lighter than the cations their momentum is negligible. It is the cations that propel the spaceship.

The negative plate is a grid, so the majority of the cations fly straight through the grid and out the other side. At this point the electrons collected at the positive plate are recombined with the charged exhaust gas to neutralise it. The neutralised gas feels no electrostatic force so it shoots off with basically the same momentum as it gained when accelerated between the plates.
The end result is simply that the propellant goes one way and the two charged plates, and the spaceship they are attached to, goes the other way.
