https://en.wikipedia.org/wiki/Electrostatic_deflection

Obviously CRTs don't contain perpetual motion machines. So there must be some way in which energy is lost in the plates.

(Note that when I say it's the same phenomenon, I'm talking about a force acting on charged particles. The initial direction of the particles is different, though. In this dipole drive, the charged particles are initially moving roughly perpendicular to the plates, while in a CRT the particles are moving parallel to the plates. But that shouldn't be relevant to the energy needed to accelerate them.)

Also, when the electrons are extracted from the cathode it gets a small positive charge, but this charge also circulate and is not accumulate. Moreover, it's the charge that eventually neutralize the charge of the electrons.

I think there are no special wires to make this happen, the current I very small and the charge is redistributed by the supporting parts.

A good comparison would be to use a CRT where the screen is split in two perfectly isolated parts. After a while the half you are sending the electrons two become too charged and it deflects the electrons.

I am NOT talking about the main electron gun. I am talking about the deflection plates.

Just to be perfectly clear, a CRT has three active components:

1. Electron gun to create a beam of electrons aimed in the direction of the screen.

2. Vertical deflection to move the beam up and down.

3. Horizontal deflection to move the beam side to side.

Here, you are talking about #1, and I'm talking about #2/#3.

If initially the position of the beam is in the center of the path , a charged #2 and #3 will change the direction and in a smaller amount the velocity of the electrons. The energy to change the velocity comes from the energy in the electric field, not from the plates.

You can say that this energy is energy of the electric field or that it is potential energy of the electrons in the electron beam. Both descriptions are accurate and give the same result. Just remember to not count it twice.

Now the question is where does the energy from the field or the potential energy of the electrons came from. The answer is that the repulsion has a small but not 0 effect outside the main part of the capacitor and this creates an electric field outside. To align the beam at the center you must overcome this small force, so you must add some energy to the electron (in this case it's easy to think in the potential energy of the electron). Perhaps the comment from ballenarosada is more clear https://news.ycombinator.com/item?id=17432975

In a normal CRT the electron bean is created from a high voltage source so it moves fast, and the electrons initially move inside wires, so you usually ignore the small potential field of the capacitor far away. If the charges move slowly and freely then you must consider the effect of the electric field outside the capacitor.

So you imagine that there is also an imaginary starships moving at a constant velocity in a constant direction. And you put an imaginary scientist doing imaginary measurements from the starship. A more serious name for the "imaginary starship" is "reference frame". All the reference frames that move at a constant speed are equivalent, they have the same laws of physics, so you can choose any of them to make the calculations.

In this case the calculations are more easy if you pick a reference frame that moves with the same speed and some direction that the solar wind (at least locally). It's only a mathematical trick, you imagine that there is an imaginary starship there just to do the calculations.

So if this device really works, it must work when you are measuring it from a imaginary starship that is moving at the same speed of the solar wind. You can't avoid the imagination. Also, all the other reference frames give the same result, but the calculation is much harder.

The advantage of using the reference frame that moves at the same speed of the solar wind is that in that reference frame the speed of the solar wind is 0. It doesn't mean that the particles doesn't move or don't interact, just that they don't move in any preferred direction. So the particles may enter the device from any direction.

If your initial setup is that the device is moving with some speed/direction that is not equal to the speed/direction of the solar wind, then in the reference frame we choose the device is initially moving in some direction with some speed.

Now, when you turn the device on, there is some kinetic energy because the device is moving and there is some energy in the electric field because the capacitor is charged. With the description in the site, the device must work even if it were a static capacitor with a fixed charge. So after a while, the ions around the device change their position and the device may start to move, but there is a maximal speed change. Initially the acceleration may be big, but then it starts to fade away. The initial energy is redistributed, part of the energy in the electric field is transformed in kinetic energy of the device, but there is only a finite amount.

When you see it from the usual reference frame, for example the one that is fixed to the Sun) the calculations are more complicated because now you have to consider the solar wind, and it's difficult to avoid some infinite amounts in the calculation of the energy. But the result must be the same that the result in the other reference frame. The device changes the speed a little and then the acceleration goes to 0.