Electron trapping by electrostatic force

[Dave G]

I'm curious about electron trapping by electrostatic force.
A simple scenario can be ...
A perforated, conductive, hollow sphere in a vacuum, connected to a negative high voltage source. I want to calculate how much electrons it can hold (total charge of it). The relation between sphere voltage, radius, total charge.

Sphere is perforated because i do not want to hold the electrons 'physically' like a gas although i don't think that is possible for conductive things.

Simple coulomb's law should be enough but I don't know how to consider individual electrons repealing each other. PLEASE HELP ME.

Thanks.

 

[mikeph]

I don't understand what the "negative high voltage" is relative to. Where is the ground?

 

[Dave G]

Alternatively u can imagine the sphere is negatively charged. Same charges suppose to repel each other therefore keeping the electrons in the centre of the sphere. Sphere structure pushing the electrons toward the centre of the sphere. Hope that will clear things up.

 

[rcgldr]

It's my understanding that a hollow charged sphere wouldn't work (the net force at any point within the sphere is zero). Wiki article about a Paul Trap, which is meant to contain ions:

http://en.wikipedia.org/wiki/Paul_trap

 

[K^2]

Earnshaw's Theorem - Basically states that you cannot suspend/trap a charge (or collection of charges) with electrostatic forces alone.

 

[Dave G]

It's my understanding that a hollow charged sphere wouldn't work (the net force at any point within the sphere is zero). Wiki article about a Paul Trap, which is meant to contain ions:

http://en.wikipedia.org/wiki/Paul_trap

Yes actually u r right...hollow sphere was a bad example..consider two parallel plates with same negative charges (ignore the boundaries) trying to trap electrons between them. The field strength should be 0 at the middle plane as both the plates will cancel each other out. So the middle section will be a preferable position for the negatively charged electrons/ions (I'm preferring electrons as they have lower mass, easier to trap). I'm aware of the conventional ion traps but why electrostatic force is not used alone...?

By means trapping ...I meant significant amount of time and density of electrons inside the configuration to do the study on them.

Thx for ur response.

 

[Dave G]

Earnshaw's Theorem - Basically states that you cannot suspend/trap a charge (or collection of charges) with electrostatic forces alone.

Yes that theorem does shed some light ...why simple electrostatic setup is not used.
But the theorem states "stable stationary equilibrium configuration ". In our case they can move as much these charged particles want to but should stay inside the setup for significant amount of time.

Bunch of electrons will act like there is nothing around them is kinda awkward don't u think ..the Coulomb force should act.

Thx

 

[K^2]

If they move in a straight line, they will escape. If they don't, then the particles are accelerating. Accelerated particles radiate and lose energy. After all the energy is lost, what state should the particle end up in? Clearly, it cannot be a stable state within the trap, as that is impossible according to theorem.

 

[Dave G]

The theorem is fine ...I don't have anything against it.
I also understand charged particles will loose their energy as Bremsstrahlung even if they change their directions. But this problem is probably applicable to other ion traps and plasma confinement methods as well.

So like other methods if I have electron/ion sources to compensate the losses then this setup should form dense charged region. If that is the case then I want to calculate the exact numbers without simulation. There probably are other variables and issues ..but any help will be greatly appreciated.

Thx.

 

[K^2]

Define dense. Electrostatic energy grows really, really fast as you add more charges. I doubt you'll do better than electromagnetic ion traps with electrostatic forces alone.

But some increase in density, sure. Simply firing a low energy electron beam at a non-grounded metal plate will result in increased density near the plate.

 

[Dave G]

electrons will repel each other so the cloud will grow fast. Now the external E-field generated from the plates (considering the parallel plate e.g) should act on them causing them to slowdown or push them away from the plates, keeping them in the middle section where E-field strength will be lowest (or 0).

So, if electron cloud growth (electrons moving away from each other) is lower than their natural rate (without external E-field) the confinement method can be called partially successful. The plates/sphere structure is their to apply E-field on the electrons.

And it is obvious that this is not the best method for ion trapping as it is not used. All the confinement methods use magnetic/oscillating E-field/inertia etc but not static e-field (at least not alone) . Earnshaw's theorem states can't keep them in "stable stationary equilibrium" which is fine with me. I just want them to slowdown and spend little more time inside the setup than they usually would do without the setup.

Now i want to do the math considering electron input, plate voltage and bremsstrahlung losses or computer simulation is the only answer.

Thanks.

 

[K^2]

I just want them to slowdown and spend little more time inside the setup than they usually would do without the setup.

That's entirely reasonable, yes. I'm not entirely sure why you want to do it with electrostatics alone, but it could be an interesting math problem. So have fun.

 

[Dave G]

Actually advice on how to do the math was my original question. Can u give me some hints on how to calculate this thing?? so I can begin the fun part. Can't figure it out. Thanks for the replies ..they were helpful.