Van de Graaff Generator in a Vacuum

[DR13]

So I am in an intro E&M class and the topic of Van de Grraaff generators came up. The instructor said that when there is enough charge on the surface then there is corona discharge. But what would happen if the generator was in a vacuum? Then there would be no corona created. So would the amount of charge on the surface be limited only by the surface area or would some other source intervene?

DR13

 

[Andrew Mason]

So I am in an intro E&M class and the topic of Van de Grraaff generators came up. The instructor said that when there is enough charge on the surface then there is corona discharge. But what would happen if the generator was in a vacuum? Then there would be no corona created. So would the amount of charge on the surface be limited only by the surface area or would some other source intervene?

The surface charge will continue to build up until the external potential difference causes the charges to leave the surface. There may be other factors, but I would think the main limiting factor is the energy required for a charge to overcome the atomic coulomb forces keeping the electrons in the conducting sphere.

In order to achieve a surface discharge, the external electrical force on the electrons on the surface of the charged sphere has to overcome the electrical force (ie atomic forces) keeping the electrons in the conducting sphere. This happens because at the atomic level there is thermal motion of charges into the external electric field. This can cause charges randomly to acquire sufficient energy to overcome the coulomb atomic forces. When that happens, charges will leave the surface and move in the direction of the external electric field (actually, by convention, negative charges - electrons - are accelerated in a direction opposite to the direction of the electric field). In doing so, they will reduce the strength of the electric field. This will tend to stop the flow of charge. So the amount of charge on the surface will be limited by the magnitude of the electric field, the temperature of the surface as well as the binding energy of the electrons to the atoms in the surface metal.

AM

 

[DR13]

So the amount of charge on the surface will be limited by the magnitude of the electric field, the temperature of the surface as well as the binding energy of the electrons to the atoms in the surface metal.

A few questions:
1. If the generator is in a vacuum then how is there a surface temperature?
2. When you say that the surface charge is limited by the electric field I am assuming that you mean the electric field created by the other electrons on the surface. Would this be correct?
3. About the binding energy. Are you saying that once the force created by the electic field from the other electrons gets above the force of the bond between the electrons and the surface then we would see discharge? I just want to make sure I'm getting this.

 

[Fish4Fun]

DR13,

As you say, in a true vacuum there is no temperature, or movement, so how are you operating your Van De Graaff? Sounds to me like you have created a paradox.

If your question involves an electrostatic charge traversing a true vacuum, the obvious answer is, "it can't". But since a "true vacuum" is physically impossible, again we are at a paradox.

So, what is it you want to know, please clarify your question.

Fish

 

[DR13]

DR13,

As you say, in a true vacuum there is no temperature, or movement, so how are you operating your Van De Graaff? Sounds to me like you have created a paradox.

If your question involves an electrostatic charge traversing a true vacuum, the obvious answer is, "it can't". But since a "true vacuum" is physically impossible, again we are at a paradox.

So, what is it you want to know, please clarify your question.

Fish

Good point. I haven't thought of that. Let's say that we are in some distant region of space that is as close to a true vacuum as physically possible.

 

[Andrew Mason]

A few questions:
1. If the generator is in a vacuum then how is there a surface temperature?

?? The surface is made of metal atoms. Why can it not have a temperature? Vacuum tubes use a hot cathode to give the electrons on the negatively charged cathode enough energy to escape the cathode and travel through the vacuum to the anode accelerated by the electric field between cathode and anode.

2. When you say that the surface charge is limited by the electric field I am assuming that you mean the electric field created by the other electrons on the surface. Would this be correct?

The surface charge is limited by the energy of the Van de Graff generator. (It requires more and more energy to add a unit of charge to the metal sphere as the charge on the conducting sphere increases).

3. About the binding energy. Are you saying that once the force created by the electic field from the other electrons gets above the force of the bond between the electrons and the surface then we would see discharge? I just want to make sure I'm getting this.

The surface of the conducting sphere is an equipotential surface. All charges are at the same potential so there can be no potential difference between the charges on the conducting sphere surface.

Suppose there is an grounded electrode placed 1 metre away from the conducting sphere. With negative charges delivered to the conducting sphere by the generator, a potential difference develops between the conducting sphere and the electrode. Suppose that potential reaches 1 million volts. That means that the electric field is 1 million joules/metre - Coulomb (1 million Newtons/Coulomb). That is a very strong electric field. An atom thrust into that electric field by thermal motion of atoms on the surface of the conducting sphere may gain enough energy to overcome the coulomb atomic force and leave the surface. It will then accelerate through the electric field to the electrode. The loss of that charge will reduce the electric field.

AM

 

[cragar]

The surface charge is limited by the energy of the Van de Graff generator. (It requires more and more energy to add a unit of charge to the metal sphere as the charge on the conducting sphere increases).

Why The E field is zero inside? Or i probably don't understand what you mean.

 

[Phrak]

The surface charge will continue to build up until the external potential difference causes the charges to leave the surface. There may be other factors, but I would think the main limiting factor is the energy required for a charge to overcome the atomic coulomb forces keeping the electrons in the conducting sphere.

In order to achieve a surface discharge, the external electrical force on the electrons on the surface of the charged sphere has to overcome the electrical force (ie atomic forces) keeping the electrons in the conducting sphere. This happens because at the atomic level there is thermal motion of charges into the external electric field. This can cause charges randomly to acquire sufficient energy to overcome the coulomb atomic forces. When that happens, charges will leave the surface and move in the direction of the external electric field (actually, by convention, negative charges - electrons - are accelerated in a direction opposite to the direction of the electric field). In doing so, they will reduce the strength of the electric field. This will tend to stop the flow of charge. So the amount of charge on the surface will be limited by the magnitude of the electric field, the temperature of the surface as well as the binding energy of the electrons to the atoms in the surface metal.

AM

This brings up one of the million unanswered questions that I've had floating around for years.

One of the answers I've been given, is that a vacuum can withstand infinite electrical stress without breakdown. I've never felt comfortable with this and assume it derives from extrapolating the dielectric strength of a gas as a function of pressure.

Ideally, the OP question is about a conductor in an ideal vacuum, without particles of any kind in the environment, including light.

Measurements obtained from photo emission would tells us the energy requirement of an electron to overcome the surface potential of any given material. Wikipedia informs me that this can range from a few electron volts to a million.

As I naively conceive of it, if the kinetic energy of an electron as a result of thermal energy of the material is greater than the surface potential, then electrons will be freed from the surface and will be swept away under any external electric potential.

But I'm not happy with this model and wish I were better informed.

 

[Andrew Mason]

...
As I naively conceive of it, if the kinetic energy of an electron as a result of thermal energy of the material is greater than the surface potential, then electrons will be freed from the surface and will be swept away under any external electric potential.

This is true, of course. But that means an electron escaping the surface leaves net positive charge behind, which makes it that much harder for the next charge to leave, so it does not keep going.

But suppose the average kinetic energy is well below the surface potential (ie the potential energy required of a unit of charge to escape the forces that are keeping the charge bound to atoms in the surface). When there is a strong external electric force pulling electrons away from the surface, the thermal motions of the charges in the surface material can cause the electrons to move a small distance into that electric field, thereby acquiring more energy ($\Delta KE = \vec{F}\cdot\vec{d} = qEd$). If the field is strong enough, that additional energy (ie. in addition to the thermal kinetic energy) can be enough to pull a charge away from the surface. If the energy source to the van de graaf can replace that charge and maintain the potential difference, a continuous current can be achieved.

AM

 

[DR13]

I think one thing that I am missing here is the source of the electric field. Is it only created by the electrons on the surface of the generator? If so, then I don't see why the electrons would project off of the surface because there is no reason for a net force to be pointing off of the surface. It would seem like the electrons should just be traveling around the surface of the generator. And on the scale of individual electrons the generator is so large that it could be considered a flat surface. This is another reason that it would not make sense for there to be a force pushing/pulling the electrons off of the surface.

 

[Phrak]

This is true, of course. But that means an electron escaping the surface leaves net positive charge behind, which makes it that much harder for the next charge to leave, so it does not keep going.

But suppose the average kinetic energy is well below the surface potential (ie the potential energy required of a unit of charge to escape the forces that are keeping the charge bound to atoms in the surface). When there is a strong external electric force pulling electrons away from the surface, the thermal motions of the charges in the surface material can cause the electrons to move a small distance into that electric field, thereby acquiring more energy ($\Delta KE = \vec{F}\cdot\vec{d} = qEd$). If the field is strong enough, that additional energy (ie. in addition to the thermal kinetic energy) can be enough to pull a charge away from the surface. If the energy source to the van de graaf can replace that charge and maintain the potential difference, a continuous current can be achieved.
AM

I did consider that model, somewhat unclearly. Later, I've seen that addition of both the surface potential and the potential due to surface charge helps. We end up with a barrier potential at the surface which takes us into the domain of solid state physics with some N-body Fermi fluid in the interior of the conductor with some of the wave leaking out--and something I know nothing about--or into quantum mechanics at the least.

Where r<=R, where R is the radius of the sphere.

$$\Phi(r) = \Phi_{charge}$$

Where r>R

$$\Phi(r) = \Phi_{surface potential} + \frac{r}{R} \Phi_{charge}$$

The height of the potential barrier at R is just $\Phi_{surface potential}$

Compared to the wavelength, the barrier is very wide, so it does appear that the solution approaches that of a classical barrier, and only that fraction of the electrons that overcome the surface potential will escape, independent of the charge placed on the sphere.

However this simple barrier condition uses the assumption that an abrupt change in potential due to charge occurs at the same radius as the surface potential. This shouldn't really happen where free electrons occupy a small thickness below the surface of the conductor.

 

[Andrew Mason]

I am sure I have over-simplified things. I am sure there are complicated quantum effects that apply. But the classical principles on which a vacuum tube operate were well developed before quantum physics was developed.

AM