# Why is the electric field inside a conductor zero in equilibrium?

My textbook says the field inside a conductor must be zero in order for the system to be equilibrium and therefore there must be no excess charge inside.

Their proof:

1) Place a gaussian surface inside the conductor. Since the system is at equilibrium, all points on the surface must have an electric field of zero.

2) Therefore the net flux is zero, implying the charge inside is zero.

3) If there is no charge inside, all excess charge must lie on the surface.

My problem:

"Since the system is at equilibrium, all points on the surface must have an electric field of zero."

Why?

Consider an conductor with no excess charge. To simplify things, say the conductor consists of a 100 free electrons and a 100 ions of charge 1$e^+$. The conductor is neutral and at equilibrium.

Now consider the field on one specific electron. There a total of 99 other electrons and 100 ions. So there would be a net field acting on that specific electron. So even though the system is at equilibrium, the net field on the individual particles is non-zero.

So a system can be at equilibrium, even if the net field inside is non-zero. What am I misunderstanding?

Another example: Consider a system consisting of one electron and one proton. If nothing is holding them back they will attract each other until they touch. After they touch, the force acting on them is still non-zero, yet they wouldn't accelerate.

• The conductor system is macroscopically at equilibrium, not necessarily microscopically. The charges and electrons may move. But the (macroscopic) $\rho$ is constant. Feb 23, 2014 at 18:05
• @Mostafa 1) Could you please give me rigorous definitions of macroscopically at equilibrium and microscopically at equilibrium? 2) How can it not be at "microscopic" equilibrium? Wouldn't the electrons keep moving until it is at equilibrium?
– user41086
Feb 23, 2014 at 18:11
• @user41086: The whole problem is notion of coarse graining. You can have a highly oscilating function (charge density on a metal) that when you 'averages it' (search for convolution), you get a zero or near zero \textbf{slowly} oscilating function. Tipical measurement process that you have on your daily life introduces some kind of averaging process as I'm describing, so it washes out most of the original (microscopic) information when you get the averaged (macroscopic) information. As long you have microscopic information that don't survives averages, it don't show up macroscopically. Mar 7, 2014 at 3:24

As Mostafa says, it is macroscopically at equilibrium, not necessarily microscopically.

There may be one misunderstanding you have, which is about "surface". I will talk about it later.

In my opinion, equilibrium should be understood as no electron moving. It is easily to show that the electric field in conductor is zero. If the electric field is non-zero, then electrons in the conductor will feel it and move, until go to the boundary of the conductor, and then stop there. Hence, the surface will accumulate charge, and finally, the distribution of charge on the surface will make the field zero in the conductor.

Now, let us talk about the surface. If it is possible, I would like to say that the charge(electrons) are on the outside surface, mathematically. The field is zero in the conductor, as well as on the inside surface. But the field on the outside surface is not zero.

However, actually, in physics, this statement is not appropriate in microcosm, "surface" is many atoms layers. The electric field changes continuously in space, and external field is not zero but internal is zero.

In fact, when we talk about macroscopic description, we can treat the surface as a surface in mathematics. Therefore, we should distinguish two sides of surface. It helps us discuss clearly.

Review the proof that you post before.

1) Place a gaussian surface inside the conductor. Since the system is at equilibrium, all points on the surface must have an electric field of zero.

2) Therefore the net flux is zero, implying the charge inside is zero.

3) If there is no charge inside, all excess charge must lie on the surface.

In 1), all points on the inside surface must have an electric field of zero.

In 3), all excess charge must lie on the outside surface.

Here, I take a short summary. You have to distinguish two sides of the surface in mathematics.

In addition, the Gaussian surface is a conception totally in mathematics, it has no thickness, and never goes through an electron or a charge. And, "surface" is different in micro and macro. In micro, surface is alway means a very thin layer including a lot of atoms, unless in macro. Only in macro, you can say inside, outside or on the surface.

Before we discuss deeper, we have to talk about "model". Electromagnetism is a theory to describe macroscopic phenomenon, or a model to show essential properties of electromagnetic macroscopic phenomenon. Actually, in this model, there is no electron, no proton or any other elemental particles, but one thing we have is charge. When we say "an electric particle", we would say "a particle with charge". This is a phenomenological model, we do not care about what charge is and why matter have charge, just care about that charge is a property of some matter. It is enough to build electromagnetism, as Maxwell did.

There is a theorem, that electric field is not stable. If particles interact with each other only by static electric force, then these particles are not force equilibrium, the system is not stable. Sorry that I forget this theorem's name, if someone knows, please edit this anwser, thank you.

I believe, your confusion comes from the contradiction between microcosmic model and macroscopical model.

I will discuss later, but now my computer run out of power. Sorry.

To be continued.

As I said, I think your confusion comes from the contradiction between microcosmic model and macroscopical model. This contradiction confuses not only you, but also everyone, because the contradiction indeed exists.

Before we going on, we must clarify the language that we use. The language of previous discussions when we discussed macroscopical model, as well as some words when we discussed microcosmic model, are based on classical physics. Here, we do not need quantum mechanics in microcosmic model, semiclassical model is enough. "Electron" and "ion" are words uesd in microcosmic model, but "field" is in macroscopical model. The conception, "Equilibrium", is totally based on classical physics. By the way, in semiclassical model, the similar conception is "detailed balance".

Let us go on. I mentioned that a static electric field is not stable, this statement is based on classical physics. If you apply this theorem to microcosmic model, you will find that the conductor is not stable, or equilibrium. Acutally, you found the same conclusion before.

It is a paradox, which is from applying a not appropriate model to describe microcosmic world. It is the origin of quantum mechanics historically, which start from explaining the spectrum of Hydrogen and why it is stable.

As for your second question, you use two different discription ways for the conductor, one is macroscopical and the other one is microscopical. In macroscopical description, there is no ions and electrons, the field in the conductor must be zero. In microscopical description, the conductor contains lots of ions and electrons, and the field in the conductor fluctuate widely, hence the field in the conductor is non-zero. But both models cannot explain the reason why the conductor is stable, or equilibrium. In macroscopical description, we usually treat conductor is rigid, or very hard with a very large modulus of elasticity. In microscopical description, "quantization" guarantees it stable, which does not appear in classical physics.

When we talk about the first question, we use macroscopical description, such as electromagnetism and classical mechanics, and we say "the field inside a conductor must be zero in order for the system to be equilibrium". This statement is alway based on the macroscopical description.

Hence, your confusion just is that you confuse two different descriptions, and forget the statement has its own conditions.

• This is an excellent response. Mar 6, 2014 at 18:47
• "If the electric field is non-zero, then electrons in the conductor will feel it and move, until go to the boundary of the conductor, and then stop there." I'm having trouble understanding this. In a neutral conductor, why doesn't a free electron have a field acting on it? Could you please address the "100 ions 99 other electrons" argument I brought up in question"
– dfg
Mar 6, 2014 at 19:11
• @dfg A free electron could have a force acting on it, but only for a very short time. The electron (and all the other electrons) would move under mutual electrostatic repulsion until they were as far away from each other as possible, then they would stop. Once they stop, there is no net force on them. If that weren't the case, they wouldn't have stopped. Mar 7, 2014 at 2:34
• @dfg I don't understand your 99/100 scenario. You start with 100 electrons and 100 ions in equilibrium, meaning none of the particles has a net force on it. Then we focus attention on one of the electrons. Now you say "there would be a net force" on the electron. But we haven't done anything to the system. We've only focussed attention on one particle. Merely choosing to "look" at one of the electrons will not cause a force. Mar 7, 2014 at 2:42
• @garyp This is what I mean: If you have an electron and you have the field from 99 other electrons acting on it, and the field from 100 ions acting on it. The fields from the other electrons can't cancel out the field from the more ions since there are ions than electrons, meaning a field will act on the electron resulting in force acting on it.
– dfg
Mar 7, 2014 at 4:47

Equilibrium in the sense of this question means there are no net forces on the objects that make up the system: the charges contained in the conductor. Note that we need a model of an ideal conductor here. A neutral ideal conductor is thought of as containing equal large amounts of unbound, infinitely small (not electrons) positive and negative charges.

Consider one of the charges, one that happens to lie on the Gaussian surface. If the field there was non-zero, the charge would experience a force and accelerate: the system is not in equilibrium.

Thus, in equilibrium, there are no forces on charges within the conductor, and the electric field is zero everywhere within the conductor.

• But in the example he gave above, the field on certain electrons are non-zero, yet the system is in equilibrium. Also, I was under the impression that electrons are constantly accelerating and decelerating in conductors, even at equilibrium.
– dfg
Feb 23, 2014 at 18:55
• @dfg If the arrangement of the 199 other charged particles is such that the one in question feels a net force, then it will accelerate. The system is not in equilibrium. The sentence about the system being in equilibrium while one of its charges feels a net force is internally inconsistent. I wondered why he concluded that "there would be a net force". I don't see the line of reasoning that leads to that conclusion. Motion: in a real conductor made out of electrons and protons: yes, there's always some motion. In the idealized world of classical electrostatics, no. Feb 23, 2014 at 19:22
• @garyp I on't see why the force has to be zero for the system to be in equilibrium. Consider a system consisting of one electron and one proton. If nothing is holding them back they will attract each other until they touch. After they touch, the force acting on them is still non-zero, yet they wouldn't accelerate.
– user41086
Mar 6, 2014 at 3:37
• @user41086 True, there will be forces on them in the case you propose. But the particles are no longer accelerating, hence the net force on them will is zero. (Note that real electrons and protons can't "touch". Your picture is a classical model for the situation.) Mar 6, 2014 at 14:01
• @garyp This might be a big part of what I'm misunderstanding - in conductors, are the free electrons bound to nucleuses or are they not attached to anything?
– dfg
Mar 6, 2014 at 19:13

Equilibrium means that there is no net change with time. A glass of water at room temperature is in equilibrium because, even though the molecules are fiddling around, their net movement is zero. Or, in another words, macroscopically, you can't see any overall change. In most practical situations, this means the state of the system after enough time has passed; and it will be reached.

Look at one electron somewhere in the interior of your material. It will see 99 other electrones, and 100 positive charges, so you argue it will be seeing a net charge, so there will be some electric field. But, where would it be pointed at? The distribution of charges + and - is uniform, so the actual net charge will be distributed all around.

Let's push it further. Take away 99 electrons. Now you have one electron surrounded by 100 cations, where would it go? If you are deep inside your solid, the attraction is going to be the same in all directions, so it would wander freely. But, if you put it near the surface, there will be much more attraction towards the centre, and there it will be attracted to. Repeat the process with a few more, and you will see they tend to go towards the centre. But in classical electromagnetism we don't know about electrons and atoms, everything is a continuum: a fluid. In the limit when we have enough electrons so they can be seen as a continuum, they will just be away from the surface.

The point where your argument fails is that, if you do take a gaussian around an specific electron, then you will see flux, but the net charge inside will also not be zero! You are looking at a far too small scale to be able to talk about classical electromagnetism or thermodynamical equilibrium. Make your surface a bit bigger, and then you can think on these terms.

If it were not, it would cause current to flow, and propagation of current involves the dissipation of energy, and this cannot occur without any external sources of energy. Hence, it follows that any charges in the conductor must be located on its surface.

You have an issue in your argument. It's not right to describe the conductor as 100 electrons and 100 ions. These are not ions. Consider ion crystals such as NaCl: that's where the ions are, and this thing's an insulator. The fact that electrons are free doesn't mean that they left the ions in cold.

It's like a cooperative, or a collective farm. All farmers share the tractor, so the tractor is free in a sense that it doesn't belong to own only farmer. But it doesn't mean that each farmer is tractor-less. A farmer has a tractor, not to his own solely, but still has it.

• So according to your analogy electrons are always bound to ions. But I thought free electrons are always moving?
– dfg
Mar 6, 2014 at 19:15
• an electron is not bound to any specific ion, rather those ions are submerged in the sea of electrons. they're not really ions in that regard. like in NaCl chrystal the electron is mostly around Cl, so Na is missing one electron most of the time, so it's called ionic crystal Mar 6, 2014 at 19:29