Why must the eletric field inside a conductor equal ZERO?

[Twukwuw]

I was told that at electrostatic equilibrium, the electric field inside a conductor will be ZERO, so that the charges will not move any more.

but why, I argue that, even when the electric field is not zero everywhere inside the conductor, there is still a chance for the charges to be stationary!

For example, imagine that, there is a spherical conductor, all the charges of positive are distributed uniformly over the surface. BUT, there is yet a positive charge, somehow placed in the middle of the sphere . All the charges are definitelt stationary in this case, yet the electric field is NOT ZERO inside the conductor!

So, isn't that this is a CONTRADICTION to the "fact" that the electric field must be zero inside the conductor?

Thanks,
Twukwu.

 

[Doc Al]

For example, imagine that, there is a spherical conductor, all the charges of positive are distributed uniformly over the surface. BUT, there is yet a positive charge, somehow placed in the middle of the sphere . All the charges are definitelt stationary in this case, yet the electric field is NOT ZERO inside the conductor!

How do you place a positive charge in the middle of a conductor (??) without causing the electrons to rearrange themselves?

If the electric field is not zero, then a force will be exerted on the electrons within the conductor causing them to rearrange themselves.

 

[Meir Achuz]

A point charge at the center of a conducting sphere would be in unstable equilibrium. If it is infinitesimally off center, it will shoot to the surface.

 

[Andrew Mason]

I was told that at electrostatic equilibrium, the electric field inside a conductor will be ZERO, so that the charges will not move any more.

but why, I argue that, even when the electric field is not zero everywhere inside the conductor, there is still a chance for the charges to be stationary!

For example, imagine that, there is a spherical conductor, all the charges of positive are distributed uniformly over the surface. BUT, there is yet a positive charge, somehow placed in the middle of the sphere . All the charges are definitelt stationary in this case, yet the electric field is NOT ZERO inside the conductor!

So, isn't that this is a CONTRADICTION to the "fact" that the electric field must be zero inside the conductor?

Thanks,
Twukwu.

The simple answer is that if you increase the positive charge inside the sphere MORE negative charges will move to the inner surface.

How do we know this will happen? Well, let's suppose it did not happen. If there is a non-zero field (potential difference) between the surface and the interior of the conductor, negative charges would keep moving to the inner surface and positives will 'move' to the outer surface (since, by definition, charges are free to move in a conductor). The effect would be to keep reducing the field inside the conductor. This would stop only when the field was 0.

AM

 

[nrqed]

I was told that at electrostatic equilibrium, the electric field inside a conductor will be ZERO, so that the charges will not move any more.

but why, I argue that, even when the electric field is not zero everywhere inside the conductor, there is still a chance for the charges to be stationary!

For example, imagine that, there is a spherical conductor, all the charges of positive are distributed uniformly over the surface. BUT, there is yet a positive charge, somehow placed in the middle of the sphere . All the charges are definitelt stationary in this case, yet the electric field is NOT ZERO inside the conductor!

So, isn't that this is a CONTRADICTION to the "fact" that the electric field must be zero inside the conductor?

Thanks,
Twukwu.

You must think of a conductor as containing a *huge* number of free electrons spread out everywhere inside the conductor. If the E field is not zero at any point, there *will* be some free electrons that will move around. You were thinking of arranging a few charges in a static configuration. But you must not think of a conductor as a just a few charges that you are trying to put in equilibirum.

 

[Twukwuw]

Agree and Disagree

Thanks to you all ladies/guys.

I agree with what Meir Achuz said, that is, if there are some points inside the conductor having NON_ZERO eletric field (or equivalently a non-uniform potential throughout the conductor), then it is NOT AT EQUILIBRIUM. Because a small disturbance will destroy the state of system and the system will reach its TRUE equilibrium after that, in which an infinitesmal disturbance can no longer destroy the system.

Hence, having a NON-ZERO electric field inside a conductor (yet the charges distribution is "static") is POSSIBLE and not contradicted with what we claim: "at equilibrium, must be ZERO E field, for the simple reason, a non-zero-E-field is not a EQUILIRIUM STATE.

Furthermore, from the state of NON-ZERO E field to the equilibrium (ZERO E field) is IRREVERSIBLE, which strongly support the point that the NON_ZERO E field is not an equilibrium state.

I disagree with some of you who said:
We can never achieve this state, because they ask :"how do you place a charge inside a conductor without disturbing all the other charges?"
This is not the point of this discussion!
We are supposed to discuss the characteristic of the state of system, not HOW we achieve that state.

This is like imagine 50 years ago, suppose somebody ask a question:
"would a person die soon if he stay in the MOON due to the tough environment. Then, many people will soon ask, HOW do you get a person to the Mon? NO WAY to achieve this mission!... and so on this kind of NO-POINT discussion.

Nice day,
Twukwuw

 

[JustinLevy]

Actually, it is not necessary for the electric field to be zero in a conductor at electrostatic equilibrium.

The true condition is not that E=0. The requirement is that the force on charges is such that they can not move in response.

Therefore the condition is that the force on charges is zero in the conductor. And for the surface, the force must be perpendicular to the surface.


For instance, even a perfect conductor would have a non-zero electric field inside of it if it is in a gravitational field. And I\'m sure you could come up with some other cases as well. The effects due to gravity are small, and the other cases probably perverse, so this is rarely discussed (although old papers have been published about this).

 

[Meir Achuz]

How to confuse a simple question:
Little green men could hold on to the charge and keep it from moving.

 

[Doc Al]

I agree with what Meir Achuz said, that is, if there are some points inside the conductor having NON_ZERO eletric field (or equivalently a non-uniform potential throughout the conductor), then it is NOT AT EQUILIBRIUM. Because a small disturbance will destroy the state of system and the system will reach its TRUE equilibrium after that, in which an infinitesmal disturbance can no longer destroy the system.

I don't get Meir's point about a charge placed in the center of a conducting sphere being in some kind of unstable equilibrium and then "shooting" to the surface if disturbed. If you put a charge in the center, the charge conductors (free electrons) will immediately begin rearranging to cancel the field until electrostatic equilibrium is restored. The net effect is for an equivalent positive charge to form on the surface.

Hence, having a NON-ZERO electric field inside a conductor (yet the charges distribution is "static") is POSSIBLE and not contradicted with what we claim: "at equilibrium, must be ZERO E field, for the simple reason, a non-zero-E-field is not a EQUILIRIUM STATE.

Huh? If the charge distribution is static, that is electrostatic equilibrium. If there's a field (from that positive charge you introduced) then the charge distribution will certainly not be static. (Realize that charges include the very mobile free electrons as well as your positive charge.)

I disagree with some of you who said:
We can never achieve this state, because they ask :"how do you place a charge inside a conductor without disturbing all the other charges?"
This is not the point of this discussion!

I was just interested in how you were thinking to put a positive charge within a conductor. You should have kept reading. My real answer came in the following sentence.

We are supposed to discuss the characteristic of the state of system, not HOW we achieve that state.

And we did. I think you may have missed it.

This is like imagine 50 years ago, suppose somebody ask a question:
"would a person die soon if he stay in the MOON due to the tough environment. Then, many people will soon ask, HOW do you get a person to the Moon? NO WAY to achieve this mission!... and so on this kind of NO-POINT discussion.

Good one!

 

[Gokul43201]

I agree with Meir's post.

Doc, a charge added to the center of a spherically symmetric charge distribution does not apply any tangential force to the charges on the surface - there will be no rearrangement. You can have an unstable equilibrium, with a "single electron" in the center of a "true" spherical conductor in the absence of perturbations but there is no such thing as a perfect sphere). So this "trivial exception" can not be achieved in any experiment and is of no physical consequence to the science.

 

[Doc Al]

Doc, a charge added to the center of a spherically symmetric charge distribution does not apply any tangential force to the charges on the surface - there will be no rearrangement.

Sure, the center charge exerts no tangential force on the surface charges. But it surely exerts a radial force on the free electrons within the conductor (which is what we are discussing here).

Are you saying (thought experiment, of course) that if one carefully puts a charge at the center of a conducting sphere (originally with no net charge) that the free electrons don't rearrange and the surface charge remains zero until that center charge is displaced from its unstable position? I find that very hard to believe. What prevents the electrons from rearranging themselves?

 

[pmb_phy]

A point charge at the center of a conducting sphere would be in unstable equilibrium. If it is infinitesimally off center, it will shoot to the surface.

Thanks for mentioning that. I knew the formula for the electric field with distance but confused it with potential energy which here has the form of U = -kx².

I appreciate the comment. Thanks Meir Achuz

Pete

ps - I'lll get back sometime with the answer but too busy now.

 

[Gokul43201]

Are you saying (thought experiment, of course) that if one carefully puts a charge at the center of a conducting sphere (originally with no net charge) that the free electrons don't rearrange and the surface charge remains zero until that center charge is displaced from its unstable position?

No, that's not what I'm saying. I'm saying it takes a rearrangement that destroys spherical symmetry in order to have a non-zero net force on the inside electron.

 

[Cyrus]

How do you place a positive charge in the middle of a conductor (??) without causing the electrons to rearrange themselves?

If the electric field is not zero, then a force will be exerted on the electrons within the conductor causing them to rearrange themselves.

I asked this question a long time ago and I think I got this answer. Please correct me if I am wrong!

Assume that you have a sphere of net charge on it. We assume that this is a large amount of charge relative to the surface area, so we can safely assume a contiuum of charge lies on the sphere, with no gaps anywhere. Now the field within this perfect sphere will be zero (Assuming the charge lies only on the surface). Now, we grab a single charged atom with a magic tweezer, and pierce through the surface and stop once we reach the perfect center on this sphere. On the way down, the charge on the surface should change as this charged particle passes through. But, eventually, the charges should dampen out and settle back to equilibrium. At this point the charge is in the perfect center. If we let go of the particle with the tweezers, nothing should happen,...in theory. It should be in metastable equilibrium; however, this is not true in real life. The charges can never be in perfect equilibrium. This is because they are not at absolute zero, so there is *some* motion of the charge on the surface. In effect, the E-field at the center is not *perfectly* zero due to this small thermal motions. So it would be impossible to place, and keep that charge in the exact center.

...that's enough making up physics as I go for one day...

I am going to run, far, far away after this sad pathetic post I just made.

 

[Doc Al]

I don't really care about the motion of charge on the surface--the real action takes place within the material of the conductor. In my view, to understand electrostatic equilibrium you must consider all the charges, both on the surface and within the interior.

I'll try once more. Imagine a solid metal sphere with some net charge. That net charge resides on the surface, but the interior is filled with freely moving electrons that are balanced by the positive charges of the metal lattice. Assume we have electrostatic equilibrium (which is always an approximation, of course). The field everwhere within the metal is zero.

Now imagine that all the electrons in the metal are frozen in position while that single positive charge is placed in the exact center of the sphere. Note that there is now (in our thought experiment) a field within the conductor due to that new positive charge. Unfreeze those electrons. What happens? They immediately rearrange themselves to cancel that field, restoring electrostatic equilibrium once again. The net result is that the surface charge increases by an amount equal to that positive charge. That positive charge placed in the center certainly doesn't make its way to the surface--that could take hours! Electrostatic equilibrium is restored in the merest fraction of a second.

I suppose that if the positive charge were placed exactly in the middle then there would be no force acting on it--but why is that interesting? That has nothing to do with the fact that electrons will immediately begin moving throughout the conductor to make the interior field zero. (The charge "cloud" shifts ever so slightly.)

 

[Cyrus]

Now imagine that all the electrons in the metal are frozen in position while that single positive charge is placed in the exact center of the sphere. Note that there is now (in our thought experiment) a field within the conductor due to that new positive charge. Unfreeze those electrons. What happens? They immediately rearrange themselves to cancel that field, restoring electrostatic equilibrium once again. The net result is that the surface charge increases by an amount equal to that positive charge. That positive charge placed in the center certainly doesn't make its way to the surface--that could take hours! Electrostatic equilibrium is restored in the merest fraction of a second.

I see what you are saying. You always have sage advice

 

[Gokul43201]

I suppose that if the positive charge were placed exactly in the middle then there would be no force acting on it--but why is that interesting? That has nothing to do with the fact that electrons will immediately begin moving throughout the conductor to make the interior field zero. (The charge "cloud" shifts ever so slightly.)

Exactly right. I got sidetracked by the secondary question of the stability of the added charge.

 

[Meir Achuz]

I agree. My statement about the unstable equilibrium does not really answer the original question. It doesn't matter where the charge is placed. If a positive charge were placed exactly in the center, conduction electrons would be drawn to it, neutralizing the area. This would leave a positive charge on the surface without the original positive charge moving.

 

[Twukwuw]

oh...

Thanks to you all ladies/guys!

You have given me a clear picture of what a conductor is and whe logic behind the ZERO-E statement.

At the time I post this dicsussion, I missed one point: there is a large amount of eletron sea residing IN the conductor.
As can be understood, these eletrons experience NO FORCE, yet they can MOVE within the conductor.

Since I forgot the existence of those electrons and their correcponding protons (which are of same amount and reside in the conductor), I used to believe that the proton in the central will be stable if the correct initial condition is given, because, all the PROTON ON THE SURFACE will push the PROTON IN THE CENTRAL and the net force is zero due to symmetry.

Note that my former point is, the electric field no need to be zero EVERYWHERE inside the conductor, but only need to be zero at the points where the "extra" charge is placed.

Now, I realize that, the electric field must be zero EVERYWHERE because there are electrons EVERYWHERE~!

Thanks you all,
Twukwuw.

 

[reilly]

Much ado about nothing, or almost.

First, the surface(s), of a conductor, by definition, must be equipotentials -- actually, sometimes this is used to be the defining property of conductors. Also, there's Earnshaw's Thrm that says, a system of charges subject only to their Coloumb interactions cannot be in stable equilibrium -- the electric potential cannot be a min or max in empty space. Thomson's Thrm says that for conductors fixed in space, charges placed on the conductors will make all the conducting surfaces equipotentials. (All this was figured out more than a century ago, and has yet to be contradicted within the realm of electrostatics and potential theory. Very uncontroversial)

If stuff can move, it will move.

Rather than beat a dead horse, read any text on E&M for details. Freshman physics? See Resnick and Halliday. Old MIT courses see Symthe, Static and Dynamic Electricity. more up-to-date, see Jackson, who gives Thomson's Thrm as a homework problem. For serious Russian physics, see Chapter I of Landau and Lifschitz's Electrodynamics of Continuous Media. for a much older discussion, see Kellogg's Foundations of Potential Theory (Dover, first published in 1929)

For a bit more tricky situation, recall the various image solutions to the problem of finding the potential for a point charge and a conducting sphere or plane or cylinder-- many distinct versions, as Jackson discusses in detail, as does Symthe, and L&L.

With all due respect, if I were still teaching, I'd suggest that you do your homework, and think clearly and precisely about the consequences of free mobility of charges within a conductor, the surfaces of which restrain charges with non-electromagnetic forces, assumed to be very short range, supplied by God knows what in classical potential theory. Free mobility requires constant potential within a conductor. The solution takes about two, maybe three sentences - the same ones used a century ago. Rather than believe me, check out the literature, which is monolithic in agreement about the properties of conductors in potential theory.

Regards,
Reilly Atkinson
(Once a professor,...)

 

[nrqed]

I don't really care about the motion of charge on the surface--the real action takes place within the material of the conductor. In my view, to understand electrostatic equilibrium you must consider all the charges, both on the surface and within the interior.

I'll try once more. Imagine a solid metal sphere with some net charge. That net charge resides on the surface, but the interior is filled with freely moving electrons that are balanced by the positive charges of the metal lattice. Assume we have electrostatic equilibrium (which is always an approximation, of course). The field everwhere within the metal is zero.

Now imagine that all the electrons in the metal are frozen in position while that single positive charge is placed in the exact center of the sphere. Note that there is now (in our thought experiment) a field within the conductor due to that new positive charge. Unfreeze those electrons. What happens? They immediately rearrange themselves to cancel that field, restoring electrostatic equilibrium once again. The net result is that the surface charge increases by an amount equal to that positive charge. That positive charge placed in the center certainly doesn't make its way to the surface--that could take hours! Electrostatic equilibrium is restored in the merest fraction of a second.

I suppose that if the positive charge were placed exactly in the middle then there would be no force acting on it--but why is that interesting? That has nothing to do with the fact that electrons will immediately begin moving throughout the conductor to make the interior field zero. (The charge "cloud" shifts ever so slightly.)

That's the point I was trying to make in my post 5. It felt to me like most people were discussing a few charges on the surface and the charge at the center, ignoring all the other free electrons filling the conductor.

 

[reilly]

In a sense, real conductors with lattices and free electrons are quite different than conductors in standard electrostatics. The positive charges are not free to move, whereas in a "potential theory" conductor, the + charges must be free to move. As I mentioned above, this is a matter that has been well understood, and extensively discussed in the literature for over a century. See, for example, Chapters II and III in the L&L text I mentioned above.

Regards,
Reilly Atkinson

 

[reilly]

Although I've taught E&M many times, I'll admit that I never much thought about putting a + charge at the center of a spherical conductor -- within the constraints of potential theory. We do set up such a possibility with an image solution for the problem of an isolated charge within a conductor -- but the standard solution solution treats the "inside charge" as if it were not subject to the forces from charges within the conductor.

It also makes a difference whether the conductor is grounded, or held at a non-zero potential, or is electrically isolated (see Jackson for details)

Let's look at a related problem, which might help explain the physics more clearly. Consider a conducting sphere with inner and outer concentric surfaces, and with a + charge, Q, in the center of the empty inner spherical volume. According to the physics of potential theory, the charged sphere will have charge -Q, uniformly distributed over the inner surface, for an isolated conductor -- clearly not the case for a grounded conductor or one at a constant potential. Now the sphere has no inside field, nor any field outside -- Gauss's Law. Within a classical potential theory conductor, there are always as many + or - charges as needed to cancel any internal field. So, with a + charge at the center, - charges will flow to neutralize the + charge; that is the conductor behaves as a polarizing medium.

Again, all this is extensively discussed in many texts. While potential theory conductors are really unphysical, they do provide a reasonable approximation to real ones, at least for low frequency fields -- see basic wave-guide theory, for example.

I suspect that an approach to this problem via energetics would provide additional insights -- all that adiabatic stuff.

Regards,
Reilly Atkinson