Electric field inside a conductor

[csdev]

Suppose there is a metal disk with an initial excess of charge represented by the four red electrons. Naturally, they reach their equilibrium position on the surface. After this, a new electron (the green one) is added exactly at the center. The repelling forces from the red charges cancel each other out, so the green charge remains there and produces an electrostatic configuration with $q_{in} \neq 0$. Is this all possible?

 

[mfb]

There is no "exactly at the center". In classical mechanics you could ask that as hypothetical question, but in quantum mechanics it does not work at all.
You'll get a smooth charge distribution over the whole surface of the object, with most of the excess charge at the rim.

 

[jtbell]

This is an example of unstable equilibrium. The tiniest perturbation in the position of any of the charges will unbalance the forces between them and set them in motion. A somewhat analogous situation, purely mechanical, is a perfect, rigid sphere balanced on top of another perfect, rigid sphere.

And as mfb noted, in QM there is no way to ensure that any electron is exactly at any specified position.

 

[csdev]

In classical mechanics you could ask that as hypothetical question

Indeed, before studying quantum mechanics I'm reviewing the contents of classical mechanics, so my question is related to it.

in quantum mechanics it does not work at all.
You'll get a smooth charge distribution over the whole surface of the object, with most of the excess charge at the rim.

I don't get this, does it mean in QM there are no point-charges?

This is an example of unstable equilibrium. The tiniest perturbation in the position of any of the charges will unbalance the forces between them and set them in motion. A somewhat analogous situation, purely mechanical, is a perfect, rigid sphere balanced on top of another perfect, rigid sphere.

And as mfb noted, in QM there is no way to ensure that any electron is exactly at any specified position.

Ok, it's unstable and unlikely but not impossible, disregarding for a moment QM. Right now I'm studying classical physics and I'm trying to fight the statement that inside a conductor in electrostatic $q_{int}=0\Leftrightarrow E=0$ (of course this "fight" is just intended to make me feel more comfortable with the theory, it's not like I'm challenging it)

 

[mfb]

I don't get this, does it mean in QM there are no point-charges?

There are no point-charges, right. Only good approximations to that, but approximations don't help if you have an unstable equilibrium.

Ok, it's unstable and unlikely but not impossible, disregarding for a moment QM.

Well, it is impossible to get such a situation. No matter how precise you are in placing the charge, the chance to hit the exact center is zero. There is also no perfect disk in reality, as a disk is made out of atoms.

q_int=0 and E=0 are conditions for ideal equilibrium positions in electrostatics - real conductors can have some electric fields and charge concentrations inside.

 

[jwinter]

There can be no static field inside a conductor because if there was then a current would immediately flow and neutralize it. Your entire excess charge (which has now increased to 5 electrons) must be situated on the outer edge. However the electrons to not have to take up discrete positions as you have shown, but rather the entire sea of electrons can displace slightly so that the excess charge is evenly distributed wherever it wants to go - which is the outer edge in this case.

If you have a slightly overfull (charged) bowl of water which has a convex meniscus around its edge, and then add another drop of water in the middle - it doesn't stay piled up there does it, neither does it go to one place on the edge! It joins in with all the other drops to still give a perfectly flat surface on the top and a slightly higher but perfectly even meniscus all around the edge. Charge on a conductor works in a very similar fashion.

 

[csdev]

qint=0 and E=0 are conditions for ideal equilibrium positions in electrostatics - real conductors can have some electric fields and charge concentrations inside.

That's exactly the reason why I'm asking this. I agree a situation like the one I described is confined to an ideal world, but Physics, as I understand it (as it is explained in textbooks), is developed around such ideal scenarios, so I was expecting this particular one to be somehow included into classical physics.

 

[mfb]

Well, don't make it too ideal. As soon as you add any element of realism (atoms, electrons in the disk, quantum mechanics, finite resolution of placement, random stray fields from the environment - one of them is sufficient) the odd situation does not appear any more. You don't even have to know how exactly those elements work, it is sufficient to look at the result: things in an unstable equilibrium don't stay there.

 

[csdev]

However the electrons to not have to take up discrete positions as you have shown, but rather the entire sea of electrons can displace slightly so that the excess charge is evenly distributed wherever it wants to go - which is the outer edge in this case.

If you have a slightly overfull (charged) bowl of water which has a convex meniscus around its edge, and then add another drop of water in the middle - it doesn't stay piled up there does it, neither does it go to one place on the edge! It joins in with all the other drops to still give a perfectly flat surface on the top and a slightly higher but perfectly even meniscus all around the edge. Charge on a conductor works in a very similar fashion.

Ok I guess this makes a good intuitive explanation of the QM phenomenon initially pointed out by mfb, thank you. I think one should have this idea clear in mind before studying electromagnetism, to understand that point charges are just an approximation. So for instance, how do you apply Coulomb's Law, which is defined for point charges, to this? Do you consider a charge distribution concentrated upon a point in an analogous manner to the center of mass for mechanics?

 

[Nugatory]

That's exactly the reason why I'm asking this. I agree a situation like the one I described is confined to an ideal world, but Physics, as I understand it (as it is explained in textbooks), is developed around such ideal scenarios, so I was expecting this particular one to be somehow included into classical physics.

The classical physics explanation says:
1) if you could set up that absolutely perfect equilibrium, with your ideal classical electron at exactly the ideal classical center point of the disk... because it is perfectly symmetrical in this idealized situation the central electron would never move because the symmetry says that the forces on it are always balanced... Then the conduction electrons in the conductor will move in response to the electrical field produced by the central electron. They will be repelled, moving towards the perimeter of the disk, and this will cause a buildup of positive charge in a spherical shell around the central electron. This positive charge will exactly counteract the charge of the central electron, leading to a zero net electric field around it.
2) In practice we cannot set up the ideal equilibrium of #1, but even if we could, it would be unstable. There will always be some small asymmetry, and this must lead to some small amount of net force on the central electron that will move it off-center - and once moved off-center it's just another conduction electron moving freely in response to any electrical field it encounters.
3) The end result of #1 and #2 is the same: Equilibrium with no net electric field within the conductor, and no net charge anywhere except at the surface. the only difference is that in the highly idealized #1 scenario, we have a single electron that because of its unique position at the ideal center doesn't move as the final equilibrium is reached. The more you think about it, the less interesting and physically relevant this difference becomes, which is why you don't see any of this discussed in intro texts.

You will, however, see an equivalent and much more physically realistic problem discussed: We glue an electric charge onto the end of a stick, and then hold it against the center of the disk. This gives us an immobile central charge without having to worry about whether it's immobile only because of an assumed unstable equilibrium... And you still get zero field within the conductive disk, and net charge only at the perimeter.

 

[Nugatory]

So for instance, how do you apply Coulomb's Law, which is defined for point charges, to this? Do you consider a charge distribution concentrated upon a point in an analogous manner to the center of mass for mechanics?

Conductors are tricky because the charge distribution is not constant - the charge carriers move around in response to any applied electrical field (and once moving, they both generate and respond to magnetic fields - any non-trivial situation will become a problem in electrodynamics and Maxwell's equations instead of just electrostatics).

However, an insulator can maintain a static charge distribution, and then Coulomb's law takes on its integral form: $\vec{F}=k\int{\frac{q(\vec{r})}{r^2}}d^3V$ where $q(\vec{r})$ is the static charge distribution.

 

[csdev]

You will, however, see an equivalent and much more physically realistic problem discussed: We glue an electric charge onto the end of a stick, and then hold it against the center of the disk. This gives us an immobile central charge without having to worry about whether it's immobile only because of an assumed unstable equilibrium... And you still get zero field within the conductive disk, and net charge only at the perimeter.

While the glued charge stick is well depicting the situation I had in mind, I don't think the law $E=0$ for electrostatics actually contemplates such situations where the structure of a conductor is this dramatically altered. For instance I could glue not just a single charge but an amount so large the disk becomes unable to neutralize it, thus having $q_{int}\neq 0$. However this doesen't apply to your point 1), since one cannot have that special equilibrium with more than one charge in the center. Thank you!

 

[Nugatory]

I don't think the law $E=0$ for electrostatics actually contemplates such situations where the structure of a conductor is this dramatically altered.

That's right - the classical idealization of a conductor assumes that we have an unlimited number of conduction elements available to move around (leaving an equally infinite amount of positive charge behind so that the net charge is finite and constant).

That's an idealization, but it's a pretty good one... It is an entertaining exercise to calculate the coulomb force that would result if you were to take all the electrons in a one-gram sample of some metal, move them one centimeter away. We are in no danger of running out of mobile charges to neutralize any reasonable applied electrical field.

 

[jwinter]

...So for instance, how do you apply Coulomb's Law, which is defined for point charges, to this?

Coulomb's law works just fine for surface and volume charge distributions - simply integrate over the surface or volume.

Do you consider a charge distribution concentrated upon a point in an analogous manner to the center of mass for mechanics?

I think so, but your "charge distribution" concentrated on "a point" is kind of contradictory don't you think?

...We glue an electric charge onto the end of a stick, and then hold it against the center of the disk. This gives us an immobile central charge without having to worry about whether it's immobile only because of an assumed unstable equilibrium... And you still get zero field within the conductive disk, and net charge only at the perimeter.

Yes you always get zero field within the conductor (otherwise charges would move in response to the field and neutralise it).

But in this case you would not only get charge imbalance at the perimeter. Rather some opposite charge would be also be displaced slightly from the sea of charges to appear on the surface of the disk very close to the held charge in an attempt to neutralize it (calculable by the method of image charges).

If the charge was held so that it actually touched the disk, then an equal and opposite charge would be drawn from the disk to touch and exactly cancel the charge being held against the disk.

 

[mfb]

For instance I could glue not just a single charge but an amount so large the disk becomes unable to neutralize it

The disk would immediately disintegrate (pun avoided).