Is Gauss' Law Wrong? Evidence & Explanation

[Twukwuw]

undefinedundefinedundefinedSuppose there is a charged HOLLOW CONDUCTOR in an Electric-field-free region. Since there is no electric field acting on that conductor, thus all the electric charges will distribute themselves on the surface, as predicted by Gauss’ law.
Gauss’ law can be interpreted as this: The charges inside the conductor will distribute themselves in such a way as to leave NO electric field inside the conductor. And, since it is a hollow conductor (and hence no internal resistance), the charges (let’s say, electrons) will be free to move to and reside on the surface.
However, I find a case where the Gauss’ law may be invalid. Let’s see:
1)Suppose there is a spherical hollow conductor, which is NEGATIVELY charged, is placed in a region of no electric field.

2)The situation is, all the electrons (the extra charge carriers that make the conductor negatively charged) will be EVENLY distributed on the surface, leaving no more electron and electric field inside the conductor.

3)Everything seems good so far…

4)Now, let’s have a greater imagination.

5)Suppose that somebody put an extra electron into that conductor somehow, without causing ANY change to the situation in (2), he finds:
The electron he added will NOT MOVE and only remain in the original position, why? Because there is NO electric field, hence it can’t experience any forces! (Gravity is neglected here)

6) At this point we find: Gauss’ law is not obeyed because for a charged hollow conductor all the charges are NOT necessary to reside on the surface!

7)Is the situation in (5) possible ? If it is not possible, why?

 

[Andrew Mason]

However, I find a case where the Gauss’ law may be invalid. Let’s see:
1)Suppose there is a spherical hollow conductor, which is NEGATIVELY charged, is placed in a region of no electric field.

2)The situation is, all the electrons (the extra charge carriers that make the conductor negatively charged) will be EVENLY distributed on the surface, leaving no more electron and electric field inside the conductor.

3)Everything seems good so far…

4)Now, let’s have a greater imagination.

5)Suppose that somebody put an extra electron into that conductor somehow, without causing ANY change to the situation in (2), he finds:
The electron he added will NOT MOVE and only remain in the original position, why? Because there is NO electric field, hence it can’t experience any forces! (Gravity is neglected here)

Is the electron added from the outside (where there is an electric field) or the inside (where there is none)?

6) At this point we find: Gauss’ law is not obeyed because for a charged hollow conductor all the charges are NOT necessary to reside on the surface!

7)Is the situation in (5) possible ? If it is not possible, why?

Your scenario only works if you can add the charge from the inside. How do you do that? If the charge was inside the sphere to begin with, Gauss' law would not give a 0 field inside the sphere (ie. when the charges distributed themselves over the sphere). If the charge was not inside to begin, how did you get it inside without going through the charged sphere?

AM

 

[whozum]

Would there not be an instantaneous field generated from the presence of the new charge, which would regulate and restore the correct formation of charge on the surface so that the internal field is zero?

 

[Twukwuw]

Oh…Oh…I’m sorry.
I think my question is not clear, because all of you had misunderstood the points in my question. Well, I would like to modify it and make it clearer.

Let’s see one more time. Suppose there is a spherical hollow conductor in somewhere a region of ELECTRIC-FIELD-FREE. The conductor is NEGATIVELY charged because it contains EXTRA electron.
My question is: CAN WE PREDICT THE DISTRIBUTION OF ELECTRON?
Note: Here we assume that electron is particle-like.

Here I suggest 2 cases which are possible. Suppose there are 10 extra electrons:
1) Case1: all the 10 electrons (the extra charge carriers that make the conductor negatively charged) will be EVENLY distributed on the surface, leaving no more electron and electric field inside the conductor.
2) Case2: There are ONLY 9 electrons EVENLY residing on the surface, and the
last one is in the centre of the conductor. There can be one electron in the centre because those 9 electrons exert no electric field on that electron. No electric field is created inside the conductor because of the SYMMETRICAL distribution of those 9 electrons.

Note: The 2 cases I mention above are possible because they are STABLE!

My question is: CAN WE PREDICT THE DISTRIBUTION OF ELECTRON?

 

[nishant]

any distribution should be possible which does not disturb the e.f inside the sphere.I think the other way to do this would be to find in which configuration the potential energy of the sphere is minimum.

 

[krab]

Gauss’ law can be interpreted as this: The charges inside the conductor will distribute themselves in such a way as to leave NO electric field inside the conductor.

It seems that you have quite the wrong conception of Gauss' Law. Gauss' Law is only that the net flux in a closed surface is equal to $\epsilon_0$ times the charge enclosed inside it. To infer from this that the electric field inside a hollow metal surface is zero, you need 2 additional assumptions: (1)Charge is continuous, i.e. non-particulate, (2)The metal is a perfect conductor. Neither of these are true in nature. But they are idealizations that work very well over large macroscopic ranges of parameters. If you only have say 10 electrons, just calculate the field based on these 10 as if they are in free space. "No field inside a conductor" is false for this few electron case even if the conductor is perfect. It is not a downfall of Gauss' Law.

Edit: BTW, if you have a perfect conductor, the particulate nature of the electrons disappears, as the electrons in a conduction band are not localized. So asking what is the distribution of electrons makes no sense.

 

[Twukwuw]

Hi, krab!
You claim that:No field inside a conductor is false for this few electron case even if the conductor is perfect. I don't think so!

When the distribution of electrons reaches a STABLE condition inside a hollow spherical conductor (which is negatively charged), it is said to have no more movement of electron.

The prerequisites for the stable condition are:
1) The electric potential at every points inside the conductor must be equal, because the
differences of electric potential will cause delocalized electron to move.
2) Since the potential is uniform everywhere, we conclude that the electric field at every
points inside the conductor must be zero! Why?
Because ELECTRIC FIELD = d(POTENTIAL) / d(POSITION VECTOR) !
3) From Gauss’ law, that is, THE NET ELECTRIC FLUX PASSING THROUGH A
CLOSED SURFACE IS PROPORTIONAL TO THE TOTAL CHARGES
ENCLOSED IN THE CLOSED SURFACE, we conclude:
There is zero electric field inside the conductor!
4)The conclusion can be proved this way, suppose we imagine an arbitrary closed surface
inside the conductor, we will get no electric flux (because of zero electric field) passing through that imagined surface, hence there is no extra charges inside that surface. And, for any larger surface which is at the same time smaller than the conductor, the same thing applies to.
5) Finally we are led to conclude one more thing: All the extra charges( here they are
Electron) MUST reside on the surface!

Here I want to point out something make me skeptical in krab’s saying : "No field inside a conductor" is false for this few electron case even if the conductor is perfect”

In my opinion, no field is inside a perfect conductor seems to be not true. I use the explanation above to support my viewpoint.

Before we continue to get into more detail of my question, we must make sure one thing first.
Assumption: 1) There is a perfect and spherical conductor with extra charges
2) The charges is particle-like
Questions: 1) Can all of the extra charges reside on the surface?
2) Can there be NO electric field once the stable condition is reached?

I also want to point out that: Gauss NEVER assume the charges MUST be CONTINUOUS so that Gauss’ law is valid.
I also give an analogy here to show once the “ charge container” is SYMMETRICL (here it is spherical ) and the charges are UNIFORMLY distribute on the surface, there can be no field inside the container. How should I do that? Easy. When we get into the Earth core , we will (roughly) experience no more gravity field exerted by the Earth crust!

 

[Chi Meson]

First of all, listen to Krab. He knows what he is talking about.

The cancellation of electric fields inside a conductor, as well as the cancellation of gravitational field inside a planet, depend on the perfect geometric location of the particles on the exterior. When considering the earth, such a large macro scale object, the small variational imperfection of distribution (a mountain here, a valley there) are insignificant over the whole.

For there to be zero net electric field inside a closed shell, the charges at the exterior must be allowed to move to the precise location that would geometrically allow total cancelation of the field. For small, real world objects, this is not perfectly possible. THis is not a violation of Gauss' Law since the surface is technically not "Gaussian" (If the surface is a "conductor," then the only way it could be completely closed is if charges could move to any location on the surface with infinitessimal precision. This gets more improbable/impossible as the scale gets smaller).

 

[erickalle]

2) Case2: There are ONLY 9 electrons EVENLY residing on the surface, and the
last one is in the centre of the conductor. There can be one electron in the centre because those 9 electrons exert no electric field on that electron. No electric field is created inside the conductor because of the SYMMETRICAL distribution of those 9 electrons.

Its impossible for either a single or more electrons to reside inside a hollow conducting sphere because of the fact that each electron will produce a positive image charge on the inside wall of the sphere. As an electron cannot possibly be exactly in the centre of the sphere, this electron will be slightly more attracted to its nearest bit of inside surface. As soon as it is going in this direction, the force in this direction will become stronger, until (very soon) the electron(s) will collide with the inside surface.

The charges on the outside of the sphere have indeed no influence on the inside.

All of the above even applies for a bad conducting sphere.

Except for the time when we (somehow) placed an electron forcefully inside the sphere and the time it took for the electron to reach the surface it follows there cannot be a charge inside this sphere.

Eric

 

[Twukwuw]

Actually I am quite blur now...
anyway,thank you all for your replies!