Problem with the physics behind charged, isolated conductors

[J Goodrich]

There is something that I do not understand completely with regards to charged, isolated conductors.

Gauss' law implies the following:
"If an excess charge is placed on an isolated conductor, that amount of charge will move entirely to the surface of the conductor. None of the excess charge will be found within the body of the conductor."

So let us suppose that we have a solid, spherical conductor made up of copper.

The previous statement says that if you were to add any quantity of negative charge, say n electrons, that they will disperse to the surface of the material in an attempt to be as distant from one another due to electrostatic repulsion. This makes sense to me, because in a conductive metal the free electrons are allowed to move about. Throughout the volume of the metal there will be the normal amount of electrons associated with the protons in the copper molecules, and the n added electrons will be dispersed across the surface area of the sphere.

However, I don't see how the converse is true: suppose that we begin with a neutral sphere of copper and then remove n electrons. Gauss' law says that there should be no excess charge will be within the conductor. The protons are fixed in place, though, there are no free protons to move about to the surface; the net charge is induced by our removal of mobile electrons, rather than the addition of protons. Therefore, in order for our theory via Gauss' law to hold, that means that there are n less electrons on the surface and still the required amount of electrons within the sphere to make the inner neutral.

My problem with this is that the electrons within the metal, the conductive electrons, are still free to move about and because of their repulsion should still want to be as far apart as possible. How/why do the electrons not act like this; why, in this scenario of electrons removed, does the entire volume of the sphere stay uniformly charged?

 

[willem2]

The electrons are also attracted to the protons. If there are more protons than electrons, the attractive force will be bigger

 

[mordechai9]

I think this may occur because: If you consider a local positive charge in the center of the volume, it has free electrons around it in every direction that will try to move to nullify the charge. Whereas if you have a local positive charge on the boundary of the volume, there are less free electrons available to move in and nullify that charge. Hence the electrons tend to be drawn inwards and the charge imbalance exists concentrated on the surface.

 

[GRDixon]

suppose that we begin with a neutral sphere of copper and then remove n electrons. Gauss' law says that there should be no excess charge will be within the conductor. The protons are fixed in place, though, there are no free protons to move about to the surface; the net charge is induced by our removal of mobile electrons, rather than the addition of protons. Therefore, in order for our theory via Gauss' law to hold, that means that there are n less electrons on the surface and still the required amount of electrons within the sphere to make the inner neutral.

My problem with this is that the electrons within the metal, the conductive electrons, are still free to move about and because of their repulsion should still want to be as far apart as possible. How/why do the electrons not act like this; why, in this scenario of electrons removed, does the entire volume of the sphere stay uniformly charged?

When electrons are removed from the conducting sphere, they are nominally all removed from surface atoms, leaving positively charged copper ions on the surface . When this supply of electrons has been exhausted, no further "positive charging" of the sphere is possible. The best way to understand the behavior of conductors, more generally, is that Ohm's law ensures that the conduction electrons/positive ions are always arranged in such a way that the electric field at interior points is zero. It isn't true that conduction electrons move to the sphere's surface in order to get as far away from one another as possible. Consider an uncharged sphere. The conduction electrons are evenly distributed within the sphere. If you could add extra electrons to a point inside the sphere (say via an insulated wire), they would engender an electric field at points within the sphere, and conduction electrons at such points will move such as to drive the electric field at interior points to zero. By Gauss' law, this indicates that they must all reside on the surface (of any conductor) once electrostatic conditions have been attained.

 

[sardar]

I can understand your confusion about the physics behind charged, isolated conductors. Let me try to explain it in simpler terms.

Gauss' law states that if we have an isolated conductor and we add a certain amount of charge to it, that charge will distribute itself evenly on the surface of the conductor. This is because the charges repel each other and want to be as far apart as possible. In the case of a solid, spherical conductor made of copper, the free electrons will move to the surface of the sphere, leaving the normal amount of electrons within the volume of the metal.

Now, if we were to remove some electrons from the conductor, the situation is a little different. The protons in the copper atoms are fixed in place, so they cannot move to the surface like the free electrons. However, the removal of electrons creates an imbalance in the charge distribution within the conductor. This imbalance creates an electric field within the conductor, which in turn causes the remaining electrons to redistribute themselves within the volume of the metal. This redistribution results in a uniform charge distribution throughout the volume of the conductor, with the net charge being zero.

So, in this scenario, the free electrons are still repelling each other, but they are also affected by the electric field created by the imbalance in charge. This is why the entire volume of the conductor remains uniformly charged, even though some electrons have been removed.

I hope this explanation helps to clarify the physics behind charged, isolated conductors. Keep asking questions and exploring the wonders of physics!