Excess Charges on Metal: Questions Answered

[acherentia]

This is an excerpt from my physics textbook.

Consider a piece of metal that is electrically isolated from the rest of the universe so that no new charge can jump on or off the metal. If we wait long enough, all mobile charges(electrons) in the metal will come to rest and be in static equilibrium.

Let's now imagine that some of these excess electrons are not on the surface but are in the interior instead. These electrons would each produce and electric field and the resulting electric field would cause other electrons in the metal to move. This result, however, contradicts our assumption that the system is in static equilibrium and means that any excess electrons must be at the surface and the electric field is zero in a metal in equilibrium.

Questions:

1. Why will all the electrons come to rest after a while?

2. Why would the electric field produced by electrons cause other electrons to move?

3. What does it mean to have a metal in equilibrium? Does the author mean electrostatic equilibrium or just zero for the sum of all forces?

 

[Eynstone]

Consider a piece of metal that is electrically isolated from the rest of the universe so that no new charge can jump on or off the metal. If we wait long enough, all mobile charges(electrons) in the metal will come to rest and be in static equilibrium.
Let's now imagine that some of these excess electrons are not on the surface but are in the interior instead. These electrons would each produce and electric field and the resulting electric field would cause other electrons in the metal to move. This result, however, contradicts our assumption that the system is in static equilibrium and means that any excess electrons must be at the surface and the electric field is zero in a metal in equilibrium.

The author could simply have assumed a static equilibrium at the outset (rather than describing how it's established approximately). Lay this aside for a while.
The author is trying to show that charge must stay wholly on the surface a conductor in electrostatic equilibrium. A conductor is, by (the author's seeming )definition, a material where the current density is proportional to the electric field at every point. If there were a point with a nonzero charge density inside the conductor, the density would generate a nonzero electric field. This, in turn, would generate a proportionate current which can't subsist in equilibrium. Therefore, the electric field & the charge density vanishes inside a conductor and all charge must stay on the surface. ( A rigourous mathematical proof requires calculus of variations).
On the assumption - accelerating charges radiate & dissipate energy.This makes moving charges slow down. One may suppose that an equilibrium is attained after a long time when charges have almost stopped moving. Besides, it's no longer customary to speak in terms of 'electrons' in basic electromagnetism It makes one think that charges are 'point-like' & quantized ,which is unwarrented in classical electromagnetism.

 

[acherentia]

Ugh. all this stuff is a little overwhelming for me.

Are you saying that the E=0 inside conductor was first established through math calculations? Or was it known experimentally first? How would you prove that experimentally?

 

[acherentia]

I understand the thread of logic derivations in your argument. At the same time I still have problems understanding how the metal is in equilibrium. Isn't the electron cloud moving at all times? Or is my question not CM related?

That's one other thing...I can't conceptualize a static electron, probably because of how I was taught the Bohr model...

 

[bjacoby]

Ugh. all this stuff is a little overwhelming for me.

Are you saying that the E=0 inside conductor was first established through math calculations? Or was it known experimentally first? How would you prove that experimentally?

The hint that E=0 inside a conductor comes out of Gauss's law. (look it up!) If E = 0 on the Gaussian surface then there should be no charge inside the surface.

However, in physics the true test is always experiment not mathematics (contrary to modern common opinion). Benjamin Franklin first noticed that there was no charge inside a metal can. Michael Faraday extended the observations: "I went into the cube and lived in it, and using lighted candles, electrometers and other tests of electrical states I could not find the least influence upon them". In 1938 Plimton and Lawton carried out a highly sensitive version finding no charge inside a conducting sphere.

Don't be overwhelmed! That thing in front of you is a computer. Use it to look up the things I just talked about and go from there!

 

[kini.Amith]

That's one other thing...I can't conceptualize a static electron, probably because of how I was taught the Bohr model...

The bohr's model is only meant for isolated atoms. When it comes to solids (collection of atoms), it is the band theory which plays the role.Only the electrons in the valance band are free from the grip of the nuclei and can be affected by an electric field.that's why the guy from ur text says "mobile charges(electrons)"
Even When a metal is electrically neutral, there are electrons in the valence band which are free to move about. but in the absence of an external electric field,the electrons move in random directions, so the net current is zero.The"mobile charges(electrons)"are at rest on an average.
In the presence of an electric field , the electrons will be biased to move in a particular direction.then all the crap about drift velocity comes up.the electrons drift in a particular direction according to the electric field and a current is set up.this current flows and charges accumulate on the opposite surfaces of the metal till the electric field inside the metal becomes zero.
BY static equilibrium, the author means no net movement of electrons, that is , no current.
It is wrong to say that the electric field inside a metal is always zero. It is also slightly misleading to say - at static equilibrium, electric field inside a metal is zero- though it is correct . The best way to put it is -
when the electric field inside a metal becomes zero, equilibrium is achieved, for there is nothing to cause the mobile electrons to drift.
So, for all purposes, if you are given that the charges in metal are in static equilibrium, u can understand that the field in it is 0.

 

[acherentia]

you just put an end to my bewilderment. thank you!