# Distribution of charge within a CONDUCTOR

## Main Question or Discussion Point

Can someone please tell me how charge (electrons is distributed inside a conductor with a circular cross section for example? how can charge be UNIformly distributed within it and why is it that the E field inside a conductor should always be zero? what dictates that?

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From experience we know that the current in a conductor is zero when no external EM field.
In this case, if E-field is not zero then there will be a current at the position in a conductor so logically E-field is zero. If there are external EM field then the field will penetrate from the surface to a certain depth so called skin depth, so near surface E-field is not zero but will decrease as getting to inside away from surface. This skin depth is a function of wave length and material conductivity etc.

Born2bwire
Gold Member
Metals are generally organized as a crystal with a band structure of that is essentially continuous. Technically, the allowed energy levels of electrons in the metal lattice is discrete, but in most situations the energy difference between energy levels is smaller than the thermal energy of the electron so the energy levels seem to be continuous. Since the lattice is nothing but a conduction band, the electrons are free to move about without any penalty in terms of the quantum energy levels (unlike say an insulator or semiconductor). The lowest energy state would be one where the charge is evenly distributed because areas with local net charge will experience coulombic repulsion between the like charges.

When you apply an external electric field, the field will create a Lorentz force on the free charges in the conductor. As long as a net field exists in the conductor, the charges will experience this force. Since the charges are free to move, they will move in response to the applied field until they disperse themselves in such a manner that the net field inside the conductor is zero. Once the net field is zero, the charges no longer experience a Lorentz force and just sit around knocking around due to thermal vibrations and such.

So an applied electric field makes a self-correcting secondary field inside the conductor that counters the incident field. If we have an electromagnetic wave, then when the wave impinges upon a conductor, eddy currents are created. These eddy currents, in a perfect electrical conductor, will create secondary electromagnetic waves that perfectly cancel out the incident wave. In an imperfect conductor, these eddy currents are limited by the resistivity of the material and so the secondary waves counter the incident wave imperfectly. This results in the incident wave propagating a small depth into the material. The characteristic depth of propagation is called the skin depth.

If there are no external fields applied, then the number of electrons inside a conductor is the same as the number of protons. For example, there are about 6 x 10^23 electrons AND 6 x 10^23 protons (plus neutrons) in 63.5 grams of copper.

If there are no external fields applied, then the number of electrons inside a conductor is the same as the number of protons. For example, there are about 6 x 10^23 electrons AND 6 x 10^23 protons (plus neutrons) in 63.5 grams of copper.
The number of electrons in a metal is always equal to the no. of protons in an electrically neutral conductor.....irrespective of the electric field.

born2bwire, from ur first paragraph, wat do u mean by local 'net charge'? and does answering this question strictly reuires knowledge of modern phsyics (quantum theory)?

Born2bwire
Gold Member
born2bwire, from ur first paragraph, wat do u mean by local 'net charge'? and does answering this question strictly reuires knowledge of modern phsyics (quantum theory)?
Quantum isn't really necessary but it in a conductor, the charges are free to move about which is not the case in say an insulator. The reason for this is due to a large energy gap between the valence and conduction bands in the inductor and this is based on quantum theoy.

The local net charge means that if you do not have a uniform distribution of charge, then in a given volume of the material you can have a net charge. The entire metal is electrically neutral but you can have small volumes in the metal that are not neutral. Since the charges are free to move about in a metal, any volume of net charge is going to have a net force from coulombic repulsion between the extra charges. In addition, if we have some local net positive charge in one area, there must be another area in the conductor that has a local net negative charge because a metal has a neutral charge (if you have not built a charge up on it artificially). So whenever you get charges building up in one area, the coulomb force will naturally redistribute the charges. So barring the random movement of charges from thermal energy, you would expect the charges to distribute in an even distribution. With a cylindrical conductor... the distribution may be a little different due to the curved boundary but I can say at least that the charge distribution would be axially symmetric.

thanks. i've been thinking about it also I arrived at this: since the electron orbits the nucleus VERY fast one could say the average position of the electron in time is right where the nucleus is so effectively the electron cancels the effect of the positive charge and the electric field produced by it and therefore the electrons effectively distribute just like the nuclei are distributed in the metal, uniformly.

Born2bwire
Gold Member
thanks. i've been thinking about it also I arrived at this: since the electron orbits the nucleus VERY fast one could say the average position of the electron in time is right where the nucleus is so effectively the electron cancels the effect of the positive charge and the electric field produced by it and therefore the electrons effectively distribute just like the nuclei are distributed in the metal, uniformly.
That is a different problem. First, the electrons never average where the nucleus is. If you solve for the wave functions of the orbitals the probability for the electron to be at the nucleus is very small (but generally non-zero). The expectation value, your "average position" as you put it, is not at the nucleus. For example, for hydrogen in the ground state, the "average position" is a spherical shell with a radius equal to the Bohr radius.

What changes here is that atoms are "unhappy" if they have partially filled orbitals. The bonds that are created in bulk material are a means to fill or empty these partially filled orbitals. This can be one by ionic bonds, where a partially filled orbital of one species is emptied and the electrons attach to another species to fill its orbital and the resulting ions are attracted via coulombic force. Or it can be covalent bonds, where the valence electrons are shared in a hybrid orbital amongst the bonded atoms.

In the metal structure, you have a system of covalent bonds. The valence electrons in the metal atoms are shared amongst eachother. However, the hold on the valence electrons in a conductor is "weak." This can be due to, for example, electron shielding. The inner electron cloud can partially shield the positive charge of the nucleus, making it have a weaker coulombic force on the outer valence electrons. If this is weak enough, then it is easy to strip the valence electrons from an atom.

With a chain of covalent bonds linking atoms together and a weak hold on the valence electrons by any given atom in the chain, then it is easy to strip the valence electrons from atom 1 send it to atom 2, strip from atom 2 send to atom 3 and so on. This is when the electrons are moved to the conduction band. In the conduction band, the electrons have enough energy to move from atom to atom in the crystal lattice without being captured back into the valence bands (unless they lose energy due to radiation or phonon interaction).

So if you are asking about how the electrons distribute in a bulk conductor, then that is simple macroscopic classical electrodynamics. They distribute in more or less a uniform manner because they are free to move about and are trying to minimize the overall potential from the coloumbic forces.

If you are asking how they distribute themselves in terms of the orbits and atoms, then this is a quantum mechanics question that pertains to how the bonds are setup and resulting energy bands that the electrons can occupy due to the combination of the atom's own orbitals and the bonds that arise.