# Conductors in Electric Fields

Berko
Every book seems that it is a no-brainer that electric fields are zero in a conductor, but I do not understand this.

First of all why assume it is zero just because, by contradiction, any electric field will produce acceleration in charges which cannot happen when there is electrostatic equilibrium. Perhaps there is equilibrium because all the charges have moved and there is no free charge that can move.

Second, how exactly does the negative charge go to the one side of the conductor and the positive charge to the other side? The negative charge are the free electrons (which can move) while the positive charge are the lattice ions (which cannot).

m1ke_
Every book seems that it is a no-brainer that electric fields are zero in a conductor, but I do not understand this.

First of all why assume it is zero just because, by contradiction, any electric field will produce acceleration in charges which cannot happen when there is electrostatic equilibrium. Perhaps there is equilibrium because all the charges have moved and there is no free charge that can move.

Second, how exactly does the negative charge go to the one side of the conductor and the positive charge to the other side? The negative charge are the free electrons (which can move) while the positive charge are the lattice ions (which cannot).

At the instant the conductor is in an electric field, there is a net field inside. However, the electrons will move to their respective side and we can approximate the opposite side as mostly positive. The electrons will distribute in such a way that if we were to now ignore the external electric field and just regard the conductor now as a dipole, the electric field vectors will be equal in magnitude but in opposite direction as that of the external field. Applying superposition principle now, the net field inside is zero, and the system is NOW in electrostatic equilibrium.

Hope this helps

yungman
Or in another light, electrons are free to move inside the conductor. Suppose we have an electric field exist inside a conductor, electrons inside the conductor will immediate move in the direction according to the polarity of the field. When electrons move towards the field, it will generate an opposite field. Electrons will keep move until every bit of the field inside the conductor is canceled out. So there will be no field inside a conductor.

Just another way to say the same thing.

lalbatros
Berko,

I like your question very much, even though it is very unrealistic.
Many other phenoma could could come into play before this limitation should be considered.
For example, charges can leave a conductor, if the field is high enough.

You could evaluate a typical density of available charge carriers in a piece of metal, and calcualte from this the external field that would move all these charge and bring you to the limit os the standard theory for conductors. The standard theory of conductors, assumes (correctly) that the available charge has no limit.

It could also be interresting to try to calculate the charge distribution and the field penetration when the amount of charge carriers is limited. It might even not be so obvious to translate this problem in clear mathematical terms. It would imply making a difference between the positively-charged parts and the negaively-charged part on the surface of the conductor, and putting then a constraint on the total charge on the positive (or negative) side. The maximum charge would then also depend on the bulk volume of the conductor. I guess that the charge would still accumulate on the surface, but this should be checked and proved again.

I like this problem because I am curious to know if it could be translated into an optimization problem. I guess that electrostatics can be derived from a variational principle, even when the amount of charge carriers is constrained.

Deric Boyle
Yes the electron don't toally go and get collected on one side( when in electric field) as the whloe distribution will become absurd (as you've rightly stated that the lattice ions can't drift but this wasn't known at the time of Coulomb thus this approximation was made).
So it is like this - when in electric field the electron distribute themselves on the conductor such that the field isn't affected by its presence, the distribution is such that we would apparently have -ve charge on one side and +ve on another (but not in real sense).
But if you want to calculate the distribution of electrons on the conductor kept in an electric field then it'll go into non-classical realms