Why is the electric field inside of a conducting sphere zero?

[Lamebert]

Homework Statement

Consider a solid conducting sphere with a radius a and charge Q1 on it. There is a conducting spherical shell concentric to the sphere. The shell has an inner radius b > a, outer radius c and a net charge Q2 on the shell. Denote the charge on the inner surface of the shell by Q′2 and that on the outer surface of the shell by Q′′2.

The Attempt at a Solution

I know the answer: you create a Gaussian surface within the outer conductor shell and somehow the electric field inside of it is zero. I'm trying to figure out why... is a charge supposed to collect on the inner surface that cancels out the charge on the inner sphere? How is that possible if the outer shell already has a charge on it? In order for the inner sphere to be canceled ( a charge of Q), the charge on the inner surface of the shell would have to be -Q, so does the negative charge basically induce a dipole on the outer shell?

 

[jackarms]

No, it's just a simple case of all of the charge building up on the surface, and the inside remains neutral. My teacher always explains it like this: if there was an electric field inside of the conductor, then the field would move charges around. Since the charges aren't moving when the conductor is at equilibrium, then there must be no electric field inside the conductor.

 

[Lamebert]

Then what prevents the charges from moving across the conductor? If there was really no force (i.e. no electric field) then entropy would favor the spread of these electrons across all atoms, not only those on the outside. There has to be some preventative force keeping these electrons on the surface, away from the central negative charge.

 

[jackarms]

Sorry, slightly misread your question. I was talking about why the charges are on the surface of a sphere. In general, the field inside of a conductor is zero if the charges aren't moving. In the case above, the charges are static on the inner and outer surfaces of the shell, so there's no field inside of the conductor. It's true that the charge on the sphere induces the shell, but there's still no field inside of it.

 

[Lamebert]

My question isn't why there is no field inside the conductor. It's why there is no field inside the Gaussian surface, where the outside exists inside the outer conductor shell: why does the inside of the shell surface match the charge on the outside of the sphere, effectively cancelling the electric field? My question is one of fundamentals.

 

[jackarms]

Careful with the wording. Gauss's law involves flux of an electric field through the surface, so the field on the inside doesn't matter -- only the enclosed charge. In your example, the Gaussian surface (I'm assuming it's a sphere) is entirely encased inside of the shell, inside of which (i.e. the shell) there is no field. Since field is zero, flux through the surface is also zero.

EDIT: My wording was a little poor with this line "the Gaussian surface (I'm assuming it's a sphere) is entirely encased inside of the shell." What I mean is that the radius of the sphere lies between the inside and outside radii of the shell such that the surface of the sphere lies within the shell.