# Maxwell's equations & Conductors

Is the fact that there could be no E-field inside a conductor purely experimental? I don't see any way to apply Maxwell's equations to prove this fact.

Dick
Homework Helper
If there is an E field in a region with mobile charge carriers then there will be a current. So after all currents have died away, either the E field must be cancelled or the medium will have run out of charge carriers. Does that help?

marcusl
Gold Member
Zero E field is an idealization that holds for perfect conductivity where exactly enough charges can move to the surface to screen out an ambient field. This is actually seen in superconductors, although related phenomena like the Meissner effect require quantum mechanics to explain. For ordinary metals of finite conductivity, a small DC field can exist inside according to Ohm's Law
$$\vec{J}=\sigma\vec{E}$$

Time varying fields, on the other hand, are screened by circulating eddy currents, and penetrate a small distance characterized by the skin depth. This is covered under discussions of wave propagation in conducting media, in upper class E&M books like Schwartz or Reitz and Milford. Here's a link that contains the derivations in sections 6.19-6.20
"www.sp.phy.cam.ac.uk/teaching/em/waves.pdf"[/URL]

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If there is an E field in a region with mobile charge carriers then there will be a current. So after all currents have died away, either the E field must be cancelled or the medium will have run out of charge carriers. Does that help?
Yeah, this makes sense. Eventhough in order to prove that there is no E-field inside the conductor we are assuming that there is no current inside the conductor, the assumption of there being no current is more pleasing.

BTW, whenever we say "inside" a conductor (in the context of there being no E-field), do we always refer to the region where the conductor is solid?

Dick
Homework Helper
If you mean inside a void in a conductor - sure, E can be non-zero in there.

If you mean inside a void in a conductor - sure, E can be non-zero in there.
No, I meant to say that in general when we say "There is no E-field inside a conductor," what does "inside" actually mean?

For example, consider spherical conductor shell. It hass two regions -- the region between the inner and outer surface of the conductor which is solid and the region between the center of the conductor and the inner surface which hollow. Which region would you call the "inside" of the conductor?

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Dick
Homework Helper
I would call 'inside the conductor' inside of the conducting shell.

I would call 'inside the conductor' inside of the conducting shell.

I am not sure which region you are referring to... the hollow region or the solid region?

Dick
Homework Helper
The solid region - where the mobile charges are.

So, in general, the E-field in the solid region of a conductor would always be zero (for perfect conductors of course), right?

So, in general, the E-field in the solid region of a conductor would always be zero (for perfect conductors of course), right?

Anyone? ........

So, in general, the E-field in the solid region of a conductor would always be zero (for perfect conductors of course), right?

Yes, it would always be zero. Just keep in mind that this, like much of classical physics, is an idealization.