# Electric field in a conductor

Is it always true that the electric field in a conductor is zero?

What would happen if I put a very big charge inside an ungrounded hollow conducting sphere like this image? The charges inside the conductor are supposed to rearrange so as to cancel the field created by the big charge. So in that case, the electrons (I believe only electrons can move freely ..?) will move towards the inner surface (because they're attracted by the big positive charge) and it creates a field in the opposite direction that somewhat cancels the existing field. But what happens if there is still a field when all the electrons are on the inner surface ? What if it's not enough to cancel the field created by the big charge?

If there were to exist an electric field in the conductor, there would also have to be a current. A radial field drives a radial current, and it is this very current that will change the charge that is on either surface, which then in turn creates its own electric field. The steady-state is when the TOTAL field is equal to zero, when there are no longer any currents.

The point is: what do you call a "conductor"? If you are talking about a medium which follows Ohm's law $\vec{j} = \sigma \vec{E}$, then $\vec{E}$ has no reason to equal zero (well, in steady state, $\vec{j}=\vec{0}$ so...).

But if you are dealing with a perfect conductor, in which $\sigma \rightarrow +\infty$, then $\vec{E} = \vec{0}$: $$\begin{cases} \vec{j} = \sigma \vec{E} \\\\ p = \vec{j} . \vec{E} \end{cases} \implies p = \sigma.\vec{E}^2$$ Where $p$ is the Joule heating, per unit volume. Then $$\begin{cases} \sigma \rightarrow +\infty \\\\ p \in \mathbb{R}\end{cases} \implies \vec{E} = \vec{0}$$

I hope I've answered your question, but I'm not aware of what you already know and thus may have stated obvious facts.

N.B.: Since I'm French, I may use different notations. Feel free to edit.

Then there would be electric field inside that shell. Well if it is still intact. Because you need an immense positive charge to do that to a typical shell, more than 100000 Coulombs (assuming you have 1 mole of conductor with about 50 electrons per atom). To bring that much charge together requires immense force and energy. In fact potential energy of 10 cm sphere with that much charge is more than $10^{20}$ Joules. It take years for a 1GW nuclear power plant to produce that much energy.

• In a typical real conductor, each atom provides one electron, but your point is taken. – garyp May 7 '15 at 0:32
• Thanks, With that much positive charge, I think they will be ioninzed (it could even move protons). Even with one electron it's still more than 10000 C. Now the energy is about 10^19 J. – Azad May 7 '15 at 4:16

If the charge is too large, it will pull electron-positron pairs out of the vacuum outside its surface, and in effect the charged sphere neutralize itself as the positrons fall to it and the electrons (from the pair creations) will fly apart in an explosion electrons.