# Is the electric field inside a conductor zero a result of Gauss's Law?

In our lecture course on Electromagnetics we were told that for a conductor charge resides on the surface and hence if we apply Gauss's Law to a surface inside the conductor the enclosed charge is zero. Hence, the electric field inside the conductor is zero (which makes sense for the charge on the conductor). However, the lecturer then proceeded to explain that this is an explanation for why Faraday's cages work, i.e. there is no electric field inside a Faraday cage, because the charge enclosed by a Gaussian surface is zero. This doesn't make an intuitive sense, as surely when you apply Gauss you have to take into account superposition. In other words, you only calculate electric field due to the charge enclosed, but there could be a charge outside of the field leading to a net electric field inside the conductor / Faraday's cage.

• The given argument works if you take "charge goes to the surface" as a postulate, but we got there in the first place by arguing that the internal field is zero for the reason Shawshank gives below. – dmckee May 11 '17 at 18:43

## 2 Answers

What you are saying can be also said for a normal conductor whose charge is entirely on the surface. But the point is that if it is a conductor then any external field would cause the movement of electrons in the conductor. These moving electrons will create their own field which will act on the electrons other than the producing one. Now since in the end the movement of electrons has stopped it means no net force acts on it. This can only happen when the net field inside is 0. So actually gauss law tells that the net field inside the metal is 0. It's is analogous to saying that a body rests on a table and gravitational force acts on it , but the body doesn't accelerate , so table must be applying an equal - opposite reaction force !

• You give the (correct) equilibrium argument for zero internal field and then claim that Guass's law is responsible for the conclusion you had already reached. – dmckee May 11 '17 at 18:08
• @dmckee I didn't understand , what is wrong with the explanation ? – Shashaank May 11 '17 at 18:13
• What is wrong is you were done before you mentioned Gauss's law-the equilibrium argument is all you needed-so your conclusion that Gauss's law tells you the field is zero is wrong. The equilibrium argument is what tells your​ that. – dmckee May 11 '17 at 18:19
• @dmckee Do you mean that I should have said that since E inside is 0 therefore net charge inside the metal is 0 by gauss law ? – Shashaank May 11 '17 at 18:26
• To get to "field is zero in the intieror" you don't need to mention Gauss's Law at all. But what you suggest is the next step in showing that the charge goes to the surface. – dmckee May 11 '17 at 18:47

You can say that loosely, but actually, the Gauss's law gives no new information that is not already present in Coulomb's law. You can look into the book of Electrodynamics by David J. Griffith.