# Why the electric field inside a metal sphere carrying a charge Q is zero?

I was solving some problem regarding the polarization during this I came across an example 4.5 in Griffith's where it is written that the Electric field inside the metal sphere carrying a charge Q is zero . The metal sphere is surrounded by a dielectric material

Is there any effect of the material which has surrounded the metal sphere due to which it's electric field inside the sphere becomes zero.

Can anyone explain me about these things? I literally confuse among the situations that where the electric field should be zero or not.

## 2 Answers

The important conditions here are that you are considering the interior of a conducting surface with the topology of a spherical shell, at equilibrium. Its spherical symmetry and surroundings turn out to be irrelevant. The spherical shell will be at an electric equipotential. (If it weren’t, the gradient of the potential would make charge redistribute.) The entire interior will have the same value of electric potential (because solutions of Laplace’s equation cannot have local maxima or minima). Any apertures in the shell would invalidate the conclusion.

The Charge Q always stays on the surface of metals because any net charge resides on the surface. So there is no charge density inside the metal sphere. i.e. ρ = 0. So applying Gauss's Law would give you: $E=0$.

In this case the surrounding dielectric doesn't play ANY role for E to be zero inside the metal. It's just one of the basic properties of conductors which are generally metals.

• I would like to know how you go from $\rho = 0$ to $E = 0$. In general there can be an electric field even in regions where there is no charge density, such as in the vicinity of stationary point charge, or in the presence of an electromagnetic wave. One has to be careful when reasoning that there is no field inside a conductor. Jul 2, 2018 at 15:35
• I am completely excluding any fringe electric fields here. For this particular Griffiths's problem that is not required. Now having said that do you think there could be an electric field well inside a "conductor" if the applied electric field is very low? Why do you think so? en.wikipedia.org/wiki/Skin_effect. And what about the Uniqueness theorems? en.wikipedia.org/wiki/Electromagnetism_uniqueness_theorem Jul 2, 2018 at 16:34