# Why are there no charges inside a conducting sphere?

Why is it necessarily true that all charges occupy themselves only on the surface of a conducting sphere, and not anywhere inside the sphere? One argument is that if a charge were to be inside a conducting sphere, then it would exert forces on other particles inside the sphere and there would be internal currents.

Now, my question is - do we have experimental evidence in every case that there are no currents or magnetic fields generated when a charged conducting sphere is held stationary with respect to another body? Or is our reason for believing that there are no charges inside the sphere of a more mathematical and theoretical nature?

I was discussing this with someone and they brought up Gauss' Law, but it seemed to me that the law is predicated on there not being any $E$ field inside the conductor for there to be no charges, which seemed like a somewhat circular argument. A counterargument was that the charges inside a conductor may exert forces, but it may not always end up producing a steady current flow.

I admit my question is of a very qualitative nature, but what are some strong reasons for why we posit that there can be no charges inside a conducting sphere?

• Well there have been several experiments and the validation of the inverse square law of Coulomb's Law depends on the fact that there is no charge inside a conducting sphere Commented Mar 6, 2018 at 14:44

The Gauss's law argument is as follows:

1) We know that there cannot be an E field inside the conductor, because if there was a net E field inside the conductor, then it would move charges, and the staticity assumption would break.

2) Now, assume that, in some region of the conductor, we have a net charge accumulated in some region.

3) Then, we can enclose that net charge in a Gaussian surface, and necessarily, it will have to obey $\oint {\vec E}\cdot {\vec dA} = q/\epsilon_{0}$. Since the RHS is nonzero, the LHS has to be nonzero, therefore, we have a net E field. We have arrived at a contradiction, so therefore, our assumptiont hat we can accumulate a net charge in the interior of the conductor must be false.

• If it is a net charge in only a region of the conductor, couldn’t there be a net charge in another region that cancels the net charge of the whole conductor? In other words, could there be some regions of the conductor where the net electric field is nonzero but it is zero for the whole conductor? Thank you Commented Aug 26, 2022 at 19:03
• @JosephSanders: this would have to be true for every gaussian surface that you could draw. If you're comfortable with vector calculus, it can be shown that Gauss's law is equivalent to the differential equation ${\vec \nabla} \cdot{\vec E} = \rho/\epsilon_0$, and that means "if $E$ is zero over any region larger than a single point, then the charge density $\rho$ has to be nonzero in that region. Commented Aug 26, 2022 at 20:09
• Thank you so much! Commented Aug 26, 2022 at 20:34

In electrostatics, we generally assume our conductors to be ideal. This indirectly assumes that charges have free mobility inside the conductor. You must remember that a system is more stable the lower its energy is. A system of free charges always tries to assume a configuration in which its potential is the lowest. (This configuration is achieved because of the interactions between the charges.) The lowest potential energy for a charge configuration inside a conductor is always the one where the charge is uniformly distributed over its surface. This is why we can assume that there are no charges inside a conducting sphere.

Also, the electric field inside a conductor is zero. (This, also, is because of the free movement of charges. If there was a net electric field inside, the charges would rearrange because of it, and cancel it out.) Using Gauss' law on a spherical surface that has the same center as your conductor, (but a radius that is smaller by an infinitesimal amount) you can conclude that the net charge inside the conductor is zero. Therefore, all the charge has to lie on the surface of the conductor. (As this is the only part of the conductor outside your Gaussian surface.)

• "The lowest potential energy for a charge configuration inside a conductor is always the one where the charge is uniformly distributed over its surface. " For non-spherical conductors, that is true only for some definitions of "uniformly". Commented Mar 6, 2018 at 16:41