# Charge distribution

## Main Question or Discussion Point

A small positively charged sphere is lowered by a nonconducting threead into a grounded metal cup without touching the inside surface of the cup.The grounding wire attached to the outside surface is disconnected and the charged sphere is then removed from cup.What is the distribution of excess charge on surface of the cup?

Answer:Negative charge resides on the outside surface and no charge resides on the inside surface

I understand tht there is only negative charge but I dont understand why the negative charge isn't also on the inside surface...is it because all charges are always on the outer surface of a conductor?
But the inside surface of the cup is also the outer surface of the conductor???

Say there were some non-zero charge on the inside surface of the sphere. Then you can construct a closed Gaussian surface lying entirely between the outer and inner surface. The charge enclosed by the Gaussian surface is $q_{enc.}=-Q$. The total electric flux through the surface is $\Phi_E=0$ because the electric field everywhere on the Gaussian surface is zero since the surface is entirely inside the conductor. Now Gauss's law says $\Phi_E=\dfrac{q_{enc}}{\epsilon_0}$. Since the left side is zero, the right side must also be zero. Therefore $q_{enc.}$ must be zero, i.e. there is no charge of any sign on the inside surface.