FIGURE 24.28 shows a Gaussian surface just barely inside the physical surface of a conductor that’s in electrostatic equilibrium. The electric field is zero at all points inside the conductor, hence the electric flux ${\text{Φ}}_{\text{e}}$ through this Gaussian surface must be zero. But if ${\text{Φ}}_{\text{e}}=0$, Gauss’s law tells us that $Q_{\text{in}}=0$. That is, there’s no net charge within this surface. There are charges—electrons and positive ions—but no net charge.

FIGURE 24.28 A Gaussian surface just inside a conductor’s surface.

If there’s no net charge in the interior of a conductor in electrostatic equilibrium, then all the excess charge on a charged conductor resides on the surface of the conductor. Any charges added to a conductor quickly spread across the surface until reaching positions of electrostatic equilibrium, but there is no net charge inside the conductor.

There may be no electric field inside a charged conductor, but the presence of net charge requires an exterior electric field in the space outside the conductor. FIGURE 24.29 shows that the electric field right at the surface of the conductor has to be perpendicular to the surface. To see that this is so, suppose $\vec{E}{}_{\text{surface}}$ had a component tangent to the surface. This component of $\vec{E}{}_{\text{surface}}$ would exert a force on the surface charges and cause a surface current, thus violating the assumption that all charges are at rest. The only exterior electric field consistent with electrostatic equilibrium is one that is perpendicular to the surface.

FIGURE 24.29 The electric field at the surface of a charged conductor.

We can use Gauss’s law to relate the field strength at the surface to the charge density on the surface. FIGURE 24.30 shows a small Gaussian cylinder with faces very slightly above and below the surface of a charged conductor. The charge inside this Gaussian cylinder is $\eta A$, where $\eta$ is the surface charge density at this point on the conductor. There’s a flux $\text{Φ}=A{E}_{\text{surface}}$ through the outside face of this cylinder but, unlike Example 24.6 for the plane of charge, no flux through the inside face because $\vec{E}{}_{\text{in}}=\vec{0}$ inside the conductor. Furthermore, there’s no flux through the wall of the cylinder because $\vec{E}{}_{\text{surface}}$ is perpendicular to the surface. Thus the net flux is ${\text{Φ}}_{\text{e}}=A{E}_{\text{surface}}$. Gauss’s law is

from which we can conclude that the electric field at the surface of a charged conductor is

FIGURE 24.30 A Gaussian surface extending through the surface has a flux only through the outer face.

In general, the surface charge density $\eta$ is not constant on the surface of a conductor but depends on the shape of the conductor. If we can determine $\eta$, by either calculating it or measuring it, then Equation 24.20 tells us the electric field at that point on the surface. Alternatively, we can use Equation 24.20 to deduce the charge density on the conductor’s surface if we know the electric field just outside the conductor.