The basic relationships between potential and field allow us to draw some interesting and important conclusions about conductors. You learned in Chapter 22 that any excess charges on a conductor in electrostatic equilibrium are always located on the surface of the conductor. Using similar reasoning, we can conclude that the electric field is zero at any interior point of a conductor in electrostatic equilibrium. Why? If the field were other than zero, then there would be a force $\vec{F}=q\vec{E}$ on the charge carriers and they would move, creating a current. But there are no currents in a conductor in electrostatic equilibrium, so it must be that $\vec{E}=\vec{0}$ at all interior points.

The two points inside the conductor in FIGURE 26.14 are connected by a line that remains entirely inside the conductor. We can find the potential difference $\text{Δ}V=V_2-V_1$ between these points by using Equation 26.3 to integrate $E_s$ along the line from 1 to 2. But $E_s=0$ at all points along the line, because $\vec{E}=\vec{0}\text{;}$ thus the value of the integral is zero and $\text{Δ}V=0$. In other words, any two points inside a conductor in electrostatic equilibrium are at the same potential.

FIGURE 26.14 All points inside a conductor in electrostatic equilibrium are at the same potential.

When a conductor is in electrostatic equilibrium, the entire conductor is at the same potential. If we charge a metal sphere, then the entire sphere is at a single potential. Similarly, a charged metal rod or wire is at a single potential if it is in electrostatic equilibrium. It’s worth noting that two conductors that come into contact form a single conductor, so they exchange charge as needed to reach a common potential.

If $\vec{E}=\vec{0}$ inside a charged conductor but $\vec{E}\ne \vec{0}$ outside, what happens right at the surface? If the entire conductor is at the same potential, then the surface is an equipotential surface. You have seen that the electric field is always perpendicular to an equipotential surface, hence the exterior electric field $\vec{E}$ of a charged conductor is perpendicular to the surface.

We can also conclude that the electric field, and thus the surface charge density, is largest at sharp points. This follows from our earlier discovery that the field at the surface of a sphere of radius R can be written $E=V_0\text{/}R$. If we approximate the rounded corners of a conductor with sections of spheres, all of which are at the same potential $V_0$, the field strength will be largest at the corners with the smallest radii of curvature—the sharpest points.

FIGURE 26.15 summarizes what we know about conductors in electrostatic equilibrium. These are important and practical conclusions because conductors are the primary components of electrical devices.

FIGURE 26.15 Properties of a conductor in electrostatic equilibrium.

We can use similar reasoning to estimate the electric field and potential between two charged conductors. As an example, FIGURE 26.16 shows a negatively charged metal sphere near a flat metal plate. The surfaces of the sphere and the flat plate are equipotentials, hence the electric field must be perpendicular to both. Close to a surface, the electric field is still nearly perpendicular to the surface. Consequently, an equipotential surface close to an electrode must roughly match the shape of the electrode.

FIGURE 26.16 Estimating the field and potential between two charged conductors.

In between, the equipotential surfaces gradually change as they “morph” from one electrode shape to the other. It’s not hard to sketch a contour map showing a plausible set of equipotential surfaces. You can then draw electric field lines (field lines are easier to draw than field vectors) that are perpendicular to the equipotentials, point “downhill,” and are closer together where the contour line spacing is smaller.