Electric Potential of a Point Charge

In the previous section, we examined the relationship between electric potential and electric field for the special case of a uniform electric field. Many real electric fields, however, are produced by localized charges such as electrons, protons, or charged conducting spheres. In these situations, the electric field is no longer uniform, and a different expression is needed to determine the electric potential.

Point charges, such as electrons and protons, are among the fundamental building blocks of matter. Even when charge is distributed uniformly over a spherical conductor—such as a charged metal sphere—the electric field and electric potential outside the sphere are exactly the same as if all the charge were concentrated at a single point at the center. This remarkable result allows us to treat many real objects as point charges whenever we are interested in points outside the charged object.

Deriving the electric potential of a point charge requires calculus because the electric field changes continuously with distance from the charge. Rather than carrying out the derivation here, we present the result. By calculating the work required to bring a small test charge q from very far away (where the electric potential is defined to be zero) to a distance r from a point charge Q, and using the relationship

we obtain the electric potential produced by a point charge:

where

Several important ideas follow directly from this equation.

First, electric potential V is a scalar quantity: it has magnitude but no direction. In contrast, the electric field E is a vector quantity, meaning that it has both magnitude and direction.

Second, notice how differently electric potential and electric field vary with distance. For a point charge,

whereas the magnitude of the electric field is

The electric field decreases with the square of the distance, while the electric potential decreases only linearly with distance. As a result, electric potential extends farther into space than the electric field. Even at large distances where the electric field has become quite weak, the electric potential may still have a significant value.

Third, because electric potential is a scalar, the total potential produced by several point charges is simply the algebraic sum of the individual potentials:

This is much simpler than finding the net electric field, since electric fields must be added as vectors and therefore require both magnitude and direction to be considered. This distinction reflects the underlying physics: electric potential is related to energy (a scalar), whereas electric field is related to force (a vector).

Example 13.2: What Is the Excess Charge on a Van de Graaff Generator?

A demonstration Van de Graaff generator has a conducting metal sphere with a diameter of 25.0 cm that is maintained at an electric potential of 100 kV. What excess charge resides on the sphere?

Strategy

Outside the conducting sphere, the electric potential is given by the point-charge equation

[latex]\text{V}=\frac{kQ}{r}[/latex]

Since the sphere's diameter is known, we first determine its radius and then solve the equation for the unknown charge.

Solution

The radius of the sphere is

[latex]r=12.5\ \text{cm}=0.125\ \text{m}[/latex]

Rearranging the point-charge equation gives

[latex]Q=\frac{r\text{V}}{k}[/latex]

Substituting the known values,

[latex]Q=\frac{(0.125\ \text{m})(100\times10^{3}\ \text{V})}{8.99\times10^{9}\ \text{N}\cdot\text{m}^{2}/\text{C}^{2}}[/latex]

[latex]Q=1.39\times10^{-6}\ \text{C}[/latex]

or

[latex]Q=1.39\ \mu\text{C}[/latex]

Discussion

The sphere carries only about 1.4 μC of excess charge, yet it reaches an electric potential of 100,000 V. This demonstrates an important principle of electrostatics: a relatively small amount of isolated charge can produce an extremely large electric potential.

Large electric potentials do not necessarily require large amounts of charge. Instead, they often result from concentrating charge on an isolated conductor. This principle is exploited in Van de Graaff generators, particle accelerators, and many high-voltage laboratory instruments.

Similar ideas also appear in biology. Cell membranes maintain potential differences of only about 70 mV, but because these voltages occur across membranes only a few nanometers thick, they produce extremely strong electric fields that regulate the movement of ions and enable nerve and muscle function.

Choosing a Reference for Electric Potential

In practice, electric potential is always measured relative to a chosen reference point. A common choice is ground (Earth), which is assigned a potential of zero volts. In other situations, infinity is used as the reference, as we have done when deriving the electric potential of a point charge.

Only differences in electric potential have physical significance. Changing the reference point changes the numerical value of the potential everywhere by the same amount, but it does not change electric fields, forces, or the motion of charged particles. This is analogous to gravitational potential energy, where sea level or the floor of a room may be chosen as the zero level without affecting the underlying physics.