25.2 The Potential Energy of Point Charges

FIGURE 25.7a shows two charges $q_{1}$ and $q_{2}$ , which we will assume to be like charges. These two charges interact, and the energy of their interaction can be found by calculating the work done by the electric field of $q_{1}$ on $q_{2}$ as $q_{2}$ moves from position $x_{i}$ to position $x_{f}$ . We’ll assume that $q_{1}$ has been glued down and is unable to move, as shown in FIGURE 25.7b.

The figure consists of two parts labeled ("a") and (b) that illustrate the interaction between two point charges.

FIGURE 25.7 Full Alternative Text

The force on $q_{2}$ is entirely in the direction of motion, so $cos θ = 1$ . Thus the work done by the electric field as $q_{2}$ moves from $x_{i}$ to $x_{f}$ is

W_{elec} = \int_{x_{i}}^{x_{f}} F_{1 on 2} d x = \int_{x_{i}}^{x_{f}} \frac{K q_{1} q_{2}}{x^{2}} d x = K q_{1} q_{2} {\frac{- 1}{x} |}_{x_{i}}^{x_{f}} = - \frac{K q_{1} q_{2}}{x_{f}} + \frac{K q_{1} q_{2}}{x_{i}}

(25.11)

The potential energy of the two charges is related to the work done by

Δ U_{elec} = U_{f} - U_{i} = - W_{elec} (i \to f) = \frac{K q_{1} q_{2}}{x_{f}} - \frac{K q_{1} q_{2}}{x_{i}}

(25.12)

By comparing the left and right sides of the equation we see that the potential energy of the two-point-charge system is

U_{elec} = \frac{K q_{1} q_{2}}{x}

(25.13)

We chose to integrate along the x-axis for convenience, but all that matters is the distance between the charges. Thus a general expression for the electric potential energy is

U_{elec} = \frac{K q_{1} q_{2}}{r} = \frac{1}{4 π ϵ_{0}} \frac{q_{1} q_{2}}{r} (two point charges)

(25.14)

This is explicitly the energy of the system, not the energy of just $q_{1}$ or $q_{2}$ .

NOTE The electric potential energy of two point charges looks almost the same as the force between the charges. One important difference is the r in the denominator of the potential energy compared to the $r^{2}$ in Coulomb’s law. Another is that Equation 25.14 does not have the absolute values of the charges, which means that $U_{elec}$ can be negative.

Three important points need to be noted:

The potential energy of two charged particles is zero only when they are infinitely far apart. This makes sense because two charged particles cease interacting only when they are infinitely far apart.
We derived Equation 25.14 for two like charges, but it is equally valid for two opposite charges. The potential energy of two like charges is positive and of two opposite charges is negative.
Because the electric field outside a sphere of charge is the same as that of a point charge at the center, Equation 25.14 is also the electric potential energy of two charged spheres. Distance r is the distance between their centers.