We started our study of electricity with electric forces and fields. But in electricity, just as in mechanics, energy is also a powerful idea. This chapter and the next will explore how energy is used in electricity, introduce the important concept of electric potential, and lay the groundwork for our upcoming study of electric circuits.

It’s been many chapters since we dealt much with work and energy, but these ideas will now be essential to our story. Consequently, the Looking Back recommendations in the chapter preview are especially important. You will recall that a system’s mechanical energy $E_{\text{mech}}=K+U$ is conserved for particles that interact with each other via conservative forces, where K and U are the kinetic and potential energy. That is,

We need to be careful with notation because we are now using E to represent the electric field strength. To avoid confusion, we will represent mechanical energy either as the sum $K+U$ or as $E_{\text{mech}}$, with an explicit subscript.

A key idea of Chapters 9 and 10 was that energy is the energy of a system, and clearly defining the system is crucial. Kinetic energy $K=\frac{1}{2}m{v}^2$ is a system’s energy of motion. For a multiparticle system, K is the sum of the kinetic energies of each particle in the system.

Potential energy U is the interaction energy of the system. Suppose the particles of the system move from some initial set of positions i to final positions f. As the particles move, the action/reaction pairs of forces between the particles—the interaction forces—do work and the system’s potential energy changes. In 《 SECTION 10.1 we defined the change in potential energy to be

where the notation means the work done by the interaction forces as the configuration changes from i to f. This rather abstract definition will make more sense when we see specific applications.

Recall that work is done when a force acts on a particle as it is being displaced. In 《 SECTION 9.3 you learned that a constant force $\vec{F}$ does work

as the particle undergoes displacement $\text{Δ}\vec{r}$, where $\theta$ is the angle between the force and the displacement. FIGURE 25.1 reminds you of the three special cases $\theta =0\text{°}\text{,}90\text{°}\text{,}\text{ and }180\text{°}$.

FIGURE 25.1 The work done by a constant force.

If the force is not constant, we can calculate the work by dividing the path into many small segments of length $dx$, finding the work done in each segment, and then summing (i.e., integrating) from the start of the path to the end. The work done in one such segment is $dW=F\left(x\right)\text{cos}\theta dx$, where $F\left(x\right)$ indicates that the force is a function of position x. Thus the work done on the particle as it moves from $x_{\text{i}}$ to $x_{\text{f}}$ is

Finally, recall that a conservative force is one for which the work done on a particle as it moves from position i to position f is independent of the path followed. We’ll assert for now, and prove later, that the electric force is a conservative force, and thus we can define an electric potential energy.