FIGURE 26.13 shows two points, 1 and 2, in a region of electric field and potential. You learned in Chapter 25 that the work done in moving a charge between points 1 and 2 is independent of the path. Consequently, the potential difference between points 1 and 2 along any two paths that join them is $\text{Δ}V=20\text{V}$. This must be true in order for the idea of an equipotential surface to make sense.

FIGURE 26.13 The potential difference between points 1 and 2 is the same along either path.

Now consider the path 1–a–b–c–2–d–1 that ends where it started. What is the potential difference “around” this closed path? The potential increases by 20 V in moving from 1 to 2, but then decreases by 20 V in moving from 2 back to 1. Thus $\text{Δ}V=0\text{V}$ around the closed path.

The numbers are specific to this example, but the idea applies to any loop (i.e., a closed path) through an electric field. The situation is analogous to hiking on the side of a mountain. You may walk uphill during parts of your hike and downhill during other parts, but if you return to your starting point your net change of elevation is zero. So for any path that starts and ends at the same point, we can conclude that

Stated in words, the sum of all the potential differences encountered while moving around a loop or closed path is zero. This statement is known as Kirchhoff’s loop law.

Kirchhoff’s loop law is a statement of energy conservation because a charge that moves around a loop and returns to its starting point has $\text{Δ}U=q\text{Δ}V=0$. Kirchhoff’s loop law and a second Kirchhoff’s law you’ll meet in the next chapter will turn out to be the two fundamental principles of circuit analysis.