KIRCHOFF’S LAW

Imagine that you are going for an afternoon hike on a nearby mountain. You drive up to the base lodge, park your car, and start up the trail. The trail goes up over a ridge, down into a ravine, up to the peak of the mountain, down the other side and then around the mountain back to the base lodge. When you get back to your car, how much gravitational potential energy have you gained from the trip? The answer is clearly zero—you are right back where you started.
If you defined gh, which is the potential energy of a unit mass, as your gravitational voltage, then as you went up the ridge, there was a voltage rise as h increased. Going down into the ravine there was a voltage drop, or what we could call a negative voltage rise. The big voltage rise is up to the top of the mountain, and the big negative voltage rise is down the back side of the mountain. When you add up all the voltage rises for the complete trip, counting voltage drops as negative rises, the sum is zero.
Consider our Figure (1) redrawn here. If we start at Point (4) where the voltage is zero, and “walk” around the circuit in the direction of the positive current i, we first encounter a voltage rise up to V = Vb due to the battery, then a voltage drop back to zero at the resistor. When we get back to the starting point, the sum of the voltage rises is zero just as in our trip through the mountains. Even in more complicated circuits with many branches and different circuit elements, it is usually true that the sum of the voltage rises around any complete path, back to your starting point, is zero. It turns out that this is a powerful tool for analyzing electric circuits, and is known as Kirchoff’s law. (Kirchoff’s law can be violated, we can get a net voltage rise in a complete circuit, if changing magnetic fields are present. We will treat this phenomenon in a later chapter. For now we will discuss the usual situation where Kirchoff’s law applies.)

Figure 1: Voltages around the circuit.