The loop law (Kirchhoff's voltage law)

Statement of the law

The loop law follows from the conservation of energy: when you traverse a loop (closed circuit) in a chosen direction, the algebraic sum of the voltages encountered is zero.

Sum of voltages around a loop = 0

Practical method

1. Choose a direction of travel for the loop (clockwise or counterclockwise).
2. For each component crossed, record the voltage with a + sign if you cross from - to + (gaining potential), and a - sign if you cross from + to - (losing potential).
3. Add up all these voltages and set the result equal to 0.

Numerical example

A loop contains a generator with e.m.f. E = 12 V and two resistors carrying the same current I, producing voltages U1 = R1 x I and U2 = R2 x I. Traversing the loop in the direction of the current:

E - U1 - U2 = 0 -> E = U1 + U2

If R1 = 2 ohm, R2 = 4 ohm, and I = 2 A: U1 = 4 V, U2 = 8 V, so U1 + U2 = 12 V = E. The law is verified.

Common pitfall

The sign of each voltage depends on the chosen direction of travel, not the actual direction of the current. Two students who choose different directions of travel get equations that look different but give the same final result. The frequent mistake is forgetting to change the sign when crossing a component backwards relative to the direction of travel.