Kirchoff's Loop Law

Two different examples are given below. The first is explained in detail. The second is more concise. Both examples involve the circuit shown in the diagram at right. Voltage values are given for every circuit element. The two black arrows represent the directions of the current through resistors D and E. The goal of the two examples is to determine the directions of the current through resistors A and B.

Detailed Example

Kirchoff's Loop Law states that the voltage rises and drops around any closed loop must sum to zero. A closed loop is a path around the circuit that starts and stops at the same location. The circuit diagram that is shown at right is identical to the one above except that a particular closed loop has been highlighted. The loop is colored green except for resistor A, which is colored red. Resistor A is colored differently to remind you that you are trying to determine the direction of the current through it.

To apply Kirchoff’s Loop Law, imagine ‘walking’ around the closed loop in one direction or the other. It doesn’t matter whether you walk clockwise or counterclockwise around the loop. It also doesn’t matter where in the loop you start because you will stop where you started. For the sake of this example, imagine that you start next to the negative terminal of battery C and walk clockwise around the highlighted loop. Before returning to your starting point you will walk past battery C, then resistor A, then resistor E. Hence, according to Kirchoff’s Loop Law:

V_C + V_A + V_E = 0

where V_C, V_A and V_E represent the voltages across C, A and E respectively. For this equation to be true, some of these voltages must be rises (i.e., positive) and some must be drops (i.e., negative).

Rules for determining whether a voltage is a rise or a drop:

• While walking around a loop, the voltage across a battery is positive if you pass by the battery from its negative terminal to its positive terminal. The voltage across a battery is negative if you pass by the battery from its positive terminal to its negative terminal. When walking clockwise around the green loop in the diagram above, notice that the loop passes by battery C from its negative to its positive terminal. Hence, V_C is positive. In particular, V_C = +14V.

• While walking around a loop, the voltage across a resistor is positive if you pass by the resistor in the direction that is opposite to the direction of the current. The voltage across a resistor is negative if you pass by the resistor in the same direction as the direction of the current. When walking clockwise around the green loop in the circuit diagram above, notice that you pass by resistor E in the same direction as the current through resistor E. Hence, V_E is negative. In particular, V_E = -8V.

The diagram indicates that the magnitude of the voltage across resistor A is 6V, but since we don’t yet know the direction of the current through resistor A, this voltage could either be +6V or -6V.

Recall that Kirchoff’s Loop Law applied to the closed loop in the circuit above resulted in the equation:

V_C + V_A + V_E = 0

In addition, we determined that:

V_C = +14V, V_A = ±6V and V_E = -8V.

Hence:

+14V + ±6V + -8V = 0.

This can only be true if V_A = -6V.

Notice that the value of V_A must be negative. This implies the direction of the current through resistor A is the same as the direction you walked along the loop as you passed by resistor A. Since you walked clockwise around the loop, you passed by resistor A while walking downward. That means the direction of the current through resistor A is also downward.