Kirchoff's Loop Law Hint

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 resistors E and H, which are colored red. They are colored differently to remind you that you are trying to determine the directions of the currents through them.

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 G and walk clockwise around the highlighted loop. Before returning to your starting point you will walk past battery G, then resistor E, then resistor H. Hence, according to Kirchoff’s Loop Law:

V_G + V_E + V_H = 0

where V_G, V_E and V_H represent the voltages across G, E and H 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:

The voltage across any wire is zero.

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 G from its negative to its positive terminal. Hence, V_G is positive. In particular, V_G = +4V.

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. In the circuit above, we don’t know the direction of the current through any of the resistors. Hence, we don’t know the sign of the voltages across any of the resistors.

Although we don’t know the sign of any of the resistor voltages, the circuit gives their absolute values. For the red resistors on the closed loop,

V_E = ±12V and V_H = ±8V.

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

V_G + V_E + V_H = 0.

Hence:

+4V + ±12V + ±8V = 0.

There is only one combination of + and – signs that makes this equation true:

V_E = -12V and V_H = +8V.

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

In addition, noticed that the value of V_H must be positive. This implies the direction of the current through resistor H is the opposite of the direction you walked along the loop as you passed by resistor H. Walking clockwise around the loop means you passed by resistor H while walking downward. This means the current through resistor H must be upward.

Concise Example

The circuit at right is identical to the circuit in the previous example, but this time a different closed loop is highlighted. As before, most of the loop is colored green except for the two resistors which are colored red to remind you that you are trying to determine the directions of the currents through them.

Neither of these examples determined the direction of the current through resistor F. To determine this direction, simply choose another closed loop that includes resistor F and apply Kirchoff’s Loop Law one last time.