# 4.2 – Kirchoff’s Laws

Many complex circuits, such as the one in Figure 4.2.1, cannot be analyzed with the series-parallel techniques developed in Resistors in Series and Parallel and Electromotive Force: Terminal Voltage (we need to describe these). There are, however, two circuit analysis rules that can be used to analyze any circuit, simple or complex. These rules are special cases of the laws of conservation of charge and conservation of energy. The rules are known as Kirchhoff’s laws.

## Kirchhoff’s Rules

• Kirchhoff’s first rule—the junction rule. The sum of all currents entering a junction must equal the sum of all currents leaving the junction.
• Kirchhoff’s second rule—the loop rule. The algebraic sum of changes in potential around any closed circuit path (loop) must be zero.

Explanations of the two laws will now be given, followed by problem-solving hints for applying Kirchhoff’s laws, and a worked example that uses them.

### Making Connections: Conservation Laws

Kirchhoff’s rules for circuit analysis are applications of conservation laws to circuits. The first rule is the application of conservation of charge, while the second rule is the application of conservation of energy. Conservation laws, even used in a specific application, such as circuit analysis, are so basic as to form the foundation of that application.

### Kirchhoff’s Second Law

Kirchhoff’s second rule (the loop rule) is an application of conservation of energy. The loop rule is stated in terms of potential, V, rather than potential energy, but the two are related since $PEelec=qV$. Recall that $\text{emf}$ is the potential difference of a source when no current is flowing. In a closed loop, whatever energy is supplied by $\text{emf$ must be transferred into other forms by devices in the loop, since there are no other ways in which energy can be transferred into or out of the circuit. Figure 4.2.3 illustrates the changes in potential in a simple series circuit loop.

Kirchhoff’s second rule requires $\text{emf} - I r - I R_1 - I R_2 = 0$. Rearranged, this is $\text{emf} = I r + I R_1 + I R_2$, which means the $\text{emf}$ equals the sum of the IR (voltage) drops in the loop.

### Applying Kirchhoff’s Rules

By applying Kirchhoff’s rules, we generate equations that allow us to find the unknowns in circuits. The unknowns may be currents, $\text{emfs}$, or resistances. Each time a rule is applied, an equation is produced. If there are as many independent equations as unknowns, then the problem can be solved. There are two decisions you must make when applying Kirchhoff’s rules. These decisions determine the signs of various quantities in the equations you obtain from applying the rules.

1. When applying Kirchhoff’s first rule, the junction rule, you must label the current in each branch and decide in what direction it is going. For example, in Figure 4.2.1, Figure 4.2.2, and Figure 4.2.3, currents are labeled $I_1$, $I_2$, $I_3$ and $I$, and arrows indicate their directions. There is no risk here, for if you choose the wrong direction, the current will be of the correct magnitude but negative.
2. When applying Kirchhoff’s second rule, the loop rule, you must identify a closed loop and decide in which direction to go around it, clockwise or counterclockwise. For example, in Figure 4.2.3 the loop was traversed in the same direction as the current (clockwise). Again, there is no risk; going around the circuit in the opposite direction reverses the sign of every term in the equation, which is like multiplying both sides of the equation by $–1$.

Figure 4.2.4 and the following points will help you get the plus or minus signs right when applying the loop rule. Note that the resistors and $\text{emfs}$ are traversed by going from a to b. In many circuits, it will be necessary to construct more than one loop. In traversing each loop, one needs to be consistent for the sign of the change in potential.

• When a resistor is traversed in the same direction as the current, the change in potential is $−IR$. (See Figure 4.2.4.)
• When a resistor is traversed in the direction opposite to the current, the change in potential is $+IR$. (See Figure 4.2.4.)
• When an $\text{emf}$ is traversed from – to + (the same direction it moves positive charge), the change in potential is $\text{+emf}$. (See Figure 4.2.4.)
• When an emf is traversed from + to – (opposite to the direction it moves positive charge), the change in potential is $\text{−emf}$. (See Figure 4.2.4.)