5.5.1. Kirchhoff's Laws

Table of Contents

Why Kirchhoff's Laws Matter

When a circuit has only one simple path, the rules for series and parallel circuits are often enough. But many real circuits have several branches and more than one loop. In those cases, Kirchhoff's laws give a general method for finding unknown currents and voltages.

Kirchhoff's laws are based on two big ideas. First, electric charge is conserved, so charge does not pile up at an ordinary junction in a steady circuit. Second, energy is conserved, so as charge goes around a closed loop, all voltage rises and drops must balance.

There are two laws. The first is the junction law, also called the current law. The second is the loop law, also called the voltage law.

The Junction Law

A junction is a point where three or more conductors meet. In a steady direct current circuit, the total current entering a junction equals the total current leaving the junction.

If currents $I_1$ and $I_2$ enter a junction and current $I_3$ leaves, then

$$
I_1 + I_2 = I_3
$$

More generally, if we assign a sign convention, we can write

$$
\sum I = 0
$$

where currents entering may be taken as positive and currents leaving as negative, or the reverse. The choice does not matter as long as it is used consistently.

Important rule: At any junction in a steady circuit,
$$
\sum I = 0
$$
This expresses conservation of charge.

The Loop Law

A loop is any closed path in a circuit. Kirchhoff's loop law says that the algebraic sum of all potential differences around any closed loop is zero.

$$
\sum \Delta V = 0
$$

This means that as you move around a loop, voltage gains and voltage drops must cancel.

For example, in a loop with a battery of emf $\mathcal{E}$ and two resistors $R_1$ and $R_2$ carrying current $I$, we get

$$
\mathcal{E} - IR_1 - IR_2 = 0
$$

$$
\mathcal{E} = I(R_1 + R_2)
$$

This is familiar for a simple series circuit, but Kirchhoff's law works even when the circuit is much more complicated.

Important rule: Around any closed loop,
$$
\sum \Delta V = 0
$$
This expresses conservation of energy in circuit form.

Sign Conventions for Using the Loop Law

The most common source of mistakes is the sign of each voltage term. To apply the loop law correctly, choose a direction to travel around the loop, clockwise or counterclockwise, and keep it fixed while writing the equation.

For a resistor, the sign depends on the direction of the current and the direction you walk through the resistor. If you move through the resistor in the same direction as the current, the potential drops by $IR$, so the term is $-IR$. If you move through the resistor opposite to the current, the potential rises by $IR$, so the term is $+IR$.

For a battery or source, if you move from the negative terminal to the positive terminal, that is a voltage rise, so the term is $+\mathcal{E}$. If you move from the positive terminal to the negative terminal, that is a voltage drop, so the term is $-\mathcal{E}$.

The current direction can be chosen freely at the start. If the final answer comes out negative, that simply means the real current flows opposite to the assumed direction.

A Visual Picture

Rendering scientific sketch…

A junction and a loop

In the left part, if $I_1$ enters while $I_2$ and $I_3$ leave, then

$$
I_1 - I_2 - I_3 = 0
$$

In the right part, going once around the loop gives a sum of voltage changes equal to zero.

Solving Circuit Problems with Kirchhoff's Laws

The method is systematic. First, assign a direction to each unknown current. Second, mark junctions and choose enough independent loop equations. Third, write junction equations using the current law. Fourth, write loop equations using the voltage law. Finally, solve the resulting algebraic equations.

The word independent is important. Some equations may repeat information already contained in others. For a circuit with several branches, you need only enough independent equations to match the number of unknown currents.

Example with Two Loops

Consider a circuit with two loops sharing one resistor. Let currents $I_1$ and $I_2$ flow in the left and right branches, and let the shared resistor carry current $I_1 - I_2$.

Suppose the left loop has battery $\mathcal{E}_1$ and resistor $R_1$, the right loop has battery $\mathcal{E}_2$ and resistor $R_2$, and the middle shared branch has resistor $R_3$.

Then the loop equations can be written as

$$
\mathcal{E}_1 - I_1R_1 - (I_1 - I_2)R_3 = 0
$$

and

$$
\mathcal{E}_2 - I_2R_2 - (I_2 - I_1)R_3 = 0
$$

These two equations can be solved together for $I_1$ and $I_2$.

Rendering scientific sketch…

Two-loop circuit

This example shows why Kirchhoff's laws are powerful. Series and parallel simplifications may not be enough, but the laws still work directly.

Interpreting Negative Answers

Sometimes you assume a current flows one way and the calculated value is negative. That is not a mistake in the physics. It means the actual current flows in the opposite direction.

For example, if you assumed $I$ to the right and solved the equations to get

$$
I = -2.0 \text{ A}
$$

then the true current is $2.0 \text{ A}$ to the left.

Important interpretation: A negative current does not mean an impossible current. It means the real current direction is opposite to the direction you assumed.

Common Sign Rules Summary

The following table is a useful quick reference when writing loop equations.

Element crossed in loop	Direction of travel	Voltage term
Resistor	Same direction as current	$-IR$
Resistor	Opposite direction to current	$+IR$
Battery	From negative to positive terminal	$+\mathcal{E}$
Battery	From positive to negative terminal	$-\mathcal{E}$

A Short Worked Example

Suppose a loop contains a battery of $12\ \text{V}$ and two resistors, $2\ \Omega$ and $4\ \Omega$, in series. Let the current be $I$.

Applying the loop law clockwise gives

$$
12 - 2I - 4I = 0
$$

$$
12 - 6I = 0
$$

and therefore

$$
I = 2\ \text{A}
$$

This is a simple case, but the same idea extends to larger circuits with many loops and branches.

Relationship Between the Two Laws

The junction law and the loop law work together. The junction law connects currents at branching points. The loop law connects voltage changes around closed paths. In multi-branch circuits, one law alone is usually not enough. Together, they produce a complete set of equations.

In practical circuit analysis, Kirchhoff's laws are the foundation for solving direct current networks. They remain valid whether the circuit is small or complicated, as long as the circuit is treated with the steady current model used in introductory physics.

Core formulas:
$$
\sum I = 0
$$
at a junction, and
$$
\sum \Delta V = 0
$$
around a closed loop.
Use a consistent sign convention throughout the problem.

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