# Kirchhoff’s Circuit Law

By wonerock •  Updated: 05/31/23 •  7 min read

Electrical circuits have become an integral part of modern society, powering homes, businesses, and technology around the world. However, the complex nature of these circuits can make them difficult to analyze and understand. This is where Kirchhoff’s Circuit Law comes in to play, allowing us to calculate the flow of electric currents and potential differences in circuits. In this article, we’ll explore the principles of Kirchhoff’s Circuit Law and how to apply it to real-world circuits.

## What is Kirchhoff’s Circuit Law?

### Definition and Explanation

Kirchhoff’s Circuit Law, named after Gustav Kirchhoff, is a set of fundamental laws that describe how electric currents and voltages behave in electrical circuits. These laws are based on the principle that electric charge must be conserved in any node or wire, and that the potential difference around any closed loop must equal zero.

### Importance in Physics and Circuits

Kirchhoff’s Circuit Law is one of the fundamental laws of physics that governs the behavior of electric circuits. It allows us to analyze and design complex circuits, from simple series and parallel circuits to more elaborate circuits with resistors, capacitors, and inductors. By understanding Kirchhoff’s Circuit Law, we can determine the current flow through each element in a circuit and predict their behavior under different conditions.

## What are the two laws of Kirchhoff’s Circuit Law?

### Kirchhoff’s Voltage Law (KVL)

#### Definition and Explanation

Kirchhoff’s Voltage Law, also known as KVL, states that the sum of the voltages around any closed loop must equal zero. This law is based on the principle of energy conservation, which states that the total energy in a closed system must remain constant.

#### Real-life Application

An example of KVL in action is a simple series circuit with a battery and a resistor. When a voltage is applied to the circuit, it creates a current flow through the resistor, generating a voltage drop across the resistor. According to KVL, the sum of the voltages across the battery and the resistor must equal zero, as the potential energy of the electrons cannot be created or destroyed.

#### Calculation Example

Let’s consider a circuit with a 9-volt battery, a 3-ohm resistor, and a 6-ohm resistor connected in series. Applying KVL, we can calculate the voltage drop across each resistor. Summing up the voltage drops will give us the total voltage across the circuit, which must equal the voltage of the battery.

We can start by assuming a clockwise direction for the loop in our circuit. Starting at the positive terminal of the battery, we encounter a voltage drop of 9 volts. Moving through the 3-ohm resistor, we encounter another voltage drop of 9 volts times 3 ohms divided by 9 ohms, or 3 volts. Finally, we move through the 6-ohm resistor, encountering a voltage drop of 9 volts times 6 ohms divided by 9 ohms, or 6 volts. According to KVL, the total sum of the voltages must equal zero, so the remaining voltage drop across the circuit must be 0 volts.

### Kirchhoff’s Current Law (KCL)

#### Definition and Explanation

Kirchhoff’s Current Law, also known as KCL, states that the sum of the currents at any junction in a circuit must equal zero. This law is based on the principle of electric charge conservation, which states that the total electric charge in a closed system must remain constant.

#### Real-life Application

An example of KCL in action is a simple parallel circuit with two resistors connected to a battery. When the battery is applied to the circuit, it creates a current flow through the resistors, which converge at a common junction before returning to the battery. According to KCL, the total current flowing into the junction must equal the total current flowing out of the junction, as electric charge cannot be created or destroyed.

#### Calculation Example

Let’s consider a circuit with a 9-volt battery, a 3-ohm resistor, and a 6-ohm resistor connected in parallel. Applying KCL, we can calculate the individual current flowing through each resistor, which must equal the total current flowing from the battery.

We can start by assuming that the current flowing from the battery splits into two branches, one through the 3-ohm resistor and the other through the 6-ohm resistor. Using Ohm’s Law, we can calculate the current flowing through each resistor by dividing the voltage of the battery by the resistance of the resistor. The current flowing through the 3-ohm resistor is 9 volts divided by 3 ohms, or 3 amps. The current flowing through the 6-ohm resistor is 9 volts divided by 6 ohms, or 1.5 amps. According to KCL, the total sum of the currents must equal zero, so the total current flowing from the battery must be 4.5 amps (3 amps plus 1.5 amps).

## How to apply Kirchhoff’s Circuit Law?

### Using Circuit Diagrams

#### Basic Circuit Elements

A circuit diagram is a graphical representation of an electrical circuit that uses symbols to represent the different components in the circuit. Some of the basic circuit elements that are commonly used in circuit diagrams include wires, resistors, capacitors, batteries, and switches. By using the appropriate symbols, we can easily represent the components of a circuit and analyze its behavior.

#### Loop Rule and Junction Rule

Once we have represented the circuit in a diagram, we can use the loop rule and junction rule to apply Kirchhoff’s Circuit Law. The loop rule states that the algebraic sum of the voltages around a closed loop must be equal to zero, while the junction rule states that the algebraic sum of the currents at any junction must be equal to zero.

### Using Algebraic Summation and Equations

#### Sum of Voltages around a Closed Loop

We can also apply Kirchhoff’s Circuit Law using algebraic summation and equations. For example, let’s consider a circuit with a battery, a resistor, and a capacitor connected in series. To calculate the voltage drop across the resistor, we can use the KVL equation, which states that the sum of the voltages around any closed loop must be equal to zero.

Starting at the positive terminal of the battery, we encounter a voltage drop of V volts. Moving through the resistor, we encounter a voltage drop of IR volts, where I is the current flowing through the resistor and R is its resistance. Finally, we move through the capacitor, which has a voltage drop of zero when fully charged. This gives us the equation V – IR = 0, which we can solve for I to get I = V/R.

#### Sum of Currents at a Node

Sometimes we may need to calculate the current flowing through a specific point in a circuit, such as a junction between two circuit elements. To calculate the current, we can use the KCL equation, which states that the algebraic sum of the currents at any junction must be equal to zero.

For example, let’s consider a circuit with two resistors connected in parallel and a battery. Applying KCL at the junction of the resistors, we can calculate the current flowing through each resistor. Let R1 and R2 be the resistances of the resistors, and I1 and I2 be the currents flowing through them. The total current flowing into the junction is given by the current of the battery, which is I = V/Rtotal, where V is the voltage of the battery and Rtotal is the equivalent resistance of the parallel resistors. Applying KCL, we get I – I1 – I2 = 0, which we can solve to get I1 = (V/R1) – (V/Rtotal) and I2 = (V/R2) – (V/Rtotal).

## What are the limitations of Kirchhoff’s Circuit Law?

### Effect of Magnetic and Electromagnetic Fields

#### Induced EMF and Currents

One limitation of Kirchhoff’s Circuit Law is that it does not account for the effects of magnetic and electromagnetic fields on circuits. In some cases, changes in magnetic fields can generate an electromotive force (EMF) and induce currents in circuits, making them more complex than predicted by Kirchhoff’s Circuit Law.

#### Effect on Real and Ideal Circuits

The effects of magnetic and electromagnetic fields are more pronounced in real circuits than in ideal circuits. In ideal circuits, all the components have zero resistance, zero inductance, and zero capacitance. In reality, these components have some non-zero values, making the circuit more complex and difficult to analyze. In such cases, we may