11.13 Kirchhoff's Laws | Electricity | Physics 11

Kirchhoff's Laws are two fundamental principles that provide a systematic way to analyze complex electrical circuits. They are essential for circuits that contain multiple loops, junctions, or sources of electromotive force (emf), where simple series and parallel resistor rules are insufficient.

Kirchhoff's First Law (The Current Law or KCL)

This law is a statement of the conservation of electric charge. It applies to any junction (or node) in a circuit, which is a point where three or more wires meet.

Statement: The algebraic sum of the currents entering a junction is equal to the algebraic sum of the currents leaving that junction.

$\sum I_{in} = \sum I_{o u t}$

Sign Convention: Currents flowing into a junction are positive (+), and currents flowing out of the junction are negative (−). Therefore, the algebraic sum of all currents at a junction is zero:

$\sum I = 0$

Example: For a junction where $I_{1}$ , $I_{2}$ , and $I_{3}$ flow in, and $I_{4}$ and $I_{5}$ flow out:

$I_{1} + I_{2} + I_{3} - I_{4} - I_{5} = 0$

A junction showing currents flowing in and out.

Kirchhoff's Second Law (The Voltage Law or KVL)

This law is a statement of the conservation of energy. It applies to any closed loop in a circuit.

Statement: The algebraic sum of all the potential differences (voltages) around any closed loop in a circuit must be zero.

$\sum Δ V = 0$

Explanation: If you start at any point in a closed loop and travel around it, by the time you return to the starting point, your electric potential must be the same. The sum of all potential rises (from EMF sources) must equal the sum of all potential drops (across resistors).

Sign Conventions for KVL

Situation	Potential Change
Traverse a resistor in the direction of current	$- I R$ (drop)
Traverse a resistor opposite to the current	$+ I R$ (rise)
Traverse an EMF source from − to + terminal	$+ ε$ (rise)
Traverse an EMF source from + to − terminal	$- ε$ (drop)

A single loop circuit used to illustrate KVL.

Application to Series and Parallel Resistors

Kirchhoff's laws can be used to derive the familiar rules for combining resistors.

Resistors in Series: KCL implies the current is the same through all components. KVL shows that the total voltage is the sum of the individual voltage drops: $V = V_{1} + V_{2} + \dots ⟹ R_{e q} = R_{1} + R_{2} + \dots$
Resistors in Parallel: KVL implies the voltage is the same across all parallel branches. KCL shows that the total current is the sum of the branch currents: $I = I_{1} + I_{2} + \dots ⟹ \frac{1}{R _{e q}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots$

Summary

Law	Based on Conservation of...	Applies to...	Formula
Current Law (KCL)	Charge	Junctions (Nodes)	$\sum I = 0$
Voltage Law (KVL)	Energy	Closed Loops	$\sum Δ V = 0$

Kirchhoff's First Law (The Current Law or KCL)

This law is a statement of the conservation of electric charge. It applies to any junction (or node) in a circuit, which is a point where three or more wires meet.

Statement: The algebraic sum of the currents entering a junction is equal to the algebraic sum of the currents leaving that junction.

$\sum I_{in} = \sum I_{o u t}$

$\sum I = 0$

Example: For a junction where $I_{1}$ , $I_{2}$ , and $I_{3}$ flow in, and $I_{4}$ and $I_{5}$ flow out:

$I_{1} + I_{2} + I_{3} - I_{4} - I_{5} = 0$

Kirchhoff's Second Law (The Voltage Law or KVL)

This law is a statement of the conservation of energy. It applies to any closed loop in a circuit.

Statement: The algebraic sum of all the potential differences (voltages) around any closed loop in a circuit must be zero.

$\sum Δ V = 0$

Sign Conventions for KVL

Situation	Potential Change
Traverse a resistor in the direction of current	$- I R$ (drop)
Traverse a resistor opposite to the current	$+ I R$ (rise)
Traverse an EMF source from − to + terminal	$+ ε$ (rise)
Traverse an EMF source from + to − terminal	$- ε$ (drop)

Application to Series and Parallel Resistors

Kirchhoff's laws can be used to derive the familiar rules for combining resistors.

Resistors in Series: KCL implies the current is the same through all components. KVL shows that the total voltage is the sum of the individual voltage drops: $V = V_{1} + V_{2} + \dots ⟹ R_{e q} = R_{1} + R_{2} + \dots$
Resistors in Parallel: KVL implies the voltage is the same across all parallel branches. KCL shows that the total current is the sum of the branch currents: $I = I_{1} + I_{2} + \dots ⟹ \frac{1}{R _{e q}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots$

Summary

Law	Based on Conservation of...	Applies to...	Formula
Current Law (KCL)	Charge	Junctions (Nodes)	$\sum I = 0$
Voltage Law (KVL)	Energy	Closed Loops	$\sum Δ V = 0$