How to Apply Kirchhoff's Voltage Law: Steps, Sign Conventions & Examples
Kirchhoff’s Second Law, also called Kirchhoff’s Voltage Law (KVL), is a fundamental principle in electric circuit analysis. This law describes how voltages behave in a closed loop or mesh within an electrical circuit. KVL is widely applied throughout physics and engineering for solving complex circuit problems and is essential for understanding current electricity.
What is Kirchhoff’s Second Law?
Kirchhoff’s Second Law states: The algebraic sum of all the electric potential differences (voltages) encountered in moving around any closed loop in a circuit is zero. In simpler words, the sum of all the emf (sources) and the sum of all the potential drops (like resistors) in a closed path always balance each other out.
This law is also called the loop rule or voltage rule.
Essential Terms in Circuit Theory
- Junction:
A point where three or more conductors meet. Also called a node or branch point. Learn about junctions.
- Loop or Mesh:
A closed path within an electric circuit. Analysis is done over such loops.
- Node:
Any point where two or more circuit elements are connected. Every junction is a node, but not all nodes are junctions.
Kirchhoff’s Second Law: Statement & Formula
According to KVL, the total sum of potential differences (voltage) around any closed loop is zero:
Alternatively, you may write:
where,
emf = electromotive force (voltage sources),
IR = voltage drop across resistors (I = current, R = resistance)
Derivation and Calculation Steps
To apply KVL, start at any point on a closed loop. Move in one direction (clockwise or anti-clockwise), and sum all emf sources and voltage drops. The result must be zero.
- If you pass through an emf source from – to +, take emf as positive (+).
- If you pass from + to –, take emf as negative (–).
- If you move across a resistor in the assumed direction of current, take voltage as (–IR).
- If moving opposite the current, take voltage as (+IR).
For example, for a loop ABCDEFA:
VA – VB = i₁R₁
VB – VC = i₂R₂
VC – VD = –E₁
VD – VE = i₃R₃
VE – VF = –i₄R₄
VB – VA = E₂
Total: i₁R₁ + i₂R₂ – E₁ + i₃R₃ – i₄R₄ + E₂ = 0
Sign Convention Table for KVL
| Condition | Sign |
|---|---|
| Through emf (battery) from – to + | + |
| Through emf from + to – | – |
| Through resistor, same as current direction | –IR |
| Through resistor, opposite current direction | +IR |
Applications of Kirchhoff’s Laws
- Calculating current, voltage, and internal resistance in DC circuits
- Solving complex circuits that cannot be reduced to simple series/parallel combinations
- Finding unknown resistors using Wheatstone Bridge and meter bridge
- Understanding and analyzing multi-loop circuits in Mesh Analysis
For practical problem solving, students often use both Kirchhoff’s Current Law (KCL) and KVL together. For further practice, visit Current Electricity.
Important Comparison: KCL vs KVL
| Law | Statement | Principle | Formula |
|---|---|---|---|
| KCL | The sum of currents at a junction is zero. | Conservation of charge | ∑I = 0 |
| KVL | The sum of voltages in a closed loop is zero. | Conservation of energy | ∑V = 0 |
Limitations of Kirchhoff’s Laws
- Not accurate for AC circuits at high frequencies.
- KVL is invalid if there’s a time-varying magnetic field (Faraday’s Law applies instead).
Changing magnetic fields create induced electric fields that are non-conservative—hence, KVL cannot be applied in such cases.
Step-by-Step: How to Solve Problems Using KVL
- Draw the circuit diagram, mark all emf sources and resistors.
- Label the assumed direction of current in each branch.
- Choose a closed loop. Start at any point.
- Move along the loop, applying the sign conventions for emfs and resistance.
- Write the KVL equation. For each element, add up voltages (with correct sign).
- Solve the resulting equation(s) for unknown values.
Worked Example (with Steps)
| Step | Description |
|---|---|
| 1 | Suppose a simple loop has a battery of 12V and a resistor of 4Ω. Assume current I flows clockwise. |
| 2 | Write KVL equation: emf – voltage drop = 0 → 12 – 4I = 0 |
| 3 | Solve for I: 4I = 12 ⇒ I = 3A |
Key Takeaways
- KVL is essential for analyzing any closed loop in a circuit.
- Always pay careful attention to sign conventions for emf and resistance.
- The principle is based on conservation of energy in electrical circuits.
- Use KVL for both simple and complex circuits, especially in combination with KCL.
Further Learning and Related Resources
- Kirchhoff’s Laws of Electric Circuits
- Mesh Analysis
- Current Electricity Concepts
- Ohm’s Law
- Electric Power in Circuits
To deepen your understanding of energy conservation in circuits, explore: Energy Conservation.
FAQs on Understanding Kirchhoff's Second Law (KVL) in Physics
1. Explain the terms: a. Junction b. Loop c. Node.
Junction: A point in an electrical circuit where three or more conductors meet. It is also called a node or branch point.
Loop: A closed path in a circuit, where one can start at a point, travel around, and return to the same point without passing any point more than once. It is often referred to as a mesh.
Node: Any point on a circuit joining two or more circuit elements. All junctions are nodes, but not all nodes are junctions.
2. Why is there a need for Kirchhoff's Law if Ohm's Law exists for the same?
Kirchhoff’s Laws are needed for solving complex circuits with multiple loops and junctions, where Ohm’s Law alone is insufficient.
• Ohm’s Law (V = IR) applies only to simple series or parallel circuits.
• Kirchhoff’s Laws (KCL and KVL) enable calculation of unknown currents, resistances, and voltages in complex networks with multiple sources and loops.
3. Give the expressions for The Junction Law and The Loop Law in just one line.
Junction Law (KCL): The sum of currents entering a junction is equal to the sum of currents leaving it: ∑I = 0.
Loop Law (KVL): The sum of all voltage changes around any closed loop is zero: ∑V = 0.
4. What is the fundamental principle behind the validation of Kirchhoff's 1st Law?
Kirchhoff’s First Law (Current Law) is based on the principle of conservation of charge—no charge is lost or accumulated at a junction in a steady-state circuit.
5. What is the fundamental principle behind the validation of Kirchhoff's 2nd Law?
Kirchhoff’s Second Law (Voltage Law) arises due to the conservative nature of electrostatic forces—the net work done by the electric field in any closed loop is zero, following conservation of energy.
6. State Kirchhoff’s Second Law.
Kirchhoff’s Second Law (also called the Voltage Law or Loop Rule) states: The algebraic sum of all the electromotive forces (emf) and potential drops in a closed circuit loop is zero. That is, ∑emf - ∑IR = 0.
7. What is conserved in Kirchhoff's Second Law?
Kirchhoff’s Second Law is based on conservation of energy—the total energy supplied to charges in a closed loop equals the total energy used or lost in resistances and other circuit elements.
8. What is the sign convention for emf and resistance in Kirchhoff's Second Law?
Emf (Battery):
• Moving from - to + (battery's direction): take emf as positive (+).
• Moving from + to - (opposite direction): take emf as negative (−).
Resistor (IR term):
• Moving with current direction: potential drop is –IR.
• Moving against current direction: potential rise is +IR.
9. What are the main limitations of Kirchhoff’s Laws?
Kirchhoff’s Laws are not strictly valid in AC circuits of very high frequency, or when changing magnetic fields are present.
• KVL fails if there is a time-varying magnetic field (Faraday’s Law), as induced emf appears in the loop.
• The laws assume lumped elements and negligible radiative/inductive effects.
10. List common mistakes students make when applying Kirchhoff’s Second Law.
Common mistakes include:
• Mixing up sign conventions for sources and resistors
• Not following the assumed loop direction consistently
• Forgetting to include all emfs and voltage drops in the loop
• Arithmetic errors when solving simultaneous equations
• Not drawing clear circuit diagrams before analysis
11. How do you apply Kirchhoff’s Second Law to solve for unknown currents in a circuit?
To solve for unknown currents:
1. Mark the direction of assumed currents in each branch.
2. Apply KCL at junctions if needed.
3. Write a KVL equation (loop equation) for each closed path: add emfs and subtract IR terms as per sign convention.
4. Solve the simultaneous equations to find current values.
12. What is the difference between Kirchhoff’s First Law and Second Law?
Kirchhoff’s First Law (KCL): Deals with the conservation of charge at a junction—total current in equals total current out.
Kirchhoff’s Second Law (KVL): Deals with conservation of energy in a closed loop—algebraic sum of all emfs and voltage drops is zero.
