Kirchhoff's Laws

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Recall Kirchhoff's first law and understand that it is a consequence of conservation of charge
  2. Recall Kirchhoff's second law and understand that it is a consequence of conservation of energy
  3. Derive, using Kirchhoff's laws, a formula for the combined resistance of two or more resistors in series
  4. Use the formula for the combined resistance of two or more resistors in series
  5. Derive, using Kirchhoff's laws, a formula for the combined resistance of two or more resistors in parallel
  6. Use the formula for the combined resistance of two or more resistors in parallel
  7. Use Kirchhoff's laws to solve simple circuit problems

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What are Kirchhoff's Laws?

Kirchhoff's Laws are two fundamental rules that help us analyze and solve electrical circuits. They are especially useful when dealing with complex circuits that have multiple batteries or multiple paths for current to flow. These laws are based on two basic principles of physics: conservation of charge and conservation of energy.

Kirchhoff's First Law (Current Law)

Statement of the law:

The sum of the currents entering a junction is always equal to the sum of the currents leaving the same junction.

In simpler terms: all the current flowing into a point must equal all the current flowing out of that point.

What is a junction?

A junction (also called a node) is any point in a circuit where three or more wires meet. It's where the current can split into different paths.

Why does this law work?

Kirchhoff's first law is a consequence of the conservation of charge. This means that electric charge cannot be created or destroyed. When current (which is moving charge) reaches a junction, it must all go somewhere — it can't just disappear or pile up at the junction. So the total current coming in must equal the total current going out.

Mathematical form:

∑Iin=∑Iout\sum I_{\text{in}} = \sum I_{\text{out}}

The symbol Σ (sigma) means "sum of" or "add up all of".

Example:

If a current of 3 A enters a junction and splits into two branches, with one branch carrying 2 A, then the other branch must carry 1 A because:

Current in = Current out
3 A = 2 A + 1 A

Important points:

  • In a series circuit (where components are connected end-to-end in a single loop), the current is the same everywhere because there are no junctions where it can split.
  • In a parallel circuit (where the circuit divides into branches), the current divides at junctions. The total current from the battery equals the sum of the currents in all the branches.

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