Kirchhoff's laws, explained with examples

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

Kirchhoff's two laws let you analyse any circuit, even ones with several batteries and loops where simple series and parallel rules are not enough. They are conservation laws in disguise: one for charge and one for energy.

The junction rule (Kirchhoff's first law)

The total current flowing into a junction equals the total current flowing out.

ΣI_in = ΣI_out. This is conservation of charge: charge cannot pile up at a junction or vanish there, so whatever arrives each second must leave. If 5 A flows into a junction and 2 A leaves along one branch, then 3 A must leave along the other.

The loop rule (Kirchhoff's second law)

Around any closed loop, the sum of the emfs equals the sum of the potential differences (IR drops).

Σε = ΣIR, or equivalently the total change in potential around a loop is zero. This is conservation of energy: a charge that travels round a loop and returns to its starting point must end with the same energy per coulomb it started with. Every volt gained in a battery is used up in the components.

Sign conventions

Choose a direction to travel round each loop and stick to it.

• Crossing a battery from − to + is a gain of ε; from + to − is a loss.
• Crossing a resistor in the direction of your assumed current is a drop of IR; against it is a rise of IR.

You do not need to guess current directions correctly. Pick any direction for each unknown current. If a current comes out negative, it simply flows the other way.

A method for any circuit

1. Label every unknown current with a direction.
2. Write junction equations to link the currents.
3. Write loop equations, one for each independent loop.
4. Solve the simultaneous equations.
5. Interpret signs: a negative current flows opposite to your arrow.

Worked examples

Example 1: the junction rule

Currents of 2.0 A and 1.5 A flow into a junction. One wire carries 0.5 A out. What is the current in the other wire leaving the junction?

• In: 3.5 A. Out: 0.5 A + I.
• I = 3.0 A.

Example 2: one loop

A 9.0 V battery with internal resistance 0.5 Ω is connected to 2.0 Ω and 1.5 Ω resistors in series. Find the current.

• Loop rule: 9.0 = I(0.5 + 2.0 + 1.5) = 4.0I.
• I = 2.25 A. This is just the series rule, derived from Kirchhoff's second law.

Example 3: parallel branches

A 12 V battery drives current through a 2 Ω resistor, then splits between 6 Ω and 3 Ω resistors in parallel. Find each current.

• Let I be the total current, I₁ through 6 Ω and I₂ through 3 Ω. Junction: I = I₁ + I₂.
• Loop through the 6 Ω: 12 = 2I + 6I₁. Loop through the 3 Ω: 12 = 2I + 3I₂. So 6I₁ = 3I₂, giving I₂ = 2I₁.
• Then 12 = 2(3I₁) + 6I₁ = 12I₁, so I₁ = 1 A, I₂ = 2 A and I = 3 A.

Example 4: two batteries

A 10 V battery and a 2 Ω resistor form the left branch; a 4 V battery and a 1 Ω resistor form the right branch; a 4 Ω resistor forms the middle branch, joining the top and bottom wires. Both batteries have their positive terminals at the top. Find the current in each branch.

• Assume I₁ flows up through the left branch, I₃ up through the right branch, and I₂ down through the middle. Junction: I₂ = I₁ + I₃.
• Left loop: 10 = 2I₁ + 4I₂. Right loop: 4 = 1I₃ + 4I₂.
• Solving: I₂ = 9/7 ≈ 1.29 A, I₁ = 17/7 ≈ 2.43 A, and I₃ = −8/7 ≈ −1.14 A.
• The negative sign means the current in the right branch actually flows down, backwards through the 4 V battery. The stronger 10 V battery is charging it.

Potential at points in a circuit

The loop rule also lets you find the potential difference between any two points. Start at one point, add each rise and subtract each drop along any path to the other point. Different paths between the same two points must give the same answer, which is a useful check on your currents. If two paths disagree, there is an error in your loop equations or your arithmetic.

Why the laws work

Both laws come from conservation principles that hold everywhere in physics. The junction rule fails only if charge could build up, which happens in capacitors while they charge; then you treat the capacitor's plates carefully. The loop rule assumes the circuit is not affected by changing magnetic fields; when it is, Faraday's law adds an induced emf. For ordinary DC circuits, both laws are exact.

Common mistakes

• Mixing sign conventions partway round a loop.
• Writing more loop equations than there are independent loops, or forgetting the junction equation.
• Thinking a negative current means a mistake; it just means the real direction is opposite.
• Forgetting internal resistance when it is given.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are Kirchhoff's laws on the AP Physics exams?

Yes. In the current course framework, DC circuits, including Kirchhoff's junction and loop rules, are part of AP Physics 2 and AP Physics C: Electricity and Magnetism. They are not part of AP Physics 1.

Are Kirchhoff's laws in A level Physics?

Yes. A level Physics uses both laws, usually stated as conservation of charge and conservation of energy, to analyse series, parallel and multi-loop circuits.

Key takeaways

• Junction rule: current in = current out (charge conservation).
• Loop rule: sum of emfs = sum of IR drops (energy conservation).
• Pick directions; negative answers mean the opposite direction.
• Use one equation per independent loop plus junction equations.
• The laws explain the series and parallel rules.

Related guides

• Ohm's law
• A level Physics electricity
• Capacitors