Kirchhoff’s Rules
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
Two rules solve any circuit. Junction rule: current into a junction equals current out (charge is conserved). Loop rule: the voltage changes around any closed loop add to zero (energy is conserved).
1. The two rules
2. A method that works
1. Call the bottom wire 0 V and an unknown junction V.
2. Write each branch current as (high V − low V) ÷ R.
3. Apply the junction rule and solve for V.
Or label a current in each branch with a guessed direction, write one junction equation and enough loop equations, and solve. A negative answer just means the current flows the other way.
3. Example, then try it
A 12 V battery runs a single loop with a 2.0 Ω and a 4.0 Ω resistor. Find the current and check the loop rule.
I = 12 ÷ (2.0 + 4.0) = 2.0 A. Drops: (2.0)(2.0) = 4.0 V and (2.0)(4.0) = 8.0 V.
Loop: +12 − 4.0 − 8.0 = 0 ✓.
In the circuit below (14 V and 4 V batteries, every resistor 2.0 Ω), find the current in each branch.
Set up: bottom wire = 0 V, top junction = V. Left branch: (14 − V)/2 in. Right branch: (4 − V)/2 in. Middle: V/2 out.
Junction rule: (14 − V)/2 + (4 − V)/2 = V/2 → 18 − 2V = V → V = 6.0 V.
Currents: left (14 − 6)/2 = 4.0 A in; middle 6/2 = 3.0 A down; right (4 − 6)/2 = −1.0 A, so 1.0 A flows out toward the 4 V battery.
Check the left loop: +14 − (4.0)(2.0) − (3.0)(2.0) = 0 ✓.
Answer: 4.0 A, 3.0 A and 1.0 A, matching the figure.
4. Common mistakes
Kirchhoff’s junction rule is a consequence of the conservation of which quantity?
Charge can’t pile up or disappear at a junction, so whatever flows in each second must flow out. The loop rule comes from conservation of energy.
Ready to practice?
5 questions on Kirchhoff’s Rules, each with a hint and an explanation for every wrong answer.