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Electricity · Physics

Kirchhoff’s laws are conservation laws for circuits.

The junction rule expresses charge conservation. The loop rule expresses energy conservation. Together, they let you solve multi-loop circuits systematically.

This topic

Kirchhoff’s Laws

Apply the junction and loop rules, choose consistent sign conventions, and solve multi-loop circuits with clean algebra.

Current
Junction (current) rule
At any junction, charge cannot pile up in steady state. That means the total current flowing into a node equals the total current flowing out.
  • Statement: ΣIin = ΣIout
  • Choose current directions arbitrarily (sign handles it)
  • Nodes: points connected by ideal wire share the same potential
  • When it fails: time-varying charge storage (preview)
Start nano-lesson At a node: in equals out.
Voltage
Loop (voltage) rule
Around any closed loop, the net change in electric potential is zero. This is energy conservation for a charge completing a loop.
  • Statement: ΣΔV = 0 around a closed loop
  • Voltage rises across ideal sources (emf)
  • Voltage drops across resistors depend on current direction
  • Interpretation: conservative electrostatic field in DC circuits
Start nano-lesson Loop sums track energy changes.
Method
Sign conventions
Sign conventions keep Kirchhoff’s equations consistent. If you follow a loop direction and define current directions, the algebra will correct wrong guesses automatically.
  • Pick loop traversal direction (clockwise/counterclockwise)
  • Across a resistor: drop in direction of current
  • Across a source: rise from − to + terminal
  • Negative results indicate actual direction is opposite
Start nano-lesson Choose directions first; trust the signs.
Solve
Multi-loop circuits
Multi-loop problems reduce to a linear system. Use junction equations for nodes and loop equations for independent loops, then solve for currents.
  • Identify independent loops (avoid redundancy)
  • Use junction rules to relate branch currents
  • Write loop equations with consistent signs
  • Solve the resulting linear equations cleanly
Start nano-lesson Write equations systematically; don’t guess.
Practice
Practice & Exercises
Practice writing Kirchhoff equations and solving multi-loop circuits with careful sign conventions.
  • Write KCL equations at labeled nodes
  • Write KVL equations for independent loops
  • Use sign conventions correctly across resistors and sources
  • Solve for unknown branch currents and voltages
  • Exam-style multi-loop circuit sets
Open practice sets Label nodes and loops before any algebra.