Kirchhoff's Laws
Two statements of pure common sense — charge doesn't vanish, energy doesn't appear — that unlock every circuit ever built.
Builds on: 1.5 Series & Parallel
KCL: what flows in, flows out
Gustav Kirchhoff (1845) wrote down the two bookkeeping rules of circuits. The current law (KCL) says: at any junction, the total current flowing in equals the total current flowing out. It has to — charge is conserved (Lesson 0.1) and it has nowhere to pile up. If 0.8 A and 0.4 A arrive at a node, exactly 1.2 A leaves.
You’ve already used KCL without the name: it is why parallel branch currents add up (Lesson 1.5), and why the current is the same everywhere in a series loop — a two-wire junction has one way in, one way out.
KVL: the energy books must balance
The voltage law (KVL) says: walk around any closed loop and the voltage gains (through sources) exactly equal the voltage drops (across loads). A coulomb arriving back where it started must have the same energy it left with — otherwise circuits would be perpetual-motion machines.
In a 9 V loop with two resistors, the drops V₁ + V₂ always total exactly 9 V — no matter what the resistors are. Change them and the shares shift, never the sum. Each resistor’s share is proportional to its resistance (V = I·R with the same I) — hold that thought for the next lesson.
Why these two laws matter so much
KCL and KVL plus Ohm’s law form a complete toolkit: write KCL at the nodes, KVL around the loops, and you get a system of simple equations that determines every voltage and current in any resistor network, no matter how tangled. Circuit simulators like SPICE do exactly this, millions of equations at a time. You’ll mostly use them informally — as sanity checks that make wrong answers obvious.
Measuring a real circuit? Voltages around a loop that don’t sum to the supply mean you’ve missed a drop — often a bad connection quietly eating volts. KVL turns a multimeter into a lie detector.
⚡ Lab — Conservation, Live
Left: a junction obeying KCL. Right: a loop obeying KVL.
- Push I₁ and I₂ around — I₃ has no choice at all.
- Make R₂ ten times R₁ and watch it grab ten times the voltage — the stack always totals 9 V.
- Try to break either law. You can’t. That’s the point.