Electronics · Math shelf · Optional — for the curious · build 2026.09.23-1644
The idea
Two conservation laws run every circuit ever built: energy (the voltage law) and charge (the current law). You've already measured both. Gustav Kirchhoff wrote them down in 1845 — as a 21-year-old student.
The version you proved in lesson 4: around any loop, the drops sum to the supply voltage. The formal dress is one notch sharper — give every voltage a sign (a rise through the supply is +, a drop across a part is −) and walk any closed loop:
around any closed loop: the signed voltages sum to 0
Why must it be zero? Voltage is energy per coulomb (the What's a Coulomb? sheet): a coulomb arriving back where it started must have spent exactly what the supply gave it. A loop that summed high would mint free energy; one that summed low would leak it into nowhere. Conservation of energy, wearing circuit clothes.
And the law says any closed loop — including a loop through two parallel branches that never touches the supply. Walk it and the sum forces a famous fact: parts in parallel share one identical voltage. Hold that thought for practice problem 4.
The version you've used since the junction dot: current into a node equals current out. Formal dress:
at any node: current in = current out
This one is conservation of charge: coulombs aren't created or destroyed at a junction, and they can't pile up on a wire — so the per-second flow in must equal the per-second flow out. It's why branch currents add (the whole parallel resistance law is KCL in disguise), and why a 0 V resistor reading condemns an entire single loop: no current here means no current anywhere on the one path.
Here's the formalization worth savoring: KVL + KCL + Ohm's law is the whole game. Every circuit-analysis trick this shelf has taught is those three laws pre-solved for a common case:
| The shortcut | What it secretly is |
|---|---|
| drops sum to the supply | KVL, one loop |
| series resistances add | KVL + Ohm, same current |
| parallel: reciprocals add | KCL + Ohm, same voltage |
| the divider formula | KVL + KCL + Ohm, solved once for two resistors |
Worked, to prove the divider claim: the 1 kΩ / 1 kΩ divider. KCL at the tap (nothing loading it): the current in R1 continues into R2 — one current, I. KVL around the loop: 5 − I·1000 − I·1000 = 0, so I = 2.5 mA. The tap sits above ground by R2's drop: 2.5 mA × 1 kΩ = 2.5 V. Same answer the divider formula hands you — because the formula is this derivation, done once and bottled.
The person
Gustav Kirchhoff (Germany, 1824–1887) published these laws in 1845 while still a student. He wasn't done: with Robert Bunsen he founded spectroscopy and explained the dark lines in sunlight — telling humanity, for the first time, what the sun is made of. The same mind behind "your drops must sum to five" also read the chemistry of the stars.
The math, for the curious — always optional
Write KCL for every node and KVL for every loop of a big circuit and you get a system of simultaneous linear equations — one unknown per current and node voltage. That's not just theory: circuit-simulation software (engineers use a family called SPICE) analyzes a design by building exactly that equation system and solving it, thousands of times over. Every chip in your laptop was rehearsed as Kirchhoff bookkeeping before it was ever manufactured.
Answers at the bottom. Problems 3 and 4 are buildable — predict, then measure.
1. 5 − 3.2 = 1.8 V — no Ohm's law needed. 2. 12 − 7 = 5 mA. 3. 5 − 0 − 1.7 = 3.3 V — KVL works on dark circuits too; the whole supply still lands somewhere. 4. Parallel parts share one voltage, and the red LED conducts at ≈ 2.0 V, clamping the shared node there. Blue needs ≈ 3.2 V to open, never gets it, stays dark: red hogs the branch. (Series red+blue: both dark. Parallel: red only. Both are Kirchhoff bookkeeping.)