Electronics · Math shelf · Optional — for the curious · build 2026.09.23-1644
The idea
The electrical units aren't a pile of arbitrary words — they're a tower, each one defined from the ones below it. Start with charge, and everything else is a rate or a ratio.
The bottom of the tower: the coulomb measures an amount of electricity, the way a liter measures an amount of water. One coulomb is the combined charge of about 6.24 × 10¹⁸ electrons — six billion billion of them. (Exponent shorthand is explained on the Milli, Kilo, Mega sheet.) A single electron's charge is unimaginably small; a coulomb is a bucketful.
A is the ampere — "amp" for short. One amp is one coulomb of charge flowing past a point every second: current isn't an amount, it's a rate, like liters per second through a pipe. Your LED's 14 mA is 0.014 coulombs per second — about 8.7 × 10¹⁶ electrons marching through every second. Eighty-seven thousand trillion. Per second. Through a wire you made at the wire station.
The joule measures energy itself — any kind. Lifting an apple one meter takes about 1 J. A fresh AA battery holds roughly 10,000 J. Energy is the thing every circuit exists to move around.
Here's the definition lessons have been dancing around: one volt means each coulomb carries one joule of energy. Your 5 V supply loads 5 joules onto every coulomb it pushes out. A 2 V drop across the LED means every coulomb passing through hands the LED 2 of its joules. "Voltage drop" is literally energy spent per unit of charge — that's why the drops around a loop must add up to the supply: every joule loaded gets spent, nothing vanishes.
Rescale both of those to a single electron and you get physics' favorite pocket units. One electron's charge — the elementary charge, written e — is exactly one 6.24-billion-billionth of a coulomb: e ≈ 1.6 × 10⁻¹⁹ C. And the energy one electron picks up crossing one volt is the electron-volt (eV):
1 eV = 1 ÷ (6.24 × 10¹⁸) J ≈ 1.6 × 10⁻¹⁹ J
The same 6.24 × 10¹⁸ converts both rungs of the tower at once: a coulomb is that many electron-charges, and a joule is that many electron-volts. Here's the payoff: since energy-per-electron is charge-of-one-electron × volts, an LED's drop in volts is the energy of each electron's landing, in eV. Your red LED's ≈ 2.0 V drop means each electron hands over ≈ 2.0 eV as it lands in a hole — and a photon of red light carries ≈ 1.9 eV: nearly the whole payment becomes light. (Blue runs less cleanly: its ≈ 3.2 V buys a ≈ 2.7 eV blue photon, with the change lost as heat inside the LED — device overhead.) The pocket formula: photon energy in eV ≈ 1240 ÷ wavelength in nm. When you measured LED drops in lesson 4, you were measuring the energy of light, one particle at a time — give or take the overhead — the How Transistors Work page (Under the hood) tells the landing-in-holes story.
The watt is the rate energy moves: one joule per second. And now watch the tower assemble itself:
amps × volts = C/s × J/C = J/s = watts
The coulombs cancel. P = IV isn't a rule somebody decreed — it's what the definitions multiply out to. Charge per second, times energy per charge, is energy per second.
| Unit | Built as | Meaning |
|---|---|---|
| ohm (Ω) | volts ÷ amps | how many volts of push one amp of flow requires — the squeeze, quantified |
| farad (F) | coulombs ÷ volts | how much charge a capacitor stores per volt across it. One farad is enormous — real capacitors live in µF, nF, pF (capacitor lessons ahead) |
| hertz (Hz) | cycles ÷ second | how often something repeats: 60 Hz wall power, GHz phone processors — and the blink rate of a 555 timer, later this course |
Nearly every electrical unit is a person — scientists honored by lowercase immortality:
| Unit | Named for | Claim to fame |
|---|---|---|
| volt | Alessandro Volta (Italy, 1745–1827) | built the first battery — the voltaic pile; Napoleon was so impressed he made him a count |
| ampere | André-Marie Ampère (France, 1775–1836) | heard that current moves compass needles, and within weeks founded the science of electromagnetism |
| ohm | Georg Ohm (Germany, 1789–1854) | schoolteacher whose V = IR was dismissed at first; decades later it was the law with his name on it |
| coulomb | Charles-Augustin de Coulomb (France, 1736–1806) | measured the tiny force between charges with a twisting wire and a very steady hand |
| watt | James Watt (Scotland, 1736–1819) | perfected the steam engine — and invented "horsepower" to advertise it; now his unit measures the horse's |
| joule | James Prescott Joule (England, 1818–1889) | a brewer who proved heat is energy; the story goes he even measured waterfall temperatures on his honeymoon |
| farad | Michael Faraday (England, 1791–1867) | bookbinder's apprentice with no formal math who became perhaps the greatest experimentalist ever — generators and transformers run on his discovery |
| hertz | Heinrich Hertz (Germany, 1857–1894) | first to make and detect radio waves — then declared them "of no use whatsoever." Radio arrived a decade later |
The capitalization rule
A unit named for a person gets a capital symbol but a lowercase word: 5 V but "five volts," 14 mA but "milliamps." Spot a capital letter in a unit, and there's usually a scientist behind it.
Answers at the bottom.
1. 5 J per coulomb × 0.5 C = 2.5 J. 2. 0.1 C per second × 10 s = 1 C. 3. Amounts: coulomb, joule. Rates: amp (coulombs/s), watt (joules/s). 4. 0.001 F × 5 V = 0.005 C — five thousandths of a coulomb, and that's a physically big capacitor. A whole coulomb is a lot of charge. 5. 3.2 eV — no conversion needed; volts-per-electron are electron-volts. (Each blue photon carries ≈ 2.7 of those eV; the rest becomes heat in the LED.)