1.2 The quantities

On the chainsaw’s pack, three numbers: 56 V, 5.0 Ah, 280 Wh. The third is the product of the other two, and none of the three says how long the saw will cut. Yet that is the only question anyone asks in front of the woodpile. It takes six quantities to answer it, and knowing which one is missing.

What flows: charge (the coulomb, C) and current (the ampere, A), charge per second. What pulls and what holds back: voltage (the volt, V) and resistance (the ohm, Ω). What the power bill reads: power (the watt, W), a rate of energy, and energy (the joule, J, or the watt-hour, Wh). Six people’s names, and that is no coincidence. Until 1881, every laboratory had its own ohm: telegraph engineers counted in miles of wire, Werner Siemens sold a column of mercury, and the numbers did not meet from one side of the Atlantic to the other. So they sat down in Paris, at the International Congress of Electricians, and voted the ohm, the volt, the ampere, the coulomb and the farad; the watt and the joule followed in 1889. The units did not fall out of nature, they were voted.

The watt has a commercial history. Boulton and Watt collected a royalty on the coal their engine saved; but the customer who had never owned an engine counted in horses, so Watt measured a mill horse, rounded up, and set the horsepower, 746 W in our units. A century later, the unit that replaced the horse carries the salesman’s name.

Power is a rate; energy is what has flowed. The joule is too small for a house, so we count a power held for an hour: the kilowatt-hour on the bill, the watt-hour on the pack, 3,600 joules. The 280 Wh on the label make one megajoule, enough to lift a tonne a hundred metres.

The amp-hour is the battery unit, and Kuphaldt jokes that it was invented to make students’ lives harder. It counts a charge, one ampere for one hour, 3,600 coulombs; not an energy, because the voltage is missing. On a pack whose voltage is known, the shortcut is harmless: 5.0 Ah at 56 V make 280 Wh. On a USB power bank, it misleads: the "10,000 mAh" are counted at 3.7 V, the cell’s voltage, which is 37 Wh; the port delivers 5 V, and 37 Wh at 5 V make 7,400 mAh at best, before the converter takes its ten to twenty percent.

And the saw. The 280 Wh are counted at 56 V; in the cut, where the pack works near 48 V, it gives back closer to 240. The saw draws around 1,800 W in hardwood. Time is energy divided by power: 240 Wh ÷ 1,800 W, eight minutes of chain in the wood. Twenty-five minutes at the woodpile, with the logs to turn and the restarts, is about eight minutes of cutting. Nothing is wrong with the pack; the label gave no time because it does not know the tool.

Prefixes read in bundles of a thousand, kilo, mega, giga going up, milli, micro, nano going down, and the common mistake is the case: 1 mΩ against 1 MΩ, a billion apart. The order of magnitude is guessed before it is calculated. A red LED, 20 mA at 2 V, 40 mW. A Raspberry Pi 4, 0.6 to 1.25 A at 5 V, 3 to 6 W. The saw, 30 to 40 A at 48 V, around 2 kW. A 100 W panel, 5.5 A at 18 V under test sunshine, 60 or 70 W at our place at noon in October. Five orders of magnitude between the LED and the saw.

The classic mistake: put the voltmeter on the pack and conclude. The voltmeter reads a voltage at one instant; energy is a power that lasts, and the only instrument that reads it is a meter, because it counts time.