We spent a century learning how hard the amber pulls. The answer was not a number but a shape: twice as close, four times as strong. That shape belongs to space, so light and gravity keep it too.
⟨ transmission ends ⟩This fragment teaches you to work out the push between 2 charges before anything moves: how strong it is, which way it points, and what happens to it when you shift them or change their charge. One short equation does all of that, and it is the first equation the archive hands you.
Fragment 01 left the size of the push unmeasured. You saw that like pushes like and opposites pull, but never how hard. Charles Coulomb settled it in 1786, and the answer turned out to rest on just 2 things: how much charge each object carries, and how far apart they are. Everything else about them, their size, their weight, what they are made of, makes no difference at all.
Below is his measurement, made easy. Two small charges sit on a rail, each carrying 1 billionth of a coulomb, written 1 nC. Both are clamped, so neither can fly away: the meter reads the push at whatever distance you set. They start 3 cm apart.
Halve the distance. Does the force double, or more?
Both charges hold 1 nC and are clamped, so nothing moves except by your hand, and each counts as a point. The meter reads against the weight of a 2 mg mosquito. The curve is the measurement itself, plotted point by point; the arrows use a squashed scale so the far end stays visible.
The force follows 1 divided by the distance squared, which is where this fragment gets its name. The reason is geometry, not electricity. Whatever a charge sends out, it sends out in every direction at once, spread over a ball of space that grows as you move away. Light dims by that rule, and so does gravity.
Distance is one half of the law. The other half is how much charge each sphere carries. On the same bench, the 2 clamps are now locked 3 cm apart, and each amount can be dialled.
Double one charge, then double the other as well. What does the meter do?
Same bench, same clamps, distance held at 3 cm. The sliders change nothing but how much charge each sphere carries. The glow around a sphere tracks what it holds; the arrows grow with the push on a squashed scale.
The force follows each charge in exact proportion, and there is no square anywhere here: the square belongs to distance alone.
Notice what the meter never showed you: a bigger charge feeling a bigger push than its small partner. Both spheres always feel the same size of force, in opposite directions, however lopsided the amounts. The pair is what matters, so the 2 amounts get multiplied together.
All of it fits on one line, the archive’s first equation. Its name is Coulomb’s law, and you have now measured every piece of it by hand:
Read it 3 ways. First, what it says: multiply the 2 charges, divide by the distance squared, multiply by k. Nothing else is allowed in. Not the size of the spheres, not their weight, not what they are made of.
Second, at the bench: the sign of the answer is the direction. Two like charges multiply to a positive number, and a positive F is a push apart. One of each kind multiplies to a negative number, and a negative F is a pull, of exactly the same size. Signs set direction, never strength.
Third, in numbers, kept simple on purpose:
One coulomb is a monstrous amount of charge: 2 of them, 1 m apart, would push with about 9 billion newtons, the weight of a million tons. That is why the bench works in nanocoulombs and reads its pushes in millionths of a newton, written µN. The opening pose, 1 nC and 1 nC at 3 cm, is 10 µN: about half that mosquito’s weight.
Two charges are the easy case. Real space is crowded. Below, a target charge is held in the middle with 1 source pushing it, and a second source is waiting off to the side.
Two pushes land on one charge at once. What force does it actually feel?
Two sources of 2 nC each push a 1 nC target, all 3 held in place. The dim arrows are the 2 pushes taken separately, the bright arrow is what the target feels. Every arrow here shares one scale, so the tip-to-tail picture is true to the numbers.
Two pushes of 20 µN each gave 28 µN, not 40. Arrows add tip to tail, and 2 arrows at a right angle make a diagonal shorter than their sum.
That sounds obvious, and it is not. Nothing forced nature to leave the deal between 2 charges alone when a third one walks up. Measurement says it does leave it alone, and the rule has a name.
Two charges feel 90 µN at 1 cm. What do they feel at 2 cm?
At what distance does that same pair feel 10 µN?
+q and −q at 1 cm: push or pull, and how strong compared with +q and +q?
You can now work out any electrostatic force there is. Take the charges 2 at a time, use Coulomb’s law on each pair, add the arrows. In principle that is the whole of electrostatics, and everything later in the archive rests on it.
In practice it turns painful fast. A thousand charges means a thousand arrows for every point you care about, redone the moment anything shifts. And the law has a hole in it: it says how hard the push is, and nothing about how the push crosses the gap. The way out is to stop asking about pairs and ask what a charge does to the space around it: Fragment 03, the field.