For 2,000 years we had the pull and no number for it. Then a soldier hung a balance from a single wire and counted the turns. After that the pull could be written down. What can be written can be sent.
⟨ transmission ends ⟩In this fragment you put the first number on the pull: how hard 2 charges push on each other. By the end you can compute the push between any 2 charges from their amounts and their distance, and add up the pushes of a whole crowd of charges, arrow by arrow.
In The Two Kinds of Charge the hanging amber told you which way: away or toward. Here a scale tells you how much. A charged metal sphere stands on a sensitive scale, zeroed before anything else came near. An identical charged sphere hangs above it on a slide, 10 cm up. Like charges push apart, so the top sphere presses the bottom one down, and the scale reads that push: 90 µN, about the weight of 4 mosquitoes. The top sphere slides. Bring it down to half the gap.
At a 10 cm gap the scale reads 90 µN. You bring the top sphere down to a 5 cm gap, half the distance. The scale will read…
Both spheres carry 10 nC, drawn as the dots on them, 0.5 nC per dot; the scale was zeroed before the top sphere was brought in, so it reads only the push. A real scale shows milligrams: 1 mg of reading is 9.8 µN of push. The slide runs from 12 cm down to 2 cm. Spheres are drawn as points for the arithmetic.
Half the gap, 4 times the push; half again, 16 times. The push does not fall with distance, it falls with distance squared: 3 times as far, 9 times weaker; 10 times as far, 100 times weaker. Charles-Augustin de Coulomb measured this shape in 1785, and it is gravity’s shape too.
Distance is half the law. The other half is the charge itself. The top sphere is back at 5 cm and the scale reads 360 µN. Beside the scale waits a twin of the top sphere, identical and uncharged. Two identical spheres touched together share their charge equally; there is no other fair split. Touch the twin to the top sphere, then set it back on its stand.
The twin takes half the top sphere’s charge away. With the gap unchanged, the scale will read…
Both spheres at 10 nC (20 dots of 0.5 nC each), centres 5 cm apart, 360 µN; same scale as above. The twin pushes too while it is near, and the scale shows that honestly; a scale reads only the up-and-down part of a push, so the twin on its stand, level with the bottom sphere, adds nothing.
Half the charge, half the push: in plain proportion, not squared. The bottom sphere carries charge too. Is it only the detector, or does its charge count the same way? A fresh twin waits. Halve the bottom sphere.
The top sphere is at half charge and the scale reads 180 µN. You now halve the bottom sphere too. The scale will read…
Top sphere 5 nC, bottom sphere 10 nC, 5 cm apart, 180 µN at the start. Same scale, same rules.
Half and half made a quarter. The push goes with the product of the 2 charges: each counts fully, and neither is merely the detector. Put the 2 halves together and you have the first equation of electricity.
Read it 3 ways. In words: the push grows with each charge and shrinks with the square of the distance. On the bench: double either charge and the scale doubles; halve the gap and it reads 4 times. In numbers, with the scale at 5 cm:
That last line is why nobody keeps a coulomb in a jar: a whole coulomb of charge would tear itself apart. Real charges, on balloons and combs and your amber, are billionths of one.
One question is left: 2 charges are only ever a pair. What happens with 3? Below, a test charge of 10 nC is held still at the centre of a bench by a thread from above. Source A, 10 nC, sits 5 cm to its west and pushes it east with 360 µN, drawn as an arrow. Source B, another 10 nC, waits far away. Drag B onto the mark 5 cm south of the test charge.
A pushes the test charge east with 360 µN. B, on the mark, pushes it north with 360 µN. With both in place the test charge feels…
All 3 charges are +10 nC, 20 dots of 0.5 nC each; A is fixed 5 cm west of the test charge. Each arrow is that source’s push computed from Coulomb’s law at the source’s actual distance, so B pushes a little even from far away and the readouts say so. The dashed arrow is B’s push copied to the tip of A’s: tip to tail. The test charge is held, so the arrows show what it feels, not where it goes.
A’s arrow did not change when B arrived, and the total was the 2 arrows laid tip to tail: 509 µN to the northeast, not 720. Every pair of charges pushes as if no other charge existed, and the pushes add as arrows. This is called superposition, and it could not have been guessed: nature might have made charges that got in each other’s way. Coulomb’s law and superposition are the 2 facts of charge at rest, and together they are the whole of it.
When you are ready, test yourself:
Your hanging amber tilts 16° against a charged stone 5 cm away. You want the same 16° tilt with the stone moved back to 10 cm. The stone must carry…
Two stones of equal charge sit 5 cm either side of a held test charge, one east, one west, both pushing it. The test charge feels…
You halve both charges and halve the distance between them. The push…
You can now compute the push between any 2 charges, and add up a crowd. But every sum you did today was about the test charge, and the pushes vanish with it. Next, Faraday asks a stranger question: is something there even when no test charge is? The Field.
Lesson 3 done.
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