Take the charge away and the space remembers it. Every point keeps a direction and a strength, waiting for whatever comes. We learned to read that waiting. It is the most useful thing we know.
⟨ transmission ends ⟩In this fragment you stop asking what 2 charges do to each other and ask what 1 charge does to the space around it. By the end you can read a field map, arrows or threads, and compute the push on any charge at any point from the field there: F = qE.
In The Inverse Square the push depended on both charges. Now keep the source fixed: a +10 nC sphere clamped to a bench, seen from above, with a mark 5 cm east of it. On a tray wait 3 test charges: 5, 10 and 20 nC. Put one on the mark and an arrow shows the push on it. A second readout divides that push by the test charge: the push per nC. The 10 nC test charge is on the mark now and feels 360 µN, 36 µN for every nC it carries. Swap it for the 20 nC one.
The 10 nC test charge feels 360 µN on the mark: 36 µN per nC. You swap in the 20 nC test charge. On the mark it feels…
Source +10 nC, clamped so the test charges cannot move it; mark 5 cm from its centre. Dots are charge, 0.5 nC each. The arrow is the push on the test charge from Coulomb’s law, 1 mm of arrow per 9 µN. Only one test charge fits the mark; a second one dropped there pushes the first off.
The push doubled; the push per nC did not move. 36 µN per nC, pointing away from the source, belongs to the mark itself, whatever you put there. That number with that direction is the electric field at the mark. And every other point around the source has its own.
Take the 10 nC probe off the mark and carry it anywhere. Wherever it goes, the bench keeps a small arrow behind it: the field at that point, drawn per nC. Start at the mark, 5 cm out, and walk straight away from the source to 10 cm.
At the mark, 5 cm out, the field arrow is 36 µN per nC pointing away from the source. At 10 cm, twice as far, the arrow is…
Source +10 nC, probe 10 nC; the stamped arrows are the field per nC, 1 mm of arrow per 0.9 µN/nC, one stamp every 2 cm of travel. The arrows stay when the probe leaves: they describe the bench, not the probe.
A quarter as long: the field falls with the square of the distance, exactly as the push did, because it is the push with the test charge divided out. The arrows you left behind are the map, and they are there whether or not a probe is. Michael Faraday drew the same map another way: threads that follow the arrows from point to point, his lines of force. Below, 2 sources, + and −, 8 cm apart, and the threads traced from the field they make together. Threads leave the + and end on the −, never cross, and crowd where the field is strong. Carry the probe to the strongest spot.
On the thread map, the push per nC is strongest…
Sources +10 nC and −10 nC, 8 cm apart; 16 threads traced from the summed field, starting on the + and ending on the − or leaving the bench. Drawn flat on the bench, the threads thin out with distance a little more slowly than the real field falls, so read the crowding as an order, not a count. The probe reads the exact field; its arrow grows with each tenfold of push, so a 100 µN/nC arrow is 2 steps longer than a 1 µN/nC one, not 100 times.
Crowded threads, strong field; spread threads, weak field; and the probe’s arrow always lies along the thread it sits on. That is the whole art of reading a field map: direction from the thread, strength from the crowding.
One last question. Back at the single source, the field on the mark is 36 µN per nC pointing away. The +10 nC probe sits there now. On the tray is a −10 nC test charge. Lift the probe off and put the − charge in its place.
You replace the +10 nC probe on the mark with a −10 nC one. The field at the mark…
Source +10 nC, mark 5 cm east; test charges +10 nC (phosphor dots) and −10 nC (amber dots, the sign you earned in The Two Kinds of Charge). The field arrow is drawn at the mark even when it is empty; the push arrow rides the visitor.
The field did not care who came. It is a property of the point, set by the sources alone; the visitor only decides how hard, and which way, it is pushed: a + charge along the field, a − charge against it. Write the 2 halves of that sentence as one definition.
Read it 3 ways. In words: the field is the push per coulomb, and it belongs to the point. On the bench: divide the arrow on the mark by the charge sitting there and the answer is the same for every test charge. In numbers, kept simple on purpose, then with the bench’s own values:
Turned round, F = qE: know the field at a point and you know the push on anything placed there, without asking what made the field. That is why the field is worth a name of its own.
When you are ready, test yourself:
The field at a spot is 50 µN per nC pointing north. You place a +4 nC charge there. It feels…
On a thread map, the threads around point A are packed twice as tightly as around point B. A 1 nC charge at A feels…
A −2 nC charge sitting at a spot is pushed west with 100 µN. The field at that spot is…
You can now read the space around a charge and predict the push on any visitor. But a map of pushes is a map of work: carry a charge across it and the field pushes you along or fights you the whole way. Counting that work gives the space a second number, a height, and the height has a name you already use every day: the volt. Next, The Potential.
Lesson 4 done.
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