Every push has a price. We walked the whole space and wrote down what each step cost, and the ledger came out simpler than the map: one number per place. You call it the volt.
⟨ transmission ends ⟩In this lesson the space around a charge gets a second number: what it costs to be there. By the end you can read potential as height on an energy map, and compute the work any charge pays or collects between two points from the voltage between them.
The bench from The Field again, seen from above: a +10 nC source, clamped, with 2 marks east of it. A is 10 cm out, B is 5 cm out. The Field told you the force at every point; it said nothing about what a journey costs. So the bench now runs an energy meter: it counts the work your hand does, the energy spent pushing, in millionths of a joule. Carry the +10 nC probe from A in to B.
You push the + probe from A in to B, closer to the + source. The energy meter will…
Source +10 nC, probe +10 nC, k = 9×10⁹; the meter integrates the hand’s work along the probe’s actual path, so sliding outward runs it down. The jar fills with what your hand pays, drop for drop, and drains as the field pays it back. Dots are charge, 0.5 nC each.
9 µJ to climb from A to B, paid step by step as you pushed. Let it slide back out and the meter runs down to 0: the field pays every µJ back. Nothing is lost either way. But a suspicion should nag you: you dragged it in along one path. Would a longer path cost more? Take the wildest detour you can draw: loop, circle the source, wander off, then land on B.
The straight path from A to B cost 9 µJ. A long wandering path from A to B costs…
The meter sums the hand’s work along the exact trail drawn; the dotted line is the straight path you already paid 9 µJ for. Steps along a circle around the source are at right angles to the force and cost nothing; steps out are refunded by steps back in. Land within the B ring and the probe settles on the mark.
9 µJ again, on a path at least 3 times as long. The wandering cancels: every step along a circle around the source is at right angles to the force and costs nothing, every step outward is refunded on the way back in. Only the in-and-out distance is ever billed, so the cost belongs to the 2 endpoints and to nothing else. That is what makes a map possible: each place has ONE cost, whatever path reached it. The bench can paint that map. It draws the rings where the cost is equal, labelled with the potential, the cost per nC of arriving there from A.
Two spots sit on the same ring. Carrying the probe from one to the other costs…
Rings at 5, 6, 7.5, 10 and 15 cm, labelled +0.9, +0.6, +0.3, 0 and −0.3 µJ per nC with A as zero; the shade lens brightens uphill. The probe rides its ring like a bead on a wire: the hand sets where along, the ring keeps the distance.
The meter held still the whole way round: each ring is one potential, and the bench is a map of potential. But which way do the rings stack? The bench can cut the map along the probe’s line and draw what it finds in profile, with the probe as a ball on the drawn ground. Predict the shape, then climb.
Read all the ring labels as one shape. The bench is…
The strip is the landscape cut along the probe’s line, drawn from the same numbers as the rings; the ball is the probe, and it climbs as rings are crossed. The −0.3 ring is downhill from A: arriving there from A is paid for by the field.
A hill, highest against the source, falling away outward: that is what the ring labels were saying all along. The rings are its contour lines, and the downhill direction at every point is exactly the field you mapped in The Field: Faraday’s threads cross these rings at right angles, the way streams cross contour lines. One thing on the map is still a private choice: where zero is. The bench planted a flag on A and measured every potential from it. Move the flag.
You move the zero flag from A to B. What happens to the numbers?
The flag sets which ring reads 0 and the tinted water fills everything below that potential, so dragging the flag uphill raises the sea. Every label shifts by the same amount; no difference between any 2 points ever moves. Circuits will hand this same choice to one wire and call it ground.
Every label moved; no difference did. Zero on this map is bookkeeping, a sea level someone picked, and picking it differently changes no journey’s bill. Circuits make the same move: they pick one wire, call it ground, and measure every potential from it. One last question and the map is yours. The potentials are per nC. What does the whole bill depend on? Put different probes on A and carry each one in.
The 10 nC probe pays 9 µJ from A to B. The 20 nC probe will pay…
Probes 5, 10 and 20 nC, all starting on the A ring, one on a mark at a time. Every delivery prints a line in the bill; the last column never moves, because the bill is the charge times the 0.9 µJ per nC potential difference, however the path wanders.
4.5, 9, 18 µJ: the bill scales with the charge, and the cost per charge never moves. That per-charge cost is the number the whole lesson has been circling: the potential difference between B and A. Its everyday unit is the joule per coulomb, and that unit has a name you have used all your life.
Read it 3 ways. In words: voltage is energy per charge, the potential difference between 2 points, and it exists only BETWEEN points; its unit is the volt, and 1 V is exactly 1 J of energy for every 1 C of charge. On the bench: divide the meter’s bill by the probe that paid it and every probe gives the same answer. In numbers, kept simple on purpose, then with the bench’s own values:
The middle line of that block is worth doing with your hands, because it is the formula read backwards. Fix the bill and change the load instead: one jar holding 9 µJ, and 3 loads to spend it on. The same money buys a different voltage every time.
The same 9 µJ is spent on each load in turn. The bigger the load…
9 µJ ÷ 10 nC = 900 V, 9 µJ ÷ 1 µC = 9 V, 9 µJ ÷ 1 mC = 9 mV: one bill, 3 voltages, 100 000 apart from top to bottom. The scale climbs a decade at a time so all 3 fit beside each other. The loads are drawn bigger as they grow, but nowhere near to scale: 1 mC really is 100 000 times 10 nC.
One bill, three voltages, 100 000 apart from top to bottom. That is the formula read backwards, and it is the thing people get wrong about volts: a volt is not an amount of energy, it is energy per coulomb. The same 9 µJ is a mountain to 10 nC and almost nothing to a millicoulomb. It is why static electricity can reach thousands of volts on a dry day and still only sting, the voltage enormous and the charge a few grains, while a modest 9 V block runs a torch all night because it has coulombs to spend. The volt is a rate: price per coulomb, never a quantity of energy.
When you are ready, test yourself:
Carrying +2 C from a to b takes 6 J of work. The voltage between b and a is…
Two spots sit on the same ring of the energy map. Carrying a charge from one to the other, by any path, costs…
A 9 V battery pushes 2 C through a lamp. The energy the lamp receives is…
You move the zero flag from A to B and every potential on the map changes. What can you still trust?
You now own both maps of the space around charge: the field, which is the force on each coulomb at every point, and the potential, which is the price. Potential differences are what make charge want to move at all. So the next question is what happens when it CAN: in some materials charge runs downhill freely, in others it stays exactly where it was put. Next, The Conductor and the Insulator.
Lesson 5 done.
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