We stopped asking what a charge feels and asked what it costs. The answer was a landscape, with every point at its own height. Everything you will ever build runs downhill on it.
⟨ transmission ends ⟩This fragment teaches you to read voltage as height. Every point in space sits at its own height on an energy landscape, and the voltage between 2 points is simply the drop between them. By the end you will be able to say exactly what the 9 V printed on a battery promises.
Fragment 03 gave you the field: at every point, an instruction saying which way a charge gets pushed and how hard. Now push back. Drag a charge the wrong way, against the instruction, and you can feel it fight you. You are spending energy, and it does not vanish: let go and the field spends it right back, hurling the charge the way it wanted to go all along. So the field keeps count. Every point holds a second number besides its instruction: how much energy it costs to get a charge there.
Below is that count, drawn as a landscape. One charge is fixed on the left, and the rings around it are heights, the way a map draws a hill. Two survey flags stand on the slope, and a test charge of 1 nC sits on flag A.
Two different roads between the same two flags. Which one costs the charge more energy?
Landscape 36 cm across; the fixed charge is +13 nC, the test charge +1 nC. Rings mark equal steps of height, so crowded rings mean a steep slope. Heights and bills are in millionths of a newton-metre, per billionth of a coulomb. The hill is flattened within 1.2 cm of the charge.
Both roads cost exactly the same, however far you wandered and however high you climbed on the way. Climbing costs, descending refunds, and when you arrive the wandering has cancelled itself out. Only the 2 endpoints count.
That is what lets the landscape exist at all. If the road mattered, a point could not have a single height: you could arrive at the same spot by 2 roads and be owed 2 different amounts. Because the bill depends only on where you start and where you stop, every point can carry one honest number, and the drop between 2 points is a fixed fact about the pair.
That drop is what the ancestors called potential difference, and it is what a voltage is. It has its own equation:
Read it 3 ways. First, as a recipe: move a charge between 2 points, measure the energy it took, divide by the charge. What is left over belongs to the pair of points, not to the charge you used. Second, turned around, it is a promise: hand a point pair a charge, and it will spend that many joules on every coulomb. Third, in numbers, kept simple on purpose:
So the flags were never really at 0.58 and 1.17. Those are the same heights in the bench’s small units, and the drop between them is 584 volts. What the meter was billing you all along was joules per coulomb.
One thing about that landscape should bother you. Heights were measured from far away, where the charge’s reach has faded to nothing. Nothing forced that choice, and the strip below lets you make a different one.
Slide the zero. What happens to the 2 labels, and to the 584 V gap between them?
The same 2 flags as the landscape, at their true heights, now in volts. The slider moves nothing but the line you measure from.
Both labels moved together and the gap never flinched. That is the honest situation: only differences between points are real. A single voltage on its own means nothing until you say what it is measured from, and picking that spot is a free choice, like calling one particular level on a coastline 0 and measuring every mountain from it.
The ancestors made that choice on every bench they built, and they called the chosen zero ground. It is why a battery terminal can be labelled 9 V at all: the label is silently measured from the other terminal. Fragment 11 puts ground into a real circuit, and Fragment 28 shows what happens when 2 machines disagree about where it is.
Which leaves the promise printed on the battery. Its 2 terminals are nothing more exotic than 2 points held a fixed drop apart, and connecting anything across them gives charge a way to fall from the high one to the low one. Below is that drop, with a dial for how much charge you send down it.
Send 3 times as much charge down the same 9 V drop. What happens to the energy spent?
A 9 V battery drawn as what it is: 2 terminals held 9 V apart. The dial moves charge down that drop; the bar is the energy released. Nothing here says how fast, only how much.
9 V never meant 9 joules. It means 9 joules for every coulomb that makes the trip, over and over, until the chemistry inside runs out. So a voltage on its own never tells you how much energy something holds. You have to ask the second question too: how much charge can it move?
A 9 V battery pushes 2 C through a lamp. How much work did it do?
Three roads on the landscape: 1 goes straight from A to B, 2 goes from A to B swinging high over the source, 3 leaves A and wanders back to A. Which costs the most?
A bird sits on a bare 10,000 V power line and is perfectly fine. Why?
You can now read any voltage as a drop between 2 points, and turn it into joules whenever you know how much charge made the trip. What you cannot yet do is get charge to make that trip. On this landscape nothing moved unless your own hand dragged it, because empty space gives charge nothing to roll along. Some materials do: in metal, charge runs free. That is Fragment 05.