Foundations — Define voltage (potential difference) and its units

Why this page exists

The parent note freely uses letters and words — a "field", "work", "charge", and a ratio that turns them into voltage — without stopping to build them. If any of those is a fog to you, every sentence after it collapses. So here we stop and earn each idea one at a time: plain words first, then a picture, then the reason it must exist before the next.

We go in this order because each idea is a brick under the next:

$\text{charge}\to \text{field}\to \text{force}\to \text{work/energy}\to \text{"per charge"}\to \text{two-point difference}\to \text{voltage}$

No letter appears in a formula until the section that defines it. If you see a symbol you don't recognise, it is a bug — flag it.

1. Charge — the symbol $q$

Picture a small bucket. The number of "electric marbles" in the bucket is $q$. One full standard scoop of $6.24\times 10^{18}$ of the smallest marbles is called one coulomb.

• Plain meaning: "how much electricity you are holding."
• The picture: a labelled bucket of marbles — see the amber dots in the figure below.
• Why the topic needs it: voltage is defined per coulomb. You cannot divide by charge until you know what charge is.

Charge comes in two flavours: positive ($+$) and negative ($-$). Like flavours repel, opposite flavours attract. Keep that sign in mind — it decides which way things get pushed.

See Electric charge and the coulomb for the full story of the coulomb.

2. The electric field — the symbol $E$

(We meet the symbol $F$ for force formally in §3; here it just means "the push the field gives.")

• Plain meaning: the "steepness of the electric slope" at a point — how hard the field shoves each coulomb.
• The picture: arrows fanning out from a charge, like slope-lines on a hillside (figure below); a longer arrow means a bigger $E$.
• Why the topic needs it: the field is what you must work against when you move charge. No field → no hill → no work → no voltage. And notice $E=F/q$ already contains the "per charge" idea that voltage will reuse.

See Electric field and potential energy for how these connect.

3. Force and distance → Work — the symbol $W$

To push a marble uphill against the tilt, you must shove it, and you shove it over a distance. Those two things multiplied give work.

• Plain meaning: how much effort you spent shoving.
• The picture: an arrow (force) sliding a marble along a track (distance) up the tilt, both pointing the same way.
• Why the topic needs it: voltage's top number is exactly this work $W$.

4. The word "per" — why we divide by charge

This is the quiet hero of the whole topic. When we say "energy per charge", the word per means divide — the same move we already made for the field ($E=F/q$).

• Plain meaning: "share the total equally among each unit of charge."
• The picture: a total pile of energy split into equal stacks, one stack per coulomb.
• Why the topic needs it: this division is literally the definition of voltage. Miss "per" and you'll confuse volts (J/C) with joules (J) — the parent lists this as a top mistake.

5. Two points, not one — what "difference" means

Voltage is never a property of a single spot. It is always measured between a point $A$ and a point $B$ — like asking "how much higher is the top of the slide than the bottom?"

• The picture: two marked heights on the same slope; voltage is the gap between them, with an arrow pointing from low to high (figure below).
• Why the topic needs it: this is why we say "voltage across" and never "voltage at a lone wire." A single point only gets a number once you name a reference (call it $0\text{V}$, i.e. "ground").

6. Putting it together — the symbol $V$

Now every piece is earned, so the definition reads cleanly:

Read it in plain words: "how many joules each coulomb gets between the two points." That is all voltage ever means.

7. Two more symbols the parent leans on: $t$ and $I$

The parent's power derivation uses time and current, so we earn those too.

• Why the topic needs it: combining $I=q/t$ with $W=qV$ gives the parent's power formula. Watch the substitution: $P=\frac{W}{t}=\frac{qV}{t}=V\cdot \frac{q}{t}=VI$

See Electric current and the ampere and Power in electric circuits P=VI.

The build, as a picture

The graph below is meant to be read like a staircase: each block sits on the ones below it, and you cannot stand on a higher step until the lower steps hold your weight. Follow the arrows upward and you rebuild voltage from charge.

Full symbol table (the earned vocabulary)

A tiny sanity example

Equipment checklist

Test yourself — cover the right side.

Connections

• Define voltage (potential difference) and its units — the parent topic these foundations feed.
• Electric charge and the coulomb — the symbol $q$ built here in full.
• Electric field and potential energy — the field $E$ and the "tilt" this page introduces.
• Electric current and the ampere — the symbol $I$.
• Power in electric circuits P=VI — where $t$ and $I$ combine with $V$.
• Ohm's Law V=IR — the next equation that uses $V$.
• Batteries and EMF — a device that maintains a fixed $V$.