Foundations — Define electric field and electric potential

Before you can read the parent note Electric Field and Electric Potential, you need every squiggle it uses to feel obvious. Below, each symbol is built from nothing: plain words → a picture → why the topic needs it. They are ordered so each one leans only on the ones above it.

1. Charge — the symbol $q$ and $Q$

The picture. Imagine two marbles. Ordinary marbles ignore each other. Charged marbles either shove apart or snap together, even without touching. That invisible willingness to push/pull is the charge.

Two flavours. Charge comes in $+$ (positive) and $-$ (negative). Same signs repel, opposite signs attract.

Why two letters? The topic separates a source charge that prepares space (we call it $Q$) from a test charge that we drop in to feel the effect (we call it $q_0$ — the little $0$ just means "the tiny visitor"). This split is the heart of the whole chapter.

Why the topic needs it :::: Every quantity in the note — force, field, potential — is per charge or between charges, so $q$ is the raw ingredient.

2. Distance — the symbol $r$ and the unit vector $\hat{r}$

The picture. Draw the source charge as a dot. Pick any point in space. The number $r$ is how far that point is. The little arrow $\hat{r}$ sits at that point and aims straight back along the line to the charge — but stretched to length 1 so it only tells direction.

Why $\hat{r}$ has length 1 :::: So that multiplying by it changes only the direction, never the strength, of whatever it multiplies.

3. Vector vs. scalar — the arrow over a symbol

The picture. Temperature is a scalar — "20°C" needs no direction. Wind is a vector — "20 km/h north-east" needs a direction, so we draw it as an arrow.

Why the topic needs this :::: The single deepest fact of the note is that field is a vector and potential is a scalar — you must feel the difference before line one.

4. Force — the symbol $\vec{F}$ and Coulomb's law

The topic gets its force from Coulomb's Law: $\vec{F}=\frac{1}{4\pi {\epsilon }_0}\frac{Q{q}_0}{r^2}\hat{r}$

Let us read every piece, since each was built above:

• $Q$ and $q_0$ — the two charges (Section 1).
• $r^2$ — distance squared (Section 2); the force weakens as you go further, and specifically as $1/r^2$.
• $\hat{r}$ — the direction of the push (Section 2).
• $\frac{1}{4\pi {\epsilon }_0}$ — a fixed conversion number, explained next.

Why the topic needs it :::: Both the field and the potential are manufactured from this one force law by dividing out the visitor charge $q_0$.

5. The constants — ${\epsilon }_0$, $\pi$, and $k$

The picture. Think of ${\epsilon }_0$ as how "thick" or "resistive" empty space is to the electric influence. The $4\pi$ appears because the influence spreads out over the surface of a sphere, and a sphere's area is $4\pi r^2$.

Why the topic needs it :::: $k$ is the single number that turns "charges and distance" into an actual force in newtons; it rides along into both $E=kQ/r^2$ and $V=kQ/r$.

6. Per unit — the fraction bar $\vec{F}/q_0$

Dividing the force by the test charge, $\vec{F}/q_0$, removes the visitor and leaves a fact about the place. Doubling the visitor doubles the force, so the ratio stays put.

Why the topic needs it :::: This is the trick that converts a two-charge story into a one-charge "property of space" — it defines both field and potential.

7. Work and energy — the symbols $W$ and $U$

The picture. Lifting a book up a shelf costs effort and stores energy (it can fall back down). Pushing a charge toward a like charge is the same: you shove against the repulsion, and that effort is stored as $U$, ready to be released.

Why the topic needs it :::: Potential $V$ is literally work per unit charge, $V=W/q_0=U/q_0$ — no work idea, no potential.

8. The integral sign — $\int$

Why an integral here and not simple multiplication? Because the force changes as $r$ changes (it follows $1/r^2$). "Force × distance" only works when the force is constant. When the force keeps changing, we must add up a fresh "force × tiny-distance" for each little slice — that is exactly what $\int$ does.

The topic uses the one result: $\int r^{-2}dr=-\frac{1}{r}$ This is why the field's $1/r^2$ becomes the potential's $1/r$: integrating (accumulating) the $1/r^2$ field over distance gives a $1/r$ potential.

Why the topic needs it :::: The potential is the accumulated field; accumulation of a changing quantity is integration.

9. The slope / derivative — $\frac{dV}{dr}$

The picture: potential is a hill. Plot $V$ (height) against position. The field is how steep the hill is under your feet, and it points downhill for a positive charge — hence the minus sign: $E=-\frac{dV}{dr}$

Why the topic needs it :::: This single equation is the bridge between the field-lens and the potential-lens — the punchline of the whole note.

10. Infinity as a starting line — $\infty$

The picture. Sea level is our chosen zero for mountain heights. "Infinity" is the sea level of electric potential — the place we agree costs nothing to sit at.

Why the topic needs it :::: The definition $V=W_{\infty \to r}/q_0$ literally starts the journey at infinity; without a chosen zero, "potential" would be a meaningless number.

How the pieces feed the topic

Equipment checklist

Self-test: can you answer each before revealing?

Connections

• Define electric field and electric potential (index 1.1.10) — the parent this page prepares you for.
• Coulomb's Law — the force law every symbol here feeds into.
• Electric Potential Energy — where work $W$ becomes stored energy $U=q_{0}V$.
• Voltage and Potential Difference — potential put to practical use.
• Equipotential Surfaces and Field Lines — the hill-and-slope picture as a map.