Foundations — Electric field of point charge, dipole, ring, disk, line charge (Gauss's law)

This page assumes you have seen nothing. We name every symbol the parent topic note uses, draw the picture behind it, and say why the topic needs it. Read top to bottom — each idea uses only the ones above it.

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1. What is "charge"? The symbol $q$ (and $Q$)

Picture two kinds of dots: positive (we draw them magenta) and negative (we draw them violet). Like kinds repel, opposite kinds attract. That is the entire personality of charge.

Why the topic needs it: charge is the source. No charge, no field. Everything on the parent page starts from either one dot ($q$) or a smear of many dots adding up to $Q$.

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2. Distance $r$ and the "how far, which way" arrow $\hat{r}$

Before we can say how strong a push is, we need how far away we are and in which direction.

Look at figure s01: the magenta charge sits at the centre, the black point $P$ is where we measure, the length of the dashed line is $r$, and the little orange arrow riding along it is $\hat{r}$ — always length 1, always pointing outward toward $P$.

Why we split "how far" from "which way": strength and direction are two different questions. $r$ answers how strong (through $1/r^2$, next section). $\hat{r}$ answers which way the push points. Keeping them separate is what lets us later say "these directions cancel, those add."

Recall Why must

$\hat{r}$ have length 1? So it only carries direction, never size ::: if it had a length, it would secretly change the strength, and we could never tell strength and direction apart.

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3. Vectors and the arrow-sum (superposition)

Two arrows are added tip-to-tail: slide the second arrow so its tail sits on the first arrow's tip; the sum is the arrow from the very first tail to the very last tip.

In figure s02 the two thin arrows are pushes from two separate charges at the same point; the thick navy arrow is their tip-to-tail sum. Notice the up-and-down parts cancel and only the sideways part survives — this cancellation is the whole trick behind the ring and dipole results.

Why the topic needs it: a real object has millions of charges. The field it makes is the arrow-sum of every one. That rule has a name.

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4. The field $\vec{E}$ itself — what the arrows mean

Picture a whole grid of little arrows filling the room, one at each point, telling a positive dot which way it would be shoved and how hard. Near a positive charge the arrows point away; near a negative charge they point toward it (see Flux and field lines).

Why "per unit charge": we want the field to describe space itself, not any particular test charge. Dividing out the test charge makes $\vec{E}$ a property of the source alone.

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

The formulas need a number that converts "amount of charge and distance" into "actual push." Two spellings of the same constant appear everywhere on the parent page.

Why two names for one thing: $k$ is compact for point-charge formulas ($kq/r^2$); the $\frac{1}{4\pi {\epsilon }_0}$ form is natural for Gauss's law because the $4\pi$ there is literally the surface of a sphere. Same number, chosen dress for the occasion.

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6. The $1/r^2$ falloff — why "spread over a sphere"

A fixed number of field arrows leave a charge and fan out. At distance $r$ they are smeared over the surface of a sphere. The surface of a sphere of radius $r$ has area

$A=4\pi r^2.$

So the density of arrows — which is the field strength — is (arrows) $÷$ (area) $\propto 1/r^2$. Figure s03 shows the same lines piercing a small near sphere densely and a big far sphere sparsely.

Why the topic needs it: this single geometric fact ($1/r^2$) is the seed of everything — Coulomb's law, and later the fact that a dipole falls faster ($1/r^3$) while a line falls slower ($1/r$).

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7. Density symbols: $\lambda$, $\sigma$ — "how packed is the charge?"

When charge is smeared over a shape, we describe how thickly it's packed.

Picture a wire coloured magenta: cut a length $\Delta \ell$ of it, and it holds charge $\lambda \Delta \ell$. Picture a sheet: cut a patch of area $\Delta A$, it holds $\sigma \Delta A$.

Why the topic needs them: the line, disk, and sheet in the parent page have their charge spread out, so we can't just write $q$ — we write "density × size of the little piece" to get each $dq$ (next section).

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8. The little-piece symbols: $dq$, $ds$, $dA$, and the integral $\int$

To arrow-sum a smeared charge we chop it into pieces so tiny each acts like a point.

Building each piece: $\text{line: }dq=\lambda ds,\text{ring of radius }s:dq=\sigma (2\pi s)ds.$ Why "$2\pi sds$": a thin ring of radius $s$ and width $ds$ has circumference $2\pi s$, so its area is (circumference)×(width) $=2\pi sds$, and its charge is $\sigma$ times that.

Why the topic needs this: the ring, disk, and line results are nothing but this chop-and-add, wearing an $\int$.

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9. Angle $\theta$, $\cos \theta$, and "which components survive"

When arrows point in slanted directions, we keep only the part that doesn't cancel.

In figure s04, a slanted field arrow from a ring element has length $dE$. The part along the axis (the direction that survives) is $dE\cos \theta$, where $\cos \theta =\frac{x}{\sqrt{R^2+x^2}}$ because the adjacent side is the axial distance $x$ and the hypotenuse is the full distance $\sqrt{R^2+x^2}$.

Why the topic needs it: on the ring's axis, the sideways parts of every element cancel by symmetry; only the axial part $dE\cos \theta$ lives. Multiplying by $\cos \theta$ is how we throw away the doomed component.

Recall On a ring's axis, which component of each

$dE$ survives? Only the axial (along-$x$) component, weighted by $\cos \theta =x/\sqrt{R^2+x^2}$ ::: the radial parts of opposite elements cancel.

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10. Flux, and the closed-loop integral $\oint$ (the Gauss machinery)

The parent uses Gauss's law as a shortcut. Two symbols must be understood first.

Why the topic needs it: when a shape is symmetric enough (sphere, cylinder, sheet), $\vec{E}$ is constant and perpendicular on a cleverly chosen surface, so the fearsome $\oint$ collapses to (field)×(area) — an instant answer. See also Coulomb's law and Flux and field lines.

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Prerequisite map

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Equipment checklist

Test yourself — you should be able to answer each before reading the parent topic.