Coulomb's law — force, comparison with gravity: Visual walkthrough

We will only use ideas a 12-year-old already owns: a dot, a distance, an arrow, the surface of a ball, and the fact that "spreading the same stuff over more area makes it thinner."

Step 1 — A charge is a dot that "reaches out" in every direction

WHAT. Draw a single tiny charged speck — a point charge. We call the amount of charge it carries $q_{1}$ (read: "q-one"). The little dot has no size; all its charge sits at one point.

WHY start here. Before two charges can push each other, we must picture what one charge does to the space around it. Coulomb's law is exact only for these idealised dots (see the parent's last mistake), so we begin honestly with a dot.

PICTURE. Arrows fan out from the dot equally in all directions — up, down, left, right, diagonally. No direction is special. This "sameness in every direction" is the seed of the whole derivation.

Figure — Coulomb's law — force, comparison with gravity

Step 2 — Put a second charge at distance $r$ : now there is a force

WHAT. Place a second dot, charge $q_{2}$ , a distance $r$ away from $q_{1}$ . The straight line joining the two dots is the only special direction now. Along that line each charge feels a push or pull — a force, which we draw as a single arrow.

WHY. A force needs two things to act between; one lonely charge feels nothing. The moment a partner appears, a direction (the joining line) and a strength (how hard) are born. Our whole job is to find that strength.

PICTURE. Two dots, a dashed line between them labelled $r$ , and a red force arrow on $q_{2}$ pointing along that line.

Reveal the symbols so far:

q_{1}

the charge on the first (source) dot.

q_{2}

the charge on the second dot, the one we watch the force on.

r

the straight-line distance between the two dots.

Step 3 — Why the strength must fall off as $1/ r^{2}$ (the spreading ball)

WHAT. Imagine the influence of $q_{1}$ as a fixed amount of "spray paint" leaving the dot and spreading outward. At distance $r$ all of it is smeared over the surface of a ball (a sphere) of radius $r$ . The surface area of that ball is $4 π r^{2}$ . So the paint's thickness at distance $r$ is

$how strong here influence density = \frac{total influence fixed, never lost}{area it spreads over 4 π r ^{2}} .$

WHY $1/ r^{2}$ and not $1/ r$ or something else? Because space is three-dimensional, and a growing sphere's surface grows like radius squared. We use area — not circumference, not volume — because the influence is conserved and it is the surface it passes through that shares it out. Double $r$ → the ball's skin is $4 \times$ bigger → the same paint is $4 \times$ thinner. That is the entire origin of the inverse-square law.

PICTURE. Two concentric spheres, radius $r$ and radius $2 r$ , the outer one four times the area, the same number of arrows piercing a patch that is four times as spread out.

Term-by-term on $A = 4 π r^{2}$ :

$A = fixed number for a full sphere 4 π radius squared = why inverse-SQUARE r^{2}$

So far: $strength \propto \frac{1}{r ^{2}}$ .

Step 4 — Why the strength doubles when either charge doubles ( $\propto q_{1} q_{2}$ )

WHAT. Now ask how the force depends on how much charge each dot carries. Split $q_{1}$ into two equal halves sitting on top of each other. Each half pulls $q_{2}$ with its own force; the two forces point the same way and simply add. So twice the charge → twice the force. The same argument works on $q_{2}$ .

WHY multiply, not add, the two charges? Because doubling $q_{1}$ doubles the force and doubling $q_{2}$ doubles the force. A quantity that doubles when either of two inputs doubles must be proportional to their product — that is what a product does. Hence $strength \propto q_{1} q_{2}$ , never $q_{1} + q_{2}$ .

PICTURE. On the left, $q_{1}$ as one dot with one force arrow on $q_{2}$ . On the right, $q_{1}$ shown as two stacked half-dots, two arrows adding into one twice-as-long arrow.

Combine Steps 3 and 4:

$strength \propto \frac{q _{1} q _{2}}{r ^{2}} .$

Step 5 — Turn "proportional to" into an equation: the constant $k$

WHAT. "Proportional to" ( $\propto$ ) means "equal up to a fixed multiplying number." Nature must supply that number so the units and the measured size come out right. Call it $k$ . Then

$F = Coulomb constant sets the size and units k the shape we derived \frac{q _{1} q _{2}}{r ^{2}} .$

WHY a constant at all? The spreading argument fixes the shape ( $q_{1} q_{2} / r^{2}$ ) but not the scale. Experiment measures the scale once and packs it into $k = \frac{1}{4 π ε _{0}} \approx 8.99 \times 1 0^{9} N\cdotm^{2} / C^{2},$ where $ε_{0}$ (the permittivity of free space) records how "willing" empty space is to carry electric influence.

PICTURE. A dial labelled $k$ turning the abstract shape $q_{1} q_{2} / r^{2}$ into a real number of newtons.

k

the constant

1/ (4 π ε_{0}) \approx 8.99 \times 1 0^{9}

N·m²/C² that turns the shape into real newtons.

Why absolute value in the scalar form?

so the magnitude is always positive; direction is handled separately.

Step 6 — The sign decides push or pull (all four cases)

WHAT. A charge can be positive or negative. Multiply the two signed charges. If the product $q_{1} q_{2}$ is positive (both + or both −), the charges repel — arrows point away from each other. If the product is negative (one +, one −), they attract — arrows point toward each other.

WHY the product's sign is enough. There are only four sign combinations, and they collapse to two outcomes:

$q_{1}$	$q_{2}$	product $q_{1} q_{2}$	result
$+$	$+$	$+$	repel
$-$	$-$	$+$	repel
$+$	$-$	$-$	attract
$-$	$+$	$-$	attract

"Like repels, unlike attracts" is just "product positive → apart, product negative → together." The vector form does this automatically:

$F_{21} = k \frac{q _{1} q _{2}}{r ^{2}} \overset{r}{^}_{21},$

where $\overset{r}{^}_{21}$ is a length-1 arrow (a unit vector) pointing from 1 to 2. If $q_{1} q_{2} > 0$ the whole thing points along $\overset{r}{^}_{21}$ (away from 1 → repel); if $q_{1} q_{2} < 0$ it flips (toward 1 → attract).

PICTURE. Four little panels, one per row of the table, each with the two dots and their force arrows.

Step 7 — Degenerate and limiting cases (never get surprised)

WHAT & WHY — walk every edge:

$r \to 0$ (charges merge). $F = k ∣ q_{1} q_{2} ∣/ r^{2} \to \infty$ . The formula blows up because two point charges can be pushed arbitrarily close. Real objects have size and structure, so this infinity is a signal that the point-charge idealisation has stopped applying.
$r \to \infty$ (very far apart). $F \to 0$ , but gently — as $1/ r^{2}$ , never abruptly. The reach never truly ends; it only fades.
$q_{1} = 0$ or $q_{2} = 0$ . $F = 0$ . A neutral partner feels and exerts no electric force — this is why large neutral bodies let feeble gravity win.
Extended (non-point) charges. The formula applies as written only to points (or perfect spheres, treated as all-charge-at-centre). For a blob you chop it into tiny points and add every contribution — again the Superposition Principle.

PICTURE. A single graph of $F$ against $r$ : a steep cliff near $r = 0$ , a long thin tail toward large $r$ , with the two limits annotated.

Recall Which case is which?

As $r \to 0$ , $F$ goes to ::: infinity (the $1/ r^{2}$ diverges). As $r \to \infty$ , $F$ goes to ::: zero, but slowly, like $1/ r^{2}$ . If either charge is zero, $F$ is ::: exactly zero — no partner, no force.

The one-picture summary

Everything at once: a point charge sprays influence over a sphere ( $4 π r^{2}$ , giving $1/ r^{2}$ ); two charges' amounts multiply ( $q_{1} q_{2}$ ); a constant $k$ sets the scale; the sign of the product picks push or pull.

$F_{21} = scale k product / sphere area (sign sets direction) \frac{q _{1} q _{2}}{r ^{2}} direction 1 \to 2 \overset{r}{^}_{21}$

Recall Feynman retelling — the whole walkthrough in plain words

Start with one charged speck. It reaches out equally in all directions, like paint sprayed from a point. Move twice as far away and that paint is spread over a ball with four times the skin, so it's four times thinner — that's the "one over r-squared." Now bring a second speck: if you double the charge on either speck you double the push, and "doubles when either one doubles" is just multiplication, so the strength goes with the two charges multiplied. Nature adds one fixed dial, called $k$ , to turn that shape into real newtons. Finally, check the signs: plus-and-plus or minus-and-minus shove apart, plus-and-minus snap together — the product's sign tells you which. Squeeze them to zero distance and the push screams to infinity; pull them infinitely far and it fades to nothing but never quite dies; make either one neutral and the force vanishes. That last fact is the whole reason gravity ever gets to win.