Fields That ClickEpisode 5 / Thinning Out Is a Ratio of Dimensions — the True Nature of 1/r²

The 1/r² that kept showing up until now is not "a property of forces" — it's the fingerprint of space being three-dimensional

Thinning Out Is a Ratio of Dimensions Gravity and electricity both go as the inverse square of distance. Why "2"? The answer isn't inside the force but in one fact —
that space is three-dimensional. Field lines merely spread over a spherical surface — the exponent has the dimension written right into it.

Tools you need: surface area of a sphere, division, "massless → 1/r" from Episode 3 This episode's ratio: force ∝ 1/r^(d−1), d = the dimension of space

Across the previous episodes, $1/r^2$ has shown its face again and again. The Coulomb force and gravity alike, as if stamped out, go as the inverse square of distance. Usually you're just made to memorize "that's the law" — but why 2? This episode's answer is startlingly deep. $1/r^2$ is not a property of forces. It's the fingerprint of our space being three-dimensional. Field lines, neither vanishing nor multiplying along the way, merely spread over the surface of a sphere of radius $r$. Because the surface area of a sphere is proportional to $r^2$, the density is $1/r^2$. If space were two-dimensional it would be $1/r$; four-dimensional, $1/r^3$. The exponent in a force law is the number $d-1$, a direct copy of the dimension of space $d$. "Why the square?" was the same question as "how many dimensions?"

01The 1/r² That Kept Showing Up — Is It "a Property of Forces"?

Coulomb force $F\propto1/r^2$, universal gravitation $F\propto1/r^2$. Two utterly different forces have the same exponent "2." If this were "the personality of the electromagnetic force," there'd be no reason for gravity to have the same 2. Would 2 line up by coincidence? — It lines up. Because 2 is not a number on the side of the force, but a number on the side of the stage (space) through which the force spreads. If the stage is the same three-dimensional space, then any "field that doesn't vanish" spreading through it becomes the same $1/r^2$. First, let's confirm that "field lines don't vanish."

02Field Lines Don't Vanish — Conservation of Flux

Let's draw the electric field as lines welling up from a source (Faraday's lines of force). These lines have an iron rule — a line can only begin on a charge and only end on a charge. In empty space along the way, a line never vanishes or springs up. So no matter where you take a surface enclosing the source, the total number of lines piercing it is the same. This is conservation of flux = Gauss's law.

Gauss's law — the total number piercing an enclosing surface is set by the enclosed charge alone

$$\text{(total lines piercing a closed surface)}=\text{constant}\ \propto\ \text{(enclosed charge }Q\text{)}$$

Make the surface bigger or smaller, make it lopsided — the total number piercing it doesn't change. Since the lines don't vanish along the way, this is only natural.

03They Just Spread Over the Sphere's Area — Hence 1/r²

If the total number $N$ is conserved, then the field's strength at distance $r$ (how crowded the lines are — the density) is $N$ divided by "the surface area of the sphere at that distance." In three-dimensional space, the surface area of a sphere of radius $r$ is $4\pi r^2$. Hence ——

Try it — total number ÷ area $$\text{field strength}=\frac{N}{\text{sphere's surface area}}=\frac{N}{4\pi r^2}\ \propto\ \frac{1}{r^2}$$

There it is. $1/r^2$ is nothing but the result of "scattering $N$ lines over a spherical surface of area $4\pi r^2$." Not the personality of the force, but the flip side of area growing as $r^2$. Scatter the same $N$ lines over a larger, more distant sphere and they thin out — that's all it is.

This argument still works if you swap out the dimension of space. In $d$-dimensional space, the surface area of a "sphere" of radius $r$ is proportional to $r^{d-1}$ (in 3D, $r^2$; in 2D the perimeter of a "circle" is $r^1$; in 4D, $r^3$). Divide the total number by the surface area, and ——

This episode's ratio — the force's exponent is the dimension of space itself

$$\text{force}\ \propto\ \frac{1}{r^{\,d-1}}\qquad(d=\text{dimension of space})$$

2D $\Rightarrow1/r$, 3D $\Rightarrow1/r^2$ (inverse square), 4D $\Rightarrow1/r^3$. The exponent $d-1$ is a unitless integer. Just by writing one line of a force law, we are declaring how many dimensions space has.

04Play With It — Change the Dimension, and the Exponent Changes

The figure below. On the left are field lines coming out of a source (their number fixed = conservation of flux). Farther arcs are longer, so even with the same number of lines the gaps widen = it thins out. On the right, the relation between force strength and distance is drawn log–log (on a doubly logarithmic plot). On log–log, $1/r^{d-1}$ becomes a straight line, and its slope is exactly $-(d-1)$.

Try changing the dimension of space $d$ with the slider. At $d=3$ the slope is $-2$ (inverse square). At $d=2$ it's $-1$; at $d=4$, $-3$. Move the dimension dial and the line's slope moves right along with it as $-(d-1)$ — you can see with your own eyes that the exponent of a force law is itself the dial that reads out the dimension of space.

Figure: left = field lines out of a source (fixed count, thinning with distance). Right = force vs distance on a log–log plot. 1/r^(d−1) becomes a line, slope = −(d−1). Change the dimension of space d and the slope moves (at d=3, −2 = inverse square)

Dimension of space d 3.0

field lines (fixed count) force ∝ 1/r^(d−1) (a line on log–log)

05The Exponent of a Force Law Is the Dimension of Space Itself

Turn this view around and you get a remarkable use — measure the force's exponent precisely and you can measure how many dimensions space has. Any deviation from the inverse square becomes evidence of a hidden dimension. In fact, experiments have doggedly checked this.

The "2" the experiments measured Writing the Coulomb-force exponent as $1/r^{2+\delta}$, the $\delta$ is $|\delta|\lesssim10^{-16}$ — "2" is 2 to 16 digits of precision. Gravity's inverse square, too, shows no breakdown down to short distances of tens of μm (sub-millimeter). If space had a small, curled-up extra dimension, then closer than its size the force ought to change to $1/r^3$ or the like — string theory predicts this, and experiments have searched down to sub-millimeter, yet it's still inverse square. "The force exponent = 2" is a declaration from experiment that our space is (at least down to that distance) exactly three-dimensional.

It meshes cleanly with Episode 3, too. A massless field has infinite range ($\lambda=\infty$), and the exponential brake comes off Yukawa's $e^{-r/\lambda}/r$, leaving only the pure geometric thinning-out — which in three dimensions is $1/r$ (the potential), and as a force, $1/r^2$. Mass sets the "range," dimension sets the "exponent." How a force reaches us can be written out completely with just these two unitless properties — mass and dimension. The numbers of units-carrying newtons and coulombs were, again, stage machinery.

◇ ◇ ◇

The honest line — the reach of "lines" and "exponent = d−1"

Gauss's law (flux through a closed surface = enclosed charge) is exact; that from the "sphere-area $\propto r^{d-1}$" in $d$ dimensions a static long-range force becomes $1/r^{d-1}$; that this is realized for massless (infinite-range) fields; and that Coulomb's inverse square is verified at the $10^{-16}$ level while gravity's inverse square is verified at the sub-millimeter level — these are established physics.

But. ① "Field lines" are a visualization aid; strictly, the divergence $\nabla\!\cdot\!\vec E=\rho/\varepsilon_0$ (the language of flux) is the real body. ② $1/r^{d-1}$ is about the static field of a point source. Radiation emitted by motion (electromagnetic waves, gravitational waves) falls off in amplitude as $1/r$ (strength $1/r^2$) far away, and this is a different mechanism from the "static-field exponent" (radiation is covered in a bonus episode). ③ Extra dimensions are a theoretical possibility; experiment has so far not found them. In the regime of strong gravity (general relativity) or quantum gravity, the flat, static discussion here is too naive. What we want you to grasp in this episode is the one point that "the exponent of a long-range, static force is counting the dimension of space."

Practice problems (solvable with just this episode's formulas)

In 3D, if you double the distance $r$ from the source, by what factor does the field strength $\propto1/r^2$ shrink? Give the reason in terms of "the area of the sphere."
Show the answer
$1/2^2=1/4$. Doubling the distance makes the sphere's surface area $2^2=4$ times larger, and the same total number of lines spreads over that fourfold area, so the density is 1/4.
If space were two-dimensional, the force would go as the inverse of distance to what power? Give the reason in terms of $d-1$.
Show the answer
With $d=2$, $1/r^{d-1}=1/r^{1}=1/r$. In 2D what encloses the source is a "circle," whose perimeter is $2\pi r\propto r^1$. Divide the total number by the perimeter, hence $1/r$.
You plot a force log–log and the line's slope is $-3$. How many dimensions does space have?
Show the answer
From slope $=-(d-1)=-3$, $d-1=3$, $d=4$. Four-dimensional space (force is $1/r^3$). Measure the exponent and you know the dimension.
Why do the Coulomb force and universal gravitation both go as the same $1/r^2$? Explain in words other than "the personality of the force."
Show the answer
$1/r^2$ is not the personality of a force; it's because both are "fields that don't vanish" spreading through the same three-dimensional space. Since the exponent $d-1=2$ is set by the dimension of space, any non-vanishing field, whatever the force, gets the same 2.

Episode 5 Summary1/r² Is Space's Fingerprint — the Exponent Is the Dimension d−1

$1/r^2$ is not "a property of forces" but the fingerprint of space being three-dimensional. Field lines only begin and end on charges and never vanish along the way (Gauss's law = conservation of flux). So the total number $N$ is conserved, and the strength at distance $r$ is $N$ divided by the sphere's surface area. In 3D the surface area is $4\pi r^2$, hence $1/r^2$. In general, in $d$ dimensions the surface area is $\propto r^{d-1}$, so force $\propto1/r^{d-1}$. The exponent $d-1$ is an integer that is a direct copy of the dimension of space.

So measuring a force's exponent is measuring the dimension of space. Coulomb's "2" is confirmed at the $10^{-16}$ level, gravity's at the sub-millimeter level, and no extra dimensions have been found yet. Combine this with "mass sets the range" from Episode 3, and how a force reaches us can be written out completely with just two unitless properties: mass (range) and dimension (exponent). That Coulomb and Newton both come out as $1/r^2$ is no coincidence but the natural consequence of the same three-dimensional stage. The units-carrying numbers are stage machinery; the integer that is the exponent is the true body — the fifth step in reading fields by ratios.

Next time — Episode 6 "Where Mass Comes From — the Higgs Field, a 'Resistance.'" In Episode 3 we said mass is a brake on a field and sets the range $\lambda=\hbar/mc$. But where does that mass $m$ come from in the first place? The answer is the Higgs field, which fills the vacuum without a gap. A particle gains mass as the resistance it feels wading through this field. And each particle's mass is set by the strength of its bond with the Higgs field (the Yukawa coupling) — yet another number without units — even mass was a ratio. On to the final page of the main "Fields That Click."

This document is Episode 5 of the "Fields That Click" series, reading for physics-loving high-schoolers and undergraduates. Gauss's law (electric flux through a closed surface = enclosed charge, differential form $\nabla\!\cdot\!\vec E=\rho/\varepsilon_0$); that in $d$-dimensional space the area of a hypersphere of radius $r$ is proportional to $r^{d-1}$, so a static long-range force of a point source becomes $1/r^{d-1}$ (inverse square in 3D); that this is realized for massless (infinite-range) fields; and that Coulomb's inverse-square law holds to $1/r^{2+\delta}$ with $|\delta|\lesssim10^{-16}$ while gravity's inverse square is verified down to tens of μm (sub-millimeter) with no extra dimensions detected — all of these are established standard physics. That "field lines" are a visualization metaphor whose real body is flux (divergence); that $1/r^{d-1}$ concerns static point sources while radiation fields obey a different law far away (amplitude $1/r$, intensity $1/r^2$); and that extra dimensions are a theoretical hypothesis and the flat, static approximation can break down in the strong-gravity / quantum-gravity regime — these are stated explicitly in the "honest line" in the body. The figure on the right is a schematic of the log–log display of $F\propto r^{-(d-1)}$ (line slope $=-(d-1)$). — To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and answers are static and hidden). Adjacent episodes: Episode 4, Strength α / Table of contents.

Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, moving the dimension of space d changes the slope −(d−1) of the log–log plot. "Show the answer" opens each solution.