The shape decides the falloff, and the force law never changes

Gauss’s law was presented on the previous rung as a counting argument that produces the inverse-square law. It produces two other laws just as readily, from the same counting, with nothing changed but the shape of the thing doing the charging.

Three geometries, three exponents #

The argument is the same each time and takes three sentences.

A point. Draw a sphere around it. Symmetry says the field is radial and equal in magnitude everywhere on the sphere, so the flux is $E \times 4\pi r^2$. That is fixed at $Q/\varepsilon_0$, and the area grows as $r^2$, so

$$E \propto \frac{1}{r^2}.$$

A long line. Draw a cylinder around it, of length $\ell$. The field is radial, outward from the line, uniform over the curved surface. The flux is $E \times 2\pi r \ell$. The charge enclosed is $\lambda \ell$ — proportional to the length, which cancels the $\ell$ in the area. What is left grows as $r$, so

$$E \propto \frac{1}{r}.$$

A wide plane. Draw a box straddling it, with faces of area $A$ parallel to the sheet. The field is perpendicular to the sheet, so only those two faces carry flux: $2EA$. The charge enclosed is $\sigma A$, and the $A$ cancels. Nothing on either side depends on the distance at all, so

$$E = \frac{\sigma}{2\varepsilon_0}, \quad \text{constant}.$$

Three different answers, three geometries, one law. The exponent came out of how the area of the enclosing surface grows with distance — as $r^2$, as $r^1$, and as $r^0$ — and that growth is a property of the source’s shape, not of electricity.

What the figure is doing that the derivation is not #

The curves in the hero figure are not plots of the three formulas. They are sums over the sources, computed independently, and the slopes are then measured off the result.

The point case is $1/r^2$ directly. The line case adds the contributions of thousands of point elements running to $\pm 4000$ units, keeping only the component perpendicular to the line, since the parallel components cancel in pairs. The plane case uses the on-axis field of a disc of finite radius, which is itself an integral of rings.

None of that summation knows about Gauss’s law. The measured slopes come out at $-2.00$, $-1.00$ and $0.00$ over the last decade, and the agreement is the assertion the figure makes: two entirely different routes to the same three exponents, one by counting flux through a symmetric surface and one by adding up Coulomb’s law over a shape.

That check is worth having because the derivations above look suspiciously easy. Each of them hangs on a symmetry claim — “the field is radial and uniform on this surface” — that Gauss’s law does not verify and cannot. The summation does verify it, by not assuming it.

Where each idealisation stops #

The three laws describe three objects that do not exist. There is no infinite line and no infinite plane, and the interesting question is not whether the idealisation is exact but at what distance it fails.

The answer is the same in both cases and it is worth stating as a rule: a finite source obeys its idealised law close up, and reverts to an inverse square far away.

From far enough off, any bounded charge distribution looks like a point. Its total charge is all that survives; its shape has been compressed to nothing by distance. So the field of a metre-long charged rod falls as $1/r$ at centimetres and as $1/r^2$ at kilometres, and the crossover is at distances comparable to the length of the rod.

For a disc of radius $R$, the on-axis field can be written down exactly and the transition is explicit. At $r \ll R$ it is the plane result. At $r = R$ it has fallen to 29 per cent of that. At $r \gg R$ it is the point result. The “constant field of an infinite plane” is therefore the small-distance limit of a perfectly ordinary function, and the alarming feature of the idealisation — a field that never weakens, carrying energy that never ends — belongs to the idealisation and not to any real sheet.

That is the distinct claim this rung makes, and it is worth separating from the arithmetic. There are not three force laws in electrostatics. There is one, and there are three regimes, each valid over a range of distances set by the size of the source — and quoting one of them without its range is the commonest way to be wrong about a field.

The plane that never weakens #

The planar case deserves separate attention, because it is the one that everybody meets, nobody quite believes, and which turns out to be the most useful of the three.

A field that does not fall off with distance offends an intuition built on point sources, and the resolution is a piece of arithmetic worth doing. Move twice as far from a small patch of the sheet and its contribution falls by four. But at twice the distance, the region of sheet that contributes at a comparable angle is four times the area. The two factors cancel exactly, at every distance, and the cancellation is what the constant is.

The line picture makes it immediate. Field strength is line density; lines from a plane run parallel; parallel lines never thin out. Nothing else needs saying, and the three cases become three statements about whether lines diverge in two dimensions, in one, or in none.

The result also produces the most-used formula in practical electrostatics. Two oppositely charged plates each give $\sigma/2\varepsilon_0$; between them the two contributions point the same way and add to $\sigma/\varepsilon_0$; outside, they oppose and cancel to zero. A parallel-plate capacitor therefore has a uniform field inside and no field outside, which is exactly what a device for storing energy in a field should have — and both halves of that statement come from a law that seems to say a field never weakens.

Reading the exponent off a measurement #

The relationship runs backwards, and that makes the three laws into a diagnostic.

Measure how a field falls off, take the slope on logarithmic axes, and the number obtained is a statement about the shape of the source, which may be invisible or inaccessible. An exponent of $-2$ says the source is compact compared with the distance; $-1$ says it is extended in one dimension; $0$ says it is extended in two.

The technique is used well outside electrostatics because the geometry is all that enters. The intensity of a sound falls as $1/r^2$ from a compact source and as $1/r$ along a line source, which is why traffic noise from a motorway falls off far more slowly with distance than noise from a single machine — a wave-intensity statement with exactly the geometry of the electrostatic one — and why noise barriers are designed against a line source rather than a point one. Light from a fluorescent tube behaves the same way at close range. And the rotation curve of a galaxy is read exactly this way — the falloff of the gravitational field reports how the mass is distributed, and its refusal to fall off as expected is the observation that dark matter was proposed to explain.

Where the symmetry is manufactured #

Two of the three idealisations are worth defending, because they are not merely convenient — they are arranged for, deliberately, in real hardware.

The parallel-plate capacitor exists to produce the planar case. Two plates close together, each much wider than their separation, give a region between them where every point is far from every edge and the plane result holds to high accuracy. The design is a machine for making an infinite plane locally true, and the guard ring is the refinement that makes it true to parts per million.

The coaxial geometry exists to produce the line case. A thin wire down the axis of a cylindrical shell gives a field that falls as $1/r$ and is therefore enormously stronger near the wire than anywhere else. A Geiger tube uses that concentration: an electron released anywhere in the volume drifts inward, and only in the last few tens of microns does the field become strong enough to start an avalanche. Every avalanche therefore happens in the same place, under the same conditions, and produces the same size of pulse regardless of where the original ionisation occurred. The $1/r$ law is what makes the instrument a counter rather than a rough meter.

Electrostatic precipitators and the coaxial cable use the same geometry for different reasons — the first for the field concentration, the second because a field confined between two conductors radiates nothing.

What the symmetry costs #

The three results are the entire useful output of Gauss’s law, and that shortness is itself the point.

Anything less symmetric — two charges, a finite rod, a charged cube, a real capacitor near its edges — leaves the law true and useless, because one equation cannot determine a field that varies over a surface in an unknown way. The field must then be found by integrating Coulomb’s law over the source, or by solving a differential equation numerically, and Gauss’s law is demoted to a check on the answer.

That demotion is worth taking seriously rather than treating as a footnote. Nearly every real electrostatic problem is in the useless category. What the three symmetric cases provide is not a method for solving problems but a set of limiting behaviours to reason with: an intuition for how a field ought to behave near an edge, along a wire, between plates. The value of the three exponents is that they bound what any answer can look like, and an answer that falls off as $1/r^{1.5}$ near a wire is wrong before it is checked.

The same three exponents in gravity #

Nothing in the counting mentioned charge, so the three results transfer to gravity with one sign changed, and the transfer is not an analogy — it is the same theorem applied to a different source.

A spherical body pulls as $1/r^2$ from outside, which is why planets can be treated as points. An extended filament of mass pulls as $1/r$. A sheet of mass pulls with a field that does not weaken with distance at all.

The middle case is the one with observational consequences. The disc of a spiral galaxy is closer to a sheet than to a point over much of its extent, so the falloff of its gravitational field is far gentler than an inverse square — and the orbital speed of a star at radius $r$ depends on that falloff. Reading a rotation curve is exactly the reverse inference described above: measure how the field falls, and deduce how the mass is arranged.

What was found is that the curves stay flat far beyond the visible disc, implying a field falling as $1/r$ where the visible matter predicts $1/r^2$. That is a statement about the shape of the mass distribution — it must be extended, roughly spherical and much larger than the light — and it is the origin of the dark matter hypothesis. The argument is the three-exponent argument on this page, run backwards, on a galaxy.

The interior case has a consequence too. Inside a uniform sphere, the field grows linearly with radius, because the enclosed mass grows as $r^3$ while the area grows as $r^2$. So a body falling down a hole through the centre of a uniform Earth would be a simple harmonic oscillator, with a period of about 84 minutes — the same 84 minutes as a low circular orbit, which is not a coincidence and is one of the more pleasing results in the subject.

What happens between the regimes #

The crossover deserves one more look, because the transition is not a switch and the behaviour in between is where real measurements live.

For the disc, the exact on-axis field is

$$E = \frac{\sigma}{2\varepsilon_0}\left(1 - \frac{r}{\sqrt{r^2 + R^2}}\right),$$

which is a single smooth function containing both limits. Expanding it for small $r$ gives the plane result minus a correction linear in $r/R$; expanding for large $r$ gives the point result. There is no distance at which the object stops being a plane and starts being a point. There is a function, and two limits, and a broad region in the middle where neither approximation is good and the exact expression is needed.

That is the honest shape of nearly every idealisation on this site. The small-angle pendulum, the paraxial lens and the ray model of light all have this structure: a limit that is exact in a corner of parameter space, an exact description that is hard to reason with, and a middle region where the interesting engineering happens and neither one is comfortable.

The practical use is the reverse reading. Knowing the crossover distance is knowing how large a plate has to be to behave like an infinite one to a stated accuracy — and for the disc, being within one per cent of the plane result requires staying inside about $r/R = 0.01$, which is a demanding aspect ratio and the reason capacitor plates are so much wider than their gaps.

The ladder from here #

Later rungs on this anchor: the divergence form, and the equivalence between a statement on a boundary and a statement at every point. The field of a uniformly charged sphere, inside and out, and the shell theorem that Newton needed a page of geometry for. Conductors and cavities. The method of images. Dielectrics, and the version of the law with polarisation folded in. The magnetic version, whose right-hand side is always zero. Gauss’s law for gravity, which is the same equation with one sign changed and which gives the interior field of the Earth. And numerical electrostatics, where the symmetric cases become the test problems that a solver has to reproduce before it is trusted on anything else.

The general lesson to carry forward is the one the crossover made: an exponent in a physical law is often a statement about geometry rather than about a force, and the same argument fixes the falloff of a wave spreading from a source without mentioning charge at all.