Illustrated Physics
physics, drawn
Concept

The inverse-square law — where it appears

The fall of a conserved flux as one over the square of the distance, following from the growth of a sphere's area and from nothing else. Any quantity that spreads without being absorbed obeys it, which is why gravity, electrostatics and sound intensity share an exponent.

Named by 18 essays across 5 fields — each of them below, with the objects they name alongside it.

The field of a dipole. Field lines traced from a positive charge toward a negative one. Every line does eventually close on the negative charge, but the outer ones loop far outside any frame, so this picture is a crop rather than the whole field.

Field lines are a choice, not a discovery

Nothing in space is arranged in lines. The lines are a drawing convention — and an unusually good one, because three separate facts about the field survive the translation.

electromagnetism · The field concept A closed surface with the charge inside. Every field line from the enclosed charge crosses the surface exactly once on its way out, so the net flux counts the charge.

Counting what comes out, and never looking inside

Draw any closed surface. The field crossing it depends only on the charge enclosed — not on where that charge sits, not on its shape, not on anything outside.

electromagnetism · Gauss's law The same law, three shapes of source. Field strength against distance on logarithmic axes, for a point, a long line and a wide plane carrying charge. The exponent is the slope, and it is set by how the area of the enclosing surface grows rather than by anything about the force law.

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

A point charge gives an inverse square, a line gives an inverse, a plane gives a constant. All three come from the same law, and the exponent belongs to the geometry of the source rather than to the physics.

electromagnetism · Gauss's law Where the electron actually is, by radius. The radial probability density of the hydrogen 1s, 2s, 2p states — the chance of finding the electron in a thin shell at each radius, in units of the Bohr radius. 1s is most likely at 1.00 Bohr radii and averages 1.50; 2s is most likely at 5.24 Bohr radii and averages 6.00; 2p is most likely at 4.00 Bohr radii and averages 5.00. Each curve integrates to one, and each has n − l − 1 radial nodes where the electron is never found.

Where the electron probably is

The Bohr atom put the electron on a circle of definite radius. What replaced it keeps the radius as the most likely place to find the electron and gives up the circle, the speed and the trajectory entirely.

quantum · Atomic structure Three geometries, three exponents. Amplitude against distance for a wave spreading in one, two and three dimensions, on logarithmic axes, over 3 decades. The same power crosses every surface round the source, so the intensity falls as one over the area of that surface and the amplitude as one over its square root. The fitted slopes are 0.000 along a line, -0.500 over a cylinder, -1.000 over a sphere, each fitted by least squares to the drawn curve rather than written on it. A ripple on water is the middle case and a sound in a room is the last, which is why a ripple stays visible so much further than a shout carries.

How a wave thins out

A wave gets weaker with distance for two quite different reasons, and only one of them is a loss. Geometry alone fixes the first exactly — three exponents for three dimensions, with nothing about the medium in them — and whatever is left over is the medium eating the wave.

waves · Wave motion The energy of a capacitor, booked as a density. The energy stored by a parallel-plate capacitor of 200 square centimetres — 0.0200 square metres — against the separation of its plates, drawn twice. Held at 15 nC the energy rises in proportion to the separation; held at 169 V it falls as the inverse. Both curves are obtained by integrating the energy density ½ε₀E² over the volume between the plates, and each agrees with ½QV to better than a part in 10¹². The two describe the same capacitor at 2.00 mm, where they cross at 1.27 µJ, and there their slopes are equal and opposite: the attraction between the plates is 635 µN, or 6.353·10⁻⁴ N, whichever quantity is held fixed. That force is Q²/2ε₀A — a property of the field in the gap and of the area it crosses, with no reference to the plates at all.

Where the energy of a field actually is

A charged capacitor holds 1.27 µJ, and two entirely different accounts agree on the number: one built from charges and potentials, one built from joules per cubic metre of empty space. They part company at a resistor, where the power arrives sideways through the surface at 1.67 W.

electromagnetism · Field energy Equipotentials, with the field lines that cross them. Contours of constant potential, traced by marching squares, with field lines traced along the gradient of the same potential. The two families meet at right angles everywhere, which is a consequence of the field being the gradient rather than a property of the drawing.

The attraction that needs no charge

Gauss's law says nothing comes out of a neutral molecule, and yet water's field one nanometre away reaches 1.1 × 10⁸ V/m. What survives when the monopole vanishes is a separation, and every step down the tower of falloffs below it is paid for with one more order of cancellation.

electromagnetism · The field concept The same start, four force laws. Orbits under 4 different central force laws, every one started at the same radius with the same fraction — 0.72 — of the local circular speed, and every one integrated for 6 radial oscillations. The paths are drawn to different scales because the excursions differ; what is comparable between the panels is whether the curve retraces itself. Under the inverse square it does: the orbit is a closed ellipse and the sixth circuit lies exactly on the first. Under the linear force it does too, and the ellipse is centred on the source rather than focused on it. Under anything in between the path is a rosette that never closes, because the angle between successive closest approaches is not a rational fraction of a full turn. Those angles are 180.0° at n = -2, 145.9° at n = -1.5, 126.4° at n = -1, 90.0° at n = 1. The closure is not a matter of degree — a rosette that nearly closes is not nearly a closed orbit, since after enough circuits it fills the annulus.

The orbit that does not come back to itself

A bounded orbit under any central force oscillates between a smallest and a largest radius for ever. That it should also return to the same point is a further demand, and only two force laws in existence meet it — the inverse square, and the linear spring.

astrophysics · Orbit stability The blow-out band has two edges, not one. The ratio of radiation force to gravity against grain radius, with the radiation-pressure efficiency included: a grain much smaller than the wavelength of the light barely interacts with it — the efficiency falls as the fourth power of the size, which is Rayleigh's law — so the ratio stops rising and turns over. The dashed line is the same ratio with the efficiency taken as one, which is the usual drawing and is right only to the right of the turnover at 115 nm. The consequence is that a grain can be too small to be blown out as well as too large. Taking the threshold at a half — the value at which a grain released from a circular orbit is unbound — the band runs from 48.2 nm to 574 nm, and the largest ratio any grain of this material reaches is 1.87. Everything outside that band stays, and what stays does not stay put: it spirals.

The size the light cannot blow away

Radiation pressure and gravity both fall as the inverse square of distance, so their ratio is a property of the grain and not of where it is. What follows is a band of sizes that get blown out — with a lower edge as well as an upper one — and a drag, on everything else, that is the same pressure read one order further in v/c.

astrophysics · Radiation pressure The shell of news, and the kink inside it. A charge that was moving at 0.6c to the right, stopped over a short interval, and has been at rest ever since. Outside the sphere of radius ct nothing has heard: the field there still points at the position the charge would have reached had it carried on, marked ahead of it. Inside, the field is that of a charge at rest. In between is a shell one deceleration-time thick, and across it the field line has to bend, because a field line cannot simply stop in empty space — the two ends are joined here to 8.5e-14 pixels. That bend is transverse to the radius, and it is the radiation. It is not an extra thing the charge emitted; it is the join between two static fields that do not line up, and it travels outward at c because that is where the news front is.

The field that points where the charge is now

The field here was set by what the charge was doing a distance over c ago, so it ought to point at where the charge used to be. For a charge moving steadily it points at where the charge is — not approximately, exactly — and nothing has outrun light. What breaks the arrangement is a change of motion, and the break is the whole of radiation.

electromagnetism · Retardation The brightest anything of a given mass can be. The Eddington luminosity against mass, with the main sequence drawn beside it. Radiation pushes outward on the electrons and gravity pulls inward on the protons, and both go as one over the distance squared — so the radius cancels out of the comparison entirely, checked here at three radii spanning four decades and coming out identical to 1e-20. What is left is a luminosity: L = 4πGMc/κ, which is 1.47 × 10³¹ watts per solar mass, or 3.8·10⁴ solar luminosities. Above it, radiation drives the outer layers away faster than gravity can hold them. The Sun is at 2.6e-5 of its own limit and in no danger; a star of 10 solar masses is at 1.2e-2; and a star of 100 is at 0.83, which is why the two lines converge at the top of the chart and why the most massive stars known are a few hundred solar masses rather than a few thousand. They do not fail to form for lack of gas; they blow away the gas that would have made them heavier, and once formed they shed mass continuously in a radiation-driven wind. The limit is the same expression for an accreting black hole, where it caps not the brightness but the rate at which mass can be taken on.

The brightness a mass cannot exceed

Light pushes outward on the electrons of a star and gravity pulls inward on its protons, and both forces fall off as one over the distance squared. The distance therefore cancels, and what is left is a limit on brightness rather than on size — 3.8 × 10⁴ times the Sun's luminosity for every solar mass, above which a body drives its own outer layers away.

astrophysics · Radiation pressure Three terms, and one distance. The three terms of an oscillating dipole's electric field against distance, in units of a reciprocal wavenumber, both logarithmic. They fall as 1/u³, 1/u² and 1/u, so they are all equal at u = 1 — a distance of λ/2π, which is a property of the frequency and of nothing about the antenna. The heavy curve is the field the three actually add up to, and it is not their sum: the static and radiation terms are in antiphase and partly cancel, which is why the total dips below every one of them just inside the crossing before settling onto the 1/u the far field is made of.

The distance where a field changes its mind

An oscillating source has three fields around it, falling as the inverse cube, the inverse square and the inverse first power of distance. They are all equal at one radius, and that radius is the wavelength over 2π — a number containing nothing about the source at all. Inside it a source mostly stores energy; outside it, mostly loses it, and the two behaviours are different technologies rather than different strengths.

electromagnetism · Retardation The same nudge in four kinds of space. A body on a circular orbit given 5 per cent more speed, integrated for up to 6 turns in the field a point mass would have in spaces of 2, 3, 4, 5 dimensions. In 2 dimensions it stays between 1.00 and 1.10 of the original radius; in 3 dimensions it stays between 1.00 and 1.23 of the original radius; in 4 dimensions it leaves after 1.93 turns; in 5 dimensions it leaves after 1.19 turns. In three dimensions the orbit is an ellipse that closes; in two it is a rosette that does not close but never wanders far; from four dimensions up there are no stable orbits at all, and a planet nudged by anything would spiral away or plunge.

The number of dimensions a planet can orbit in

Gauss's law says that a point source's field is its strength spread over the surface around it, and the size of that surface depends on how many dimensions space has. In three it grows as the square of the distance, and the field falls as the inverse square. Count differently and the field falls differently — and above three dimensions it falls so steeply that no planet could hold a stable orbit and no atom could hold its electron. The inverse-square law is not a fact about gravity or electricity. It is a fact about space, and it is one of the conditions for there being anything in space to orbit.

electromagnetism · Gauss's law Two ways to be wrong, and the apparatus that sees each. The fractional sag of the potential at the centre of a charged spherical shell below the shell's own, on a logarithmic axis, against the shell's radius in metres, also logarithmic, for two different departures from Coulomb's law. If the force fell as 1/r^(2+ε) with ε = 10⁻¹⁶, the sag is 3.07·10⁻¹⁷ whatever the shell's size (dashed): the law has no length in it, so neither does its failure. If the photon had a mass of 10⁻¹⁴ eV, the sag is (μR)²/6 (solid): 4.3·10⁻²⁰ for a centimetre shell and 4.3·10⁻¹⁶ for a metre. A mass introduces a length, and the signal grows as the square of the apparatus compared with it — so the strongest bounds on a photon mass come from the largest fields, not the most sensitive laboratories.

The mass that would make Gauss's law leak

Gauss's law says the flux out of any closed surface counts the charge inside, whatever the surface's size. If the photon had even a tiny mass, that would stop being true: the field of a charge would die off exponentially beyond a length ħ/mc, the flux through a large sphere would fall short of the charge, and the inside of a charged shell would no longer be field-free. There are two different ways Coulomb's law could fail, and they are caught by opposite strategies. A wrong exponent has no length in it and shows equally in any apparatus; a mass has a length, and shows as the square of the apparatus's size. That is why the tightest limit on the photon's mass comes not from a laboratory but from the magnetic field the solar wind carries past Pluto.

electromagnetism · Gauss's law Four cosines between a lens and the edge of its picture. Light from a direction 30° off the axis passing through a lens and landing at the edge of the image, with the four factors that dim it relative to the centre. The aperture, seen from that direction, is foreshortened to cos θ of its area. The image point is further from the lens, f/cos θ instead of f, and the light spreads as the square of the distance: cos²θ. The light arrives at the image plane at a slant, spreading over 1/cos θ more area: one more cos θ. Together, cos⁴θ: at 30° the edge receives 0.563 of the light at the centre, 0.83 photographic stops darker. Nothing here is a fault of the glass. A perfect thin lens, or a pinhole, does it, because it is geometry.

The four cosines at the edge of a photograph

Photograph a plain, evenly lit wall with a wide-angle lens and the corners of the picture come out darker than the middle — by more than three photographic stops with a 14 mm lens on a full-frame camera. It is usually blamed on the lens. A perfect lens does it too, and so does a pinhole, because the light reaching the edge of an image is reduced by four separate cosines of the angle it came from: three of them pure geometry, and one the slant at which it lands. Fisheye lenses avoid most of the loss by refusing to keep straight lines straight. Every wide lens is a choice between the two.

optics · Imaging An orbit, and the circle its velocities make. Left: an orbit of eccentricity 0.6 round a central mass (dot), integrated step by step, with the velocity drawn as an arrow at twelve points equally spaced in time. Right: the same twelve velocities drawn from a single point, the origin of velocity. Their tips lie on a circle — the hodograph — of radius GM/h, whose centre is displaced from the origin by e times that radius, here 0.6: the integrated velocities stay on it to within one part in ten thousand. The fastest velocity, at the closest approach, is 1.60 times GM/h and the slowest, at the furthest point, 0.40 times. The tips are bunched near the slow end, because the body spends most of its time far away.

The velocity that always lies on a circle

A planet on an elliptical orbit speeds up and slows down, its velocity swinging round as it goes. Draw all its velocities from a single point and their tips fall exactly on a circle — not an ellipse, a circle — offset from the point by the orbit's eccentricity. Hamilton noticed this in 1846; it holds for every orbit under an inverse-square force and for no other force; and it turns the ellipse into something that can be proved with a ruler and compass, as Feynman did in a lecture that was lost for thirty years.

astrophysics · Orbit stability A cloud that grows wider instead of denser. The density of rubidium atoms in a magneto-optical trap against distance from its centre, for 10⁵, 10⁶, 10⁷, 10⁸ atoms, computed from the trap's spring (200 Hz), the atoms' temperature (100 μK) and the repulsion of rescattered light, with σᵣ/σₗ = 1.30. The smallest cloud is the trap's Gaussian, 78 μm in radius, with a peak of 0.91 × 10¹⁰ per cm³. Larger clouds stop getting denser: each fills a sphere at the same 2.5 × 10¹⁰ atoms per cm³ (dashed), out to 0.21, 0.46, 0.99 mm, with a soft edge about a Debye length, 45 μm, wide. More atoms make a bigger ball, not a denser one.

The light that makes a cold cloud repel itself

Load more atoms into a magneto-optical trap and the cloud does not get denser. It gets bigger, at a density that stays fixed whatever the number. The cause is the trap's own light. A photon one atom scatters can be absorbed by another, which is pushed away, and the push falls off as the inverse square of the distance, like the force between two charges. A cloud of laser-cooled atoms is therefore a plasma of neutral particles. Each behaves as if it carried a ten-thousandth of an electron's charge, and the trap fills with them as a harmonic well fills with charge: at one density, out to whatever radius the number requires.

astrophysics · Radiation pressure How much a tiny imbalance of charge would take off a body's weight. The electric repulsion between the parts of any body, as a fraction of their mutual gravity, if every nucleon carried an unbalanced charge δ (in units of the electron's), on logarithmic axes. It is (δ/δc)², with δc = 8.93 × 10⁻¹⁹, for a grain of sand, a planet or a galaxy alike. At the imbalance Lyttleton and Bondi proposed, 2 × 10⁻¹⁸, repulsion would beat gravity 5.0 times over, everywhere — their mechanism for an expanding universe. Experiments with gas let out of tanks had bounded the imbalance near 10⁻²⁰ within a year, where the ratio is 1.3·10⁻⁴, and modern ones near 10⁻²¹, 1.3·10⁻⁶.

The imbalance of charge that would outweigh gravity

Gravity and electricity both fall as the inverse square, so Gauss's law treats them identically: the field of any round body counts what is inside it. If every nucleon in ordinary matter carried a tiny unbalanced charge, the body's charge would follow its mass everywhere, and its electric repulsion would be a fixed fraction of its gravity at every size from a grain of sand to the universe. Below about one part in a billion billion of an electron's charge, gravity always wins; above it, repulsion always does. In 1959 two astronomers proposed that the universe expands because the true imbalance is just above that line.

electromagnetism · Gauss's law

Named alongside it

The objects these essays reach for when they reach for this one.

Gauss's lawElectric fieldElectric potentialFalloff exponentField linesFluxRadiation pressureAngular momentumEffective potentialOrbitPoynting vectorRadiation

All concepts