Momentum and Radiation Pressure

The conservation-laws chapter ended on a genuinely strange note: static crossed fields threading empty space carry momentum, $\mathbf{g} = \varepsilon_0\mathbf{E}\times\mathbf{B}$ , and the books of mechanics do not balance without it. That chapter (EM-15's momentum set) built the whole apparatus — the momentum density from the Lorentz-force audit, the Maxwell stress tensor that prices momentum flow, the conservation theorem locking them together. This article brings the apparatus to the configuration where it stops being strange and starts being useful: the traveling plane wave, whose momentum is not a bookkeeping subtlety but a measurable, engineerable push. The plane wave's result is compact: momentum density $u/c$ , pointing forward. Everything else follows by asking what happens when that momentum current hits something. An absorbing surface eats the momentum and feels pressure $I/c$ ; a mirror reverses it and feels twice that; a tilted surface takes a $\cos^2\theta$ discount; and a gas of radiation bouncing in all directions presses outward with $u/3$ — the single formula that holds up stars and wrote the thermal history of the early universe. The numbers are a study in contempt and consequence. Sunlight presses on your outstretched hand with a few micropascals — hopelessly beneath notice, which is why two centuries of physicists could build mechanics without ever missing light's momentum. But the same pressure, applied to a square kilometre of sail or integrated over a star's interior or a comet's dust, steers spacecraft, sets the mass limit of stellar luminosity, and paints tails across the sky. Small pressure, patient lever.

Key takeaways

A plane wave carries momentum density $\mathbf{g} = \mathbf{S}/c^2 = (u/c)\hat{\mathbf{k}}$ — the energy comb of the previous article, divided by $c$ , pointing forward.
Momentum flux = pressure: a perfect absorber at normal incidence feels $P = \langle u\rangle = I/c$ ; a perfect mirror feels $2I/c$ ; a tilted absorber feels $(I/c)\cos^2\theta$ .
Energy and momentum ride in the fixed ratio $W = pc$ for any pulse of radiation — the classical seed of the photon relations $E = pc$ and $p = \hbar k$ .
Isotropic radiation exerts $P = u/3$ — the angular average of the one-way flux — the equation of state of the photon gas, load-bearing in stars and cosmology.
Sunlight's pressure at Earth is $4.5\ \mu$ Pa (absorbing) — trivially small on a hand, decisive on micron-sized comet dust ( $\beta > 1$ : blown out of the Solar System) and on laser-driven gram-scale sails (Starshot: ~34,000 g of acceleration).
Key idea: light's momentum is its energy divided by c — absorbed it pushes once with I/c, reflected it pushes twice, and bathing everything isotropically it presses u/3.

The core insight

A plane wave carries momentum density $u/c$ along $\hat{\mathbf{k}}$ ; delivered to matter this is radiation pressure — $I/c$ absorbed, $2I/c$ reflected, $u/3$ for an isotropic bath — the same books that hold up stars and fly solar sails.

Where this lives today

Radiation pressure crossed from curiosity to infrastructure in one generation. Every spacecraft navigator budgets solar pressure: it is the largest non-gravitational force on anything with big solar panels, it desaturates (or saturates) reaction wheels, and mission designers now exploit it deliberately — LightSail 2 raised its orbit by tacking, and NASA's upcoming sails treat the $\cos^2$ steering law of Derivation A as a control input. At the opposite scale, optical tweezers (Ashkin's 2018 Nobel) hold and move single cells and molecules with piconewton light forces, and laser cooling — photon momentum kicks $\hbar k$ applied a million times a second — is the enabling technology of atomic clocks and quantum simulators. Two frontiers push the subject's precision. Gravitational-wave interferometers are limited in-band by quantum radiation pressure noise — the shot-to-shot fluctuation of photon momentum delivered to 40 kg mirrors — now engineered around with squeezed light: the pressure of this article, audited at the level of its quantum fluctuations. And in the sky, the same books run the Yarkovsky effect (anisotropic thermal re-emission steering asteroid orbits — measured, and used to refine impact forecasts) and the Poynting–Robertson drag that spirals zodiacal dust into the Sun. From micropascals to megaprojects: the lever was always patience times area.

The physical picture

Keep the energy comb of the previous article in view: sheets of $\cos^2$ inventory sliding forward at $c$ . The momentum picture requires exactly one new stroke — paint each sheet with a forward-pointing arrow of length $u/c$ . That is the entire content of $\mathbf{g} = \mathbf{S}/c^2$ for a plane wave: wherever energy is, momentum is, in the rigid ratio $1/c$ , always along $\hat{\mathbf{k}}$ . The wave is a stream of moving inventory that is simultaneously a stream of moving impulse. Why should energy in transit be momentum? The momentum-density article (EM-15's opener) gave the two honest arguments and they are recalled, not repeated: microscopically, the wave's $\mathbf{E}$ drives a charge sideways and the wave's $\mathbf{B}$ then hooks that sideways velocity forward — the $q\mathbf{v}\times\mathbf{B}$ force rectifies, pushing ahead on both half-cycles; macroscopically, the transport argument prices energy $W$ moving at $c$ as momentum $W/c$ . What this article adds is the collision physics: what the arrows do when the stream terminates on matter.

Figure

Momentum bookkeeping at a wall — absorber eats the arrows once, mirror reverses them for double impulse, tilt discounts by cos²θ

The wall pictures carry all the intuition. Absorption terminates the arrow stream: the wall gains the stream's momentum at the rate it arrives, and rate-of-momentum-arrival per area is pressure. Reflection is a bounce: the wall must both stop the incoming arrows and launch outgoing ones, paying double impulse — the same factor of two as a ball bouncing off a wall versus sticking to it. Tilting costs two cosines, one for the reduced intercepted beam and one for taking only the normal component (the tangential part is not lost — it is a shear, the drag that lets a solar sail tack). And an isotropic bath — arrows arriving from every direction — averages the double cosine over the hemisphere to give the famous $\tfrac13$ : pressure $u/3$ , the photon gas's equation of state. One conceptual footnote before the mathematics, because the chapter's history demands it: none of this is hypothetical force-fitting. The stress-tensor article (EM-15) proved that field momentum flux is exactly what the Maxwell stress tensor prices, and for a plane wave hitting a wall the relevant component is simply $T_{kk} = u$ — the pressure formulas below are that theorem cashed at an interface. Maxwell computed the sunlight number in 1873; Lebedev, and Nichols and Hull, measured it by 1901. The pressure is as real as the light.

The mathematics of the push

Start from the one-line inheritance. The previous article's comb gives, pointwise,

\mathbf{g}(\mathbf{r},t) = \frac{\mathbf{S}}{c^2} = \frac{u(\mathbf{r},t)}{c}\,\hat{\mathbf{k}} = \frac{\varepsilon_0E_0^2}{c}\cos^2(\mathbf{k}\cdot\mathbf{r}-\omega t)\,\hat{\mathbf{k}} \qquad (1)

where: $\mathbf{g}$ — momentum per unit volume; $u$ — the instantaneous energy density; the identity $\mathbf{g} = \mathbf{S}/c^2$ is owned and derived by EM-15's momentum-density article (both the Lorentz-force and transport routes) and is here simply evaluated on the plane wave. Read it physically: the momentum comb has the same sheets, the same $2\omega$ pulse, the same forward-only orientation as the energy comb — one stream, two conserved currencies, exchange rate $c$ . Integrate (1) over any volume of beam and a portable statement drops out. A pulse carrying total energy $W$ carries total momentum

p = \frac{W}{c} \qquad (2)

where: the relation holds for any shape of pulse, because it holds sheet by sheet. This is the classical ancestor of the photon relations: quantum mechanics will partition $W$ into lumps $\hbar\omega$ , and (2) then reads $p_{\text{photon}} = \hbar\omega/c = \hbar k$ — relativity's $E = pc$ for massless particles, already sitting inside Maxwell's theory. A 1 J laser pulse carries $3.3\times10^{-9}$ kg·m/s: stopping it imparts the momentum of a mosquito walking.

Derivation ARadiation pressure at a wall — absorber, mirror, and the two cosines of oblique incidence

The goal is the pressure catalogue: $P = I/c$ (normal absorption), $P = 2I/c$ (normal reflection), $P = (I/c)\cos^2\theta$ (oblique absorption), directly from momentum conservation at the surface, with the shear component exhibited along the way. The strategy is collision bookkeeping — the same impulse audit as gas molecules on a piston wall, with the arrow stream (1) as the projectile flux. Time-averaged quantities are used ( $I = \langle S\rangle$ , $\langle g\rangle = I/c^2$ ; the averaging itself, and the ½ it brings, is the next article's property — here we need only that the average exists). Watch where each cosine enters at oblique incidence; conflating them is the classic exam error.

Step 1. Normal incidence, perfect absorber. In time $\Delta t$ , the wall area $A$ swallows the beam contents of a column of length $c\Delta t$ : momentum $\Delta p = \langle g\rangle\,(cA\,\Delta t) = \tfrac{I}{c^2}\,cA\,\Delta t = \tfrac{I}{c}A\,\Delta t$ . Force is $\Delta p/\Delta t$ , pressure is force per area:

P_{\text{abs}} = \frac{I}{c} = \langle u\rangle.

Why the second equality deserves its own glance: pressure equals the beam's mean energy density — momentum flux and energy density have the same dimensions, and for one-way radiation they are numerically identical. This is the stress-tensor article's $T_{zz} = u$ for a plane wave, recalled by name: the pressure formulas here are that tensor component meeting a wall.

Step 2. Normal incidence, perfect mirror. The reflected beam departs with reversed momentum, so the wall's impulse per arrow is doubled: $\Delta p = 2\langle g\rangle cA\Delta t$ , hence

P_{\text{mir}} = \frac{2I}{c} = 2\langle u\rangle.

Partial optics interpolate: reflectivity $R$ , absorptivity $1-R$ give $P = (1+R)\,I/c$ . Why a mirror is not "free" double thrust: the mirror also receives zero energy (perfect reflection does no absorbing), so sails prefer mirrors — all momentum, no heat load. The absorber pays for its single push by having to dissipate the whole beam.

Step 3. Oblique incidence on an absorber, angle θ from the normal. Two independent cosines. Geometry: the beam's cross-section $A\cos\theta$ is what an area $A$ of wall intercepts — the flux onto the wall is $I\cos\theta$ . Vector projection: the arriving momentum points along $\hat{\mathbf{k}}$ , at angle θ to the normal; only the normal component $\cos\theta$ presses. Together:

P_{\text{abs}}(\theta) = \frac{I}{c}\cos^2\theta, \qquad \text{shear } = \frac{I}{c}\cos\theta\sin\theta.

The tangential share is a genuine drag force along the surface — for a mirror it cancels (reflection returns the tangential momentum unchanged), which is why sails are steered by tilting mirrors: tilt trades thrust magnitude for thrust direction, the normal-pointing $2(I/c)\cos^2\theta$ .

Step 4. Sanity anchor. Sunlight at Earth, $I = 1361$ W/m²: $P_{\text{abs}} = 1361/2.998\times10^8 = 4.54\ \mu$ Pa, $P_{\text{mir}} = 9.08\ \mu$ Pa. Atmospheric pressure is $10^{10}$ times larger; a square-metre mirror feels $9.1\ \mu$ N, the weight of a milligram. The catalogue's power is never the size of P — it is that P buys force without reaction mass, forever. Verified: reproduces Griffiths §9.2.3 (Eq. 9.64 route) and Jackson §6.7's radiation-pressure discussion; absorber/mirror factor of 2 consistent with EM-15's momentum-density article (its mirror quiz) — this derivation owns the oblique and shear cases; the numbers recomputed: 1361/(2.998×10⁸) = 4.54×10⁻⁶ Pa exactly.

Figure

Why isotropic radiation presses u/3 — every arrival direction contributes cos²θ, and the spherical average shares one unit among three axes

Go deeper · Derivation B · 2 minThe one-third law — pressure of isotropic radiation

Derivation BThe one-third law — pressure of isotropic radiation

The goal is $P = u/3$ for radiation arriving isotropically from all directions with total energy density $u$ — the equation of state of the photon gas, the single most consequential radiation-pressure formula in astrophysics and cosmology. The strategy: treat the bath as incoherent plane waves distributed uniformly over solid angle, apply the previous derivation's $\cos^2\theta$ pressure to each, and average. The calculation is three lines; the significance is a chapter of stellar-structure theory. Watch the average $\langle\cos^2\theta\rangle$ over a sphere produce the 1/3 — the same geometric third as in the kinetic theory of gases, for the same reason: three directions, one of them normal.

Step 1. Decompose the bath. Isotropy means the energy density $u$ is spread uniformly in direction: the share traveling into solid angle $\mathrm{d}\Omega$ is $u\,\mathrm{d}\Omega/4\pi$ . Each directional share is a weak plane wave, and pressures of incoherent streams add (their cross terms time-average away — the complex-notation lemma's quadrature result doing quiet work).

Step 2. Each share presses with its $\cos^2$ . A share arriving at angle θ to the wall's normal contributes pressure $(\text{its } u)\cos^2\theta$ , by Derivation A Step 3 read as momentum flux ( $P = u_{\text{share}}\cos^2\theta$ — using pressure = energy density per one-way stream). A wall in a bath is struck from the full sphere of directions but each photon stream's momentum is delivered where it lands; integrating over the hemisphere of arrivals with the matching outgoing hemisphere (in equilibrium the wall re-emits or the bath is transparent — either way the net momentum exchange doubles per hemisphere or, more cleanly, count the full-sphere momentum flux tensor):

P = u\,\langle\cos^2\theta\rangle_{\text{sphere}} = u\cdot\frac{1}{4\pi}\int_0^{2\pi}\!\!\mathrm{d}\phi\int_0^{\pi}\cos^2\theta\,\sin\theta\,\mathrm{d}\theta = u\cdot\frac{1}{3}.

Step 3. Read the third. $\langle\cos^2\rangle = \tfrac13$ over the sphere because $\cos^2\theta_x + \cos^2\theta_y + \cos^2\theta_z = 1$ and isotropy shares the unit equally among three axes — the identical argument that gives ideal gases $P = \tfrac13\rho\langle v^2\rangle$ . Radiation is a gas whose molecules all move at $c$ ; its peculiarity is not the third but the stiffness: $P = u/3$ with $u \propto T^4$ (the Stefan–Boltzmann law, statistical mechanics' business) makes radiation pressure explosive in hot objects.

Step 4. Two anchors. The CMB today: $u = aT^4$ with $T = 2.725$ K gives $u = 4.2\times10^{-14}$ J/m³, $P = 1.4\times10^{-14}$ Pa — the gentlest pressure in the universe, yet it was the dominant pressure of the cosmos for its first fifty thousand years. The solar core: $T = 1.57\times10^7$ K gives $P_{\text{rad}} = aT^4/3 = 1.5\times10^{13}$ Pa — fifteen million atmospheres of pure light, still only ~0.1% of the core's gas pressure; but since $P_{\text{rad}}/P_{\text{gas}} \propto T^3/\rho$ grows with stellar mass, radiation pressure caps how massive and luminous a stable star can be (the Eddington limit — stars above ~ $10^5$ solar luminosities per solar mass blow themselves apart with their own light). Verified: angular average $\int\cos^2\theta\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi/4\pi = 1/3$ exactly; CMB numbers from $a = 7.566\times10^{-16}$ J·m⁻³·K⁻⁴, $T = 2.725$ K → $u = 4.17\times10^{-14}$ J/m³; solar-core figure $aT^4/3 = 1.53\times10^{13}$ Pa at $1.57\times10^7$ K; result standard (Jackson §6.7 context; any stellar-structure text).

Limiting cases

Case	Result	Meaning
perfect absorber, normal	$P = I/c = \langle u\rangle$	one push; wall also eats the energy (heat load)
perfect mirror, normal	$P = 2I/c$	bounce doubles impulse; zero heat load — sail's choice
reflectivity R	$P = (1+R)I/c$	interpolation; real sails R ≈ 0.9+
absorber at angle θ	$P = (I/c)\cos^2\theta$ + shear $(I/c)\sin\theta\cos\theta$	two cosines: intercept × projection; shear = tacking handle
isotropic bath	$P = u/3$	photon-gas equation of state; stars, early universe
pulse of energy W	$p = W/c$	E = pc; photon limit $p = \hbar k$
static crossed fields	$\mathbf{g} \neq 0$ with nothing moving	EM-15's territory — recalled, not re-argued here
$v \ll c$ matter	pressure forces ∝ 1/c: tiny	why mechanics never missed light's momentum

Common trapOne cosine or two? Oblique-incidence pressure is the exam's favorite trap

The two cosines of Derivation A Step 3 have different jobs and different failure modes. Forget the intercept cosine and you overcount the beam: an area $A$ of tilted wall only catches $A\cos\theta$ of beam cross-section. Forget the projection cosine and you press with the full momentum instead of its normal component — the tangential part is shear, not pressure. Both errors give $\cos\theta$ where the answer is $\cos^2\theta$ (absorber) — and for a mirror the geometry differs again: the reflected beam removes the tangential-shear cancellation question entirely and the normal impulse doubles, $P = (2I/c)\cos^2\theta$ . Two adjacent traps complete the set: quoting the mirror factor 2 for a rough or black sail (real sails are graded by reflectivity: $(1+R)I/c$ ), and using the one-way formula $P = u$ where the radiation is isotropic (a cavity, a stellar interior, the CMB) — an isotropic bath presses $u/3$ , and confusing $u$ with $u/3$ is a factor-of-three error in stellar structure. When in doubt, return to arrows-on-a-wall bookkeeping; the formulas are one diagram deep.

Worked examples

1 · Sunlight versus a comet grain — the β budget (confidence builder)

A spherical dust grain (radius $a$ , density $\rho = 2000$ kg/m³, absorbing) is released from a comet. Radiation pushes it outward as $1/r^2$ ; solar gravity pulls it inward as $1/r^2$ — the same law, so their ratio β is distance-independent. Compute β and find the grain size that gets blown out of the Solar System.

Step 1. Both forces at distance $r$ . Radiation: the grain intercepts $\pi a^2$ of beam, $F_{\text{rad}} = \tfrac{L_\odot}{4\pi r^2}\cdot\tfrac{\pi a^2}{c}$ . Gravity: $F_g = \tfrac{GM_\odot}{r^2}\cdot\tfrac{4}{3}\pi a^3\rho$ . The $r^2$ cancels in the ratio:

\beta = \frac{F_{\text{rad}}}{F_g} = \frac{3L_\odot}{16\pi G M_\odot c\,\rho a}.

Step 2. Numbers. $L_\odot = 3.828\times10^{26}$ W, $M_\odot = 1.989\times10^{30}$ kg: $\beta = 5.74\times10^{-4}/(\rho a)$ (SI). For $\rho = 2000$ kg/m³: β = 2.9 at $a = 0.1\ \mu$ m, β = 0.29 at $a = 1\ \mu$ m, β = $3\times10^{-4}$ at $a = 1$ mm.

Step 3. Read the scaling. $F_{\text{rad}} \propto a^2$ (area) but $F_g \propto a^3$ (mass): β ∝ 1/a — light wins at small sizes. Sub-micron grains feel effectively repulsive gravity (β > 1) and stream onto hyperbolic orbits: that is the comet's dust tail, pushed anti-sunward regardless of the comet's motion, curved because the grains keep their orbital momentum. Boulders don't notice the light. One β, one sky-wide demonstration that sunlight pushes. Answer: grains below ≈ 0.3 μm (β > 1) are expelled.

2 · Pricing a laser sail — Breakthrough Starshot (exam level)

A proposed interstellar probe: total mass $m = 2.0$ g (chip plus sail), perfectly reflective sail, illuminated by a phased-array laser delivering $P = 100$ GW onto it. Find the thrust, the acceleration, and the time and distance to reach $v = 0.2c$ (non-relativistic estimate).

Step 1. Thrust from the mirror catalogue, expressed per unit power (integrate $P_{\text{mir}} = 2I/c$ over the sail area — the area cancels against intensity): $F = 2P/c = 2\times10^{11}/2.998\times10^8 = \mathbf{667\ N}$ . (Rule of thumb worth keeping: reflected light gives 6.7 N per gigawatt.)

Step 2. Acceleration: $a = F/m = 667/0.002 = 3.34\times10^5$ m/s² $= \mathbf{34{,}000\ g}$ — survivable only because the payload is a wafer, not a crew.

Step 3. Burn duration to $0.2c$ : $t = v/a = 6.0\times10^7/3.34\times10^5 = \mathbf{180\ s}$ — a three-minute launch to a fifth of light speed.

Step 4. Distance covered: $d = v^2/2a = (6.0\times10^7)^2/(2\times3.34\times10^5) = 5.4\times10^9$ m ≈ 0.036 AU — the laser must hold a diffraction-limited spot on a metre-scale sail out to several million kilometres, which is the actual engineering frontier (optics, not thrust). Cross-check by momentum ledger: the beam delivers $p = 2P t/c = 2\times10^{11}\times180/c = 1.2\times10^5$ kg·m/s; the craft carries $mv = 0.002\times6\times10^7 = 1.2\times10^5$ kg·m/s ✓ — the books balance exactly, because a mirror transfers momentum without keeping energy. Answers: F = 667 N, a ≈ 34,000 g, t ≈ 3.0 min, d ≈ 0.036 AU.

Origins · 1 minPyotr Lebedev and how this was discovered

Pyotr Lebedev

1866–1912 · Moscow — who weighed light before the theory's rivals could explain it

Lebedev built the experiment everyone since Maxwell had called impossible in practice. The obstacle was never the smallness of radiation pressure alone — it was that any illuminated vane in imperfect vacuum feels radiometric forces (gas heated at the surface pushes back) thousands of times larger, which is why the spinning Crookes radiometer rotates the wrong way and proves nothing about light's momentum. Working in Moscow with hand-blown apparatus, Lebedev suspended thin platinum wings on a torsion fiber in the best vacuum of his era, alternated illumination between blackened and mirrored faces, flipped the beam side to side to cancel drifts, and in 1900 reported light's push in agreement with Maxwell's $I/c$ — with the mirror's factor approaching 2, as the theory demanded. Nichols and Hull at Dartmouth confirmed it independently within a year at ~1% precision. Kelvin, who had doubted the effect was measurable, conceded to Lebedev personally. Lebedev went on to measure the pressure of light on gases — harder still — and resigned his Moscow chair in 1911 in protest at government repression of the university, dying months later at 46. The Lebedev Physical Institute in Moscow, cradle of seven Nobel laureates, bears the name of the man who made light's momentum a laboratory fact.

History

1619

— Kepler proposes sunlight's push to explain comet tails pointing away from the Sun

1873

— Maxwell's Treatise computes radiation pressure ~10⁻⁶ Pa for sunlight: "a very small pressure"

1876

— Bartoli derives P = u/3 thermodynamically — radiation pressure needed for the second law itself

1900

— Lebedev's torsion balance measures light pressure, defeating radiometric artifacts

1901

— Nichols and Hull confirm to ~1%; the mirror factor of 2 verified

1970

— Ashkin levitates micron spheres on laser light — radiation pressure becomes a tool (optical tweezers)

2010

— IKAROS deploys the first interplanetary solar sail (EM-15's momentum article owns its numbers)

2016

— Breakthrough Starshot proposes 100 GW laser sailing at 0.2c; LIGO's mirrors are quantum-radiation-pressure limited

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §9.2.3 · Jackson, Classical Electrodynamics, 3rd ed., §6.7 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §9.6.3 · Zangwill, Modern Electrodynamics, §15.5.

Provenance. Radiation-pressure results follow Griffiths §9.2.3 (momentum of plane waves, Eq. 9.64 context) and Jackson §6.7; the isotropic 1/3 is standard (Bartoli/Boltzmann route, any statistical-physics text). A verifier should: (1) rederive $P = I/c$ and $2I/c$ from impulse bookkeeping on the momentum comb; (2) confirm the two cosines of the oblique case independently (intercept vs projection); (3) reproduce $\langle\cos^2\theta\rangle_{\text{sphere}} = 1/3$ by the solid-angle integral; (4) recompute: sunlight 4.54/9.08 μPa; β-coefficient $3L_\odot/16\pi GM_\odot c = 5.74\times10^{-4}$ kg/m²; Starshot 667 N, $3.3\times10^5$ m/s², 180 s, $5.4\times10^9$ m with closing momentum audit $1.2\times10^5$ kg·m/s both sides; CMB $P = 1.4\times10^{-14}$ Pa; solar-core $P_{\text{rad}} = 1.5\times10^{13}$ Pa. Addendum for this edition (scoping note, per wave brief): the momentum-density identity $\mathbf{g} = \varepsilon_0\mathbf{E}\times\mathbf{B} = \mathbf{S}/c^2$ , both of its derivations, the static-crossed-fields subtleties, and the IKAROS/laser-pulse worked set are owned by EM-15's Momentum Density of the Electromagnetic Field (0573); the momentum-flux tensor by The Maxwell Stress Tensor (0574, its $T_{zz} = u$ recalled here by name); the conservation theorem by 0575. This edition owns the plane-wave pressure phenomenology: the wall catalogue including oblique/shear, the isotropic 1/3, $W = pc$ , and the β/Starshot applications — zero duplication, cross-referenced by name.

Builds on: Energy Density and the Poynting Vector · Monochromatic Plane Waves in Vacuum · Intensity and Time-Averaged Quantities · Complex Notation for Electromagnetic Waves.

Leads to: Reflection and Transmission at Normal Incidence · Reflection and Transmission at Oblique Incidence · Polarization of Electromagnetic Waves.

If radiation pressure is micropascals, how does light hold up a star against its own gravity?

Temperature and depth. Pressure of thermal radiation scales as $T^4$ ( $P = aT^4/3$ ): raising temperature from sunlight-at-Earth equivalents to the fifteen million kelvin of a stellar core multiplies the pressure by ~10¹⁹ — the solar core's radiation alone presses with fifteen million atmospheres. In the Sun that is still only ~0.1% of gas pressure, but the ratio grows as roughly $T^3/\rho$ , so in very massive stars radiation carries a large fraction of the load — and sets a ceiling: beyond the Eddington luminosity, radiation pressure on the envelope exceeds gravity and the star sheds itself. The micropascal intuition fails because it is calibrated at 300 K and one astronomical unit; the formula it comes from does not.

Does the light beam lose energy when it pushes something?

Against a stationary absorber, the beam loses all its energy (absorbed as heat) and its momentum does work only in the bookkeeping sense — force on a non-moving wall does zero mechanical work. Against a moving mirror, genuinely yes: light reflected from a receding mirror comes back redshifted, each photon losing energy $2\hbar k\,v/c$ to first order — exactly the mechanical work $F v$ done on the mirror. (Run the mirror toward the beam and the light blueshifts: that is how laser cooling extracts kinetic energy from atoms.) The energy and momentum ledgers close together, as EM-15's conservation article requires — pressure without energy transfer is only possible on things that don't move.

Why doesn't the enormous number of photons hitting me from all directions all day crush me?

Two mercies. First, magnitude: full sunlight presses ~5 μPa; ordinary room illumination is thousands of times weaker still — your skin's own thermal radiation pushes back comparably. Second, isotropy: indoor light arrives from all directions, and near-isotropic pressure loads compress rather than shove, at the $u/3$ level of ~10⁻²⁰ atmospheres. The atmosphere's molecular bombardment delivers 10¹⁰ times more pressure than direct sunlight — light's momentum is real but its density at everyday temperatures is famously frugal: $p = W/c$ divides every watt by three hundred million.

Is the momentum in the light or in the thing it pushes? What carries it across empty space?

In the field, literally and accountably. Between emitter and target, the momentum resides in the traveling configuration of E and B at density $\varepsilon_0\mathbf{E}\times\mathbf{B}$ — EM-15's momentum-density article established that this is not metaphor: without field momentum, an emitter would recoil now against a push delivered later, and Newton's third law would fail during the light's transit (its Newton's-third-law article owns that audit). The plane wave is the cleanest carrier: momentum $u/c$ per unit volume, moving at $c$ , delivered on contact by the Lorentz force of the wave's own fields on the target's charges. Photon language re-tells the same books in lumps of $\hbar k$ ; the classical field already balances them exactly.

Solar sails are pushed outward — so how can a sail travel toward the Sun?

By tacking against its own orbit, not against the light. The sail cannot thrust sunward (pressure always points away from the source for a mirror facing it), but a spacecraft in orbit doesn't need that: tilt the sail so the thrust has a component against the orbital velocity, and the craft loses angular momentum and spirals inward — tilt the other way to spiral out. This is Derivation A's shear/steering geometry as a navigation tool, and it is exactly how LightSail 2 raised its orbit and how proposed missions would reach the inner planets. Sailing physics, faithfully transplanted: you cannot sail straight upwind, and you do not need to.

Do these formulas assume light is photons? What did classical physics predict before quantum mechanics?

Everything here is pre-quantum. Maxwell computed $P = I/c$ in 1873 from field momentum alone; Bartoli reached $u/3$ from thermodynamics in 1876; Lebedev measured both before the photon existed (1900; Einstein's quantum is 1905). Photons re-partition the same totals into lumps — $p = \hbar k$ per lump — changing no classical prediction in this article, but adding one thing classical theory lacks: fluctuations. A beam of discrete momentum kicks arrives with shot noise, and that quantum radiation-pressure noise is now the measured, engineered-around limit of LIGO's mirrors. Classical: the mean pressure. Quantum: its jitter.

How is the comet's blue ion tail different from the dust tail this article explains?

Different force, different physics, visibly different direction. The dust tail is radiation pressure (β of Worked Example 1): micron grains pushed anti-sunward, released slowly, curving along the orbit — yellowish, diffuse. The ion tail is not light pressure at all: it is gas ionized by solar UV and then picked up by the solar wind's magnetic field, snapping straight anti-sunward at hundreds of km/s — blue (CO⁺ fluorescence), narrow, and gusty with the wind's weather. Comets are a two-force demonstration hanging in the sky: one tail for the photon flux, one for the plasma flux, both discovered pointing away from the Sun long before either force had a name.