Momentum Density of the Electromagnetic Field

This chapter has been auditing energy: how fields store it, how it flows, who pays for the work done on charges. Now the audit turns to the other conserved currency of mechanics — momentum — and immediately hits a scandal. Take two point charges and send them past each other on perpendicular tracks. Compute the electromagnetic forces they exert on one another and you will find the forces are not equal and opposite. Newton's third law, the axiom that guaranteed momentum conservation since the Principia, simply fails. Either momentum conservation is dead in electrodynamics, or something besides the charges is carrying momentum. This article names that something. Wherever an electric field and a magnetic field coexist and are not parallel, the field itself carries linear momentum, distributed through space with density $\mathbf{g} = \varepsilon_0(\mathbf{E}\times\mathbf{B})$ — the Poynting vector again, divided by $c^2$ . This is not a bookkeeping metaphor. Sunlight striking your hand delivers momentum at a measurable rate; a spacecraft has sailed the solar system on nothing else; and when the accounts of any electromagnetic collision are totalled, the field's momentum is exactly what closes the books. The claim earns its keep twice over here: once dynamically (a wave shoving a charge delivers momentum equal to the energy absorbed divided by $c$ — computed directly from the Lorentz force), and once structurally (energy in transit at speed $c$ is momentum, by the same relativity that gave $E = mc^2$ ). The stranger consequences — momentum stored in fields where nothing moves at all, and the hidden mechanical momentum that must balance it — are this chapter's next four articles. This one establishes the currency.

Key takeaways

The electromagnetic field carries linear momentum with density $\mathbf{g} = \varepsilon_0\,\mathbf{E}\times\mathbf{B} = \mathbf{S}/c^2$ — the Poynting vector rescaled by $1/c^2$ : momentum density and energy flux are the same vector field wearing different units.
The momentum is physical and exchangeable: a wave absorbed by matter delivers momentum $\Delta p = \Delta U/c$ ; the resulting radiation pressure is $\langle S\rangle/c$ on an absorber and $2\langle S\rangle/c$ on a mirror.
Sunlight at Earth ( $1361$ W/m²) pushes with $4.5\ \mu$ Pa — femto-scale per cubic metre, yet enough that solar sails (IKAROS, LightSail 2) have flown on it.
Without field momentum, momentum conservation fails for moving charges: the interparticle forces need not balance. The field is the third participant that closes every ledger.
Even static crossed fields carry $\mathbf{g} \neq 0$ — a charge sitting beside a magnet stores net field momentum while nothing moves. That surprise drives the rest of this chapter.
KEY IDEA: wherever $\mathbf{E}$ and $\mathbf{B}$ coexist and are not parallel, the field carries momentum density $\mathbf{g} = \varepsilon_0\,\mathbf{E}\times\mathbf{B} = \mathbf{S}/c^2$ — real momentum, exchangeable with matter, required for conservation to survive.

The core insight

Wherever electric and magnetic fields coexist and are not parallel, the field carries linear momentum density $\mathbf{g} = \varepsilon_0\,\mathbf{E}\times\mathbf{B} = \mathbf{S}/c^2$ — physical momentum, exchangeable with matter, required by conservation laws.

Where this lives today

Field momentum is a working commodity across modern physics. In the laboratory, optical tweezers trap and steer micron-scale beads — and single biomolecules tethered to them — with piconewton forces bookkept entirely by $\mathbf{g}$ ; laser cooling subtracts atomic momentum one $\hbar k$ photon-kick at a time, the technology beneath every atomic clock and quantum-gas experiment. Metrologists now run the logic in reverse: since $F = 2P_{\text{laser}}/c$ exactly, a calibrated laser hitting a mirror is a force standard, used to measure both small thrusts and, inverted, laser power by its push. At the largest scales, the momentum budget of light shapes structure: radiation pressure supports the interiors of the most massive stars against gravity and sets the Eddington limit — the luminosity at which a star's own light unbinds infalling matter — which in turn caps the feeding rate of the black holes powering quasars. Comet dust tails streak away from the Sun on radiation pressure; interplanetary dust spirals inward as it re-radiates (the Poynting–Robertson drag, this chapter's namesake vector at work on solar-system timescales). And gravitational-wave interferometry has made $\mathbf{g}$ a noise term: LIGO's mirrors are measurably kicked by quantum fluctuations of the circulating beam's momentum delivery — radiation-pressure noise — now a design constraint on the most sensitive instruments ever built. The dilute density of equation (1), given a kilowatt of circulating power and a 40 kg mirror, competes with spacetime strain at the $10^{-19}$ m scale.

Origins · 1 minJ.J. Thomson and how this was discovered

J.J. Thomson

1856–1940 · Cambridge

Joseph John Thomson — "J.J." to a generation of physicists he trained — ran the Cavendish Laboratory for 35 years and is remembered as the discoverer of the electron (1897, cathode rays bent by fields, charge-to-mass ratio measured). But four years before that, his Recent Researches in Electricity and Magnetism (1893) — written as a sequel volume to Maxwell's Treatise — introduced the idea this article is built on: that the field surrounding moving charges carries momentum, which Thomson located in the "Faraday tubes" of force he used as his working picture, in a density proportional to $\mathbf{E}\times\mathbf{B}$ . He used it to explain why a charged sphere resists acceleration beyond its bare mass — electromagnetic inertia, the first hint that mass itself might be partly field. His momentum bookkeeping was refined by Poynting's energy work and formalized by Abraham and Lorentz, but the physical claim — the field is a reservoir of momentum, not a mathematical fiction — is his. Seven of his research assistants won Nobel prizes, as did his son George (for showing the electron J.J. proved a particle is also a wave — the family owning both sides of the duality). His own prize came in 1906.

History

1873

— Maxwell's Treatise computes the pressure of light from stress in the field; sunlight's push at Earth estimated

1893

— J.J. Thomson's Recent Researches assigns moving charges a field momentum via Faraday tubes — the concept is born

1900

— Poincaré treats E cross B over c squared as the momentum of a "fluide fictif," wrestling with recoil paradoxes

1901

— Lebedev, and independently Nichols and Hull, measure radiation pressure on suspended vanes — the fingerprint confirmed

1903

— Abraham gives the field momentum density its modern form and name in electron theory

1905

— Einstein's light quantum makes p = E/c particulate; 1906, the box argument ties energy transport to momentum universally

2010

— IKAROS sails interplanetary space on sunlight; LIGO era makes radiation-pressure noise a precision-instrument design constraint

The physical picture

Start with the cleanest experiment nature runs continuously: light hitting an absorber. A plane wave arrives, the absorber's electrons are shaken by the wave's $\mathbf{E}$ , and the shaking electrons — now moving in the wave's $\mathbf{B}$ — feel a magnetic force $q\mathbf{v}\times\mathbf{B}$ that points, always, along the propagation direction. The wave does not merely warm the wall; it shoves it. Work through the forces (Derivation A) and the shove is exactly one part in $c$ : absorb energy $U$ , receive momentum $U/c$ . That momentum did not materialize at the wall. It rode in with the wave, spread through it at some density — and the only candidate with the right direction, the right magnitude, and the right units is $\varepsilon_0(\mathbf{E}\times\mathbf{B})$ .

Figure

Field momentum in one picture: a wave delivers momentum U over c, and the density that carried it is epsilon-zero E cross B

Now zoom out from the wave and read the formula as geometry. The cross product $\mathbf{E}\times\mathbf{B}$ points perpendicular to both fields — in a wave, straight down the propagation direction, so field momentum points where the energy is going. That is no coincidence: $\mathbf{g}$ is the Poynting vector divided by $c^2$ , and the proportionality is the field-theory edition of the deepest kinematic fact in relativity — energy in transit is momentum. A crate of energy $U$ moving at speed $v$ carries momentum $Uv/c^2$ ; field energy moves at $c$ ; divide once by $c^2$ and the Poynting flux becomes a momentum density (Derivation B makes this precise). One vector field, $\mathbf{E}\times\mathbf{B}$ , keeps both books: multiplied by $1/\mu_0$ it is the energy in motion, multiplied by $\varepsilon_0$ it is the momentum in residence. The formula also answers the scandal in the overview. When two moving charges push on each other with unbalanced forces, mechanical momentum alone drifts — but tally $\int\mathbf{g}\,d^3r$ over the fields around them and the drift is exactly cancelled, instant by instant. The field is not a passive stage on which charges act out their forces; it is a full participant with its own momentum account. Newton's Third Law in Electrodynamics (two articles ahead) prosecutes that case in detail, and Conservation of Linear Momentum in Electrodynamics proves the general theorem. One warning before the mathematics, because it is this chapter's favorite trap. Nothing in $\mathbf{g} = \varepsilon_0\,\mathbf{E}\times\mathbf{B}$ requires anything to change. A point charge held at rest beside a bar magnet produces static $\mathbf{E}$ and static $\mathbf{B}$ whose cross product circulates through space: the configuration stores momentum — genuinely, by the same formula that pushes solar sails — while nothing moves and nothing flows. Conservation then demands a compensating momentum somewhere in the matter, and finding it (Hidden Momentum in Magnetic Systems) is one of the subtlest stories in classical physics. Hold the puzzle; this article establishes the currency it is denominated in.

Figure

The trap made visible: a charge at rest beside a magnet — static fields, yet epsilon-zero E cross B threads the space between them

The momentum ledger's new column

The central definition, first. Alongside the energy density $u$ that the sibling article Energy Density of the Electromagnetic Field established, every region of space with crossed fields holds momentum:

\mathbf{g} = \varepsilon_0\,(\mathbf{E}\times\mathbf{B}) = \mu_0\varepsilon_0\,\mathbf{S} = \frac{\mathbf{S}}{c^2} \qquad (1)

where: $\mathbf{g}$ — momentum per unit volume (kg·m⁻²·s⁻¹; equivalently N·s per m³); $\mathbf{S} = (\mathbf{E}\times\mathbf{B})/\mu_0$ — the Poynting vector, energy flux in W/m²; and $c^2 = 1/\mu_0\varepsilon_0$ converts between them. Read the units physically: in bright sunlight ( $S = 1361$ W/m²) the momentum density is $S/c^2 = 1.5\times10^{-14}$ kg·m⁻²·s⁻¹ — a cubic kilometre of noon sunshine holds about $1.5\times10^{-5}$ kg·m/s, the momentum of a crawling ant. The factor $1/c^2$ makes field momentum density almost comically small; what makes it consequential is that light rearms it at speed $c$ , so the delivery rate — the pressure — is only one factor of $c$ down from the energy flux. Integrate over a region and the field's total momentum is

\mathbf{p}_{\text{field}} = \int_V \mathbf{g}\,d^3r = \varepsilon_0\int_V (\mathbf{E}\times\mathbf{B})\,d^3r, \qquad (2)

where: $V$ — the region being audited (all space, for an isolated system). This is the quantity that joins $\sum m\mathbf{v}$ in the conservation law: the statement that survives electrodynamics is not "mechanical momentum is conserved" but $\mathbf{p}_{\text{mech}} + \mathbf{p}_{\text{field}} =$ constant, for isolated systems. The precise local form of that statement — a continuity equation with a momentum-flux tensor playing the role currents play for charge — is built over the next two articles (The Maxwell Stress Tensor supplies the flux object, Conservation of Linear Momentum in Electrodynamics proves the theorem); equation (1) is quoted here on that credit, and Derivations A and B below pay the physical deposit. For a plane wave the density is worth having in closed form. With $\mathbf{E} = E_0\cos(kz-\omega t)\,\hat{\mathbf{x}}$ and $\mathbf{B} = (E_0/c)\cos(kz-\omega t)\,\hat{\mathbf{y}}$ ,

\mathbf{g} = \frac{\varepsilon_0 E_0^2}{c}\cos^2(kz-\omega t)\,\hat{\mathbf{z}}, \qquad \langle\mathbf{g}\rangle = \frac{\varepsilon_0 E_0^2}{2c}\,\hat{\mathbf{z}} = \frac{\langle u\rangle}{c}\,\hat{\mathbf{z}}, \qquad (3)

where: $\langle u\rangle = \tfrac12\varepsilon_0E_0^2$ — the wave's average energy density with its equal electric and magnetic shares (the crown article's fifty-fifty split). The relation $g = u/c$ is the plane-wave fingerprint: for field energy travelling at $c$ , momentum equals energy over $c$ , point by point — the classical shadow of the photon's $p = E/c$ . Feed it a detector and you get pressure:

P_{\text{abs}} = c\,\langle g\rangle = \frac{\langle S\rangle}{c}, \qquad P_{\text{mirror}} = \frac{2\langle S\rangle}{c}, \qquad (4)

where: the absorber collects the incoming momentum column ( $c$ metres of it per second — hence $c\langle g\rangle$ ), and the mirror reverses it, doubling the transfer. These two lines are the entire theory of radiation pressure at normal incidence; the systematic wave treatment (oblique incidence, partial reflection, time-averaging technology) is E8 business, owned by Momentum and Radiation Pressure — this article needs only the fingerprint.

Derivation AA wave shoves what it shakes — momentum transfer from the Lorentz force

The goal is to show, from nothing but the Lorentz force, that an electromagnetic wave absorbed by a charge delivers momentum equal to the absorbed energy divided by $c$ — establishing that the wave carries momentum in the propagation direction at density $u/c$ , with no conservation postulate assumed. The strategy: put a single mobile charge in the path of a plane wave, let the wave's electric field do work on it, and audit the magnetic half of the Lorentz force, which is where the push lives. Watch how the two field components split the labor — $\mathbf{E}$ pays energy, $\mathbf{B}$ converts payment into push — and watch $B = E/c$ turn the ratio into exactly $1/c$ .

Step 1. Set the stage. A plane wave travels along $+\hat{\mathbf{z}}$ with $\mathbf{E} = E\,\hat{\mathbf{x}}$ and $\mathbf{B} = (E/c)\,\hat{\mathbf{y}}$ at the location of a charge $q$ that is free to respond (an electron in the absorbing wall). The wave drives the charge; friction in the wall (collisions, re-radiation) keeps its velocity bounded, so the charge settles into some response velocity $\mathbf{v}$ dominated by the driving field: $\mathbf{v} = v\,\hat{\mathbf{x}}$ , oscillating in phase with whatever the dissipation allows.

Step 2. The energy account. Only $\mathbf{E}$ does work (magnetic forces never do — the E4 doctrine), so the charge absorbs power

\frac{dU}{dt} = q\,\mathbf{E}\cdot\mathbf{v} = qEv.

Whatever the phase relation between $v$ and $E$ , this is the instantaneous rate at which the wave's energy becomes the wall's energy.

Step 3. The momentum account. The charge moving at $\mathbf{v} = v\,\hat{\mathbf{x}}$ in the wave's magnetic field feels

\mathbf{F}_{\text{mag}} = q\,\mathbf{v}\times\mathbf{B} = qv\,\frac{E}{c}\,(\hat{\mathbf{x}}\times\hat{\mathbf{y}}) = \frac{qEv}{c}\,\hat{\mathbf{z}}.

Why this always points forward: when the wave's $\mathbf{E}$ reverses (half a cycle later), the driven $\mathbf{v}$ reverses with it — but so does $\mathbf{B}$ . The product $v B$ keeps its sign. For a charge whose response is in phase with the drive — which is what the dissipation in Step 1 enforces — the magnetic force rectifies the oscillation into a steady shove along $+\hat{\mathbf{z}}$ , every half cycle.

Step 4. Divide the two accounts. Comparing Steps 2 and 3,

\frac{dp_z}{dt} = F_z = \frac{qEv}{c} = \frac{1}{c}\,\frac{dU}{dt} \qquad\Longrightarrow\qquad \boxed{\ \Delta p = \frac{\Delta U}{c}\ }

— absorb energy $\Delta U$ , receive momentum $\Delta U/c$ , with all the messy material physics ( $q$ , $v$ , phases, collision rates) cancelling in the ratio. The wall gains momentum it did not have; the wave loses it; so the wave carried it. Since the wave transports energy at density $u$ moving at speed $c$ , the momentum it must carry to honor this exchange rate is density $u/c$ pointing along the travel direction — precisely equation (3), and with $u = \varepsilon_0 E^2 = \varepsilon_0 c\,|\mathbf{E}\times\mathbf{B}|$ for a wave, precisely $\mathbf{g} = \varepsilon_0\,\mathbf{E}\times\mathbf{B}$ . Why the ratio is universal: Step 4's identity $F_z = (1/c)\,dU/dt$ holds instant by instant for any response phase, so the push over a cycle is always $1/c$ times the energy actually absorbed; the ratio $1/c$ matches both the photon relation $p = E/c$ and the plane-wave result (3) computed from the field formula directly.

Go deeper · Derivation B · 2 minEnergy in transit is momentum — the transport argument for g equals S over c squared

Derivation BEnergy in transit is momentum — the transport argument for g equals S over c squared

The goal is the general statement — $\mathbf{g} = \mathbf{S}/c^2$ for any field configuration, not just plane waves — obtained from the relativistic kinematics of energy transport rather than from any particular force calculation. The strategy is Einstein's 1906 box argument run in reverse: relativity's consistency (specifically, that an isolated system's center of energy cannot spontaneously accelerate) forces any flow of energy to be accompanied by momentum, in the exact ratio flow over $c^2$ . Watch that the argument never mentions electromagnetism — it fixes the momentum of any energy current, which is why the same $1/c^2$ appears in every field theory.

Step 1. The thought experiment. A closed, isolated box of mass $M$ and length $L$ floats at rest. A lamp on its left wall fires a pulse of light with energy $U$ toward the right wall, where it is absorbed. By Derivation A the lamp recoils: the pulse carries some momentum $p$ , so the box slides left at $v = p/M$ while the light is in flight, then stops when the pulse is absorbed (time of flight $t \approx L/c$ ).

Step 2. The consistency requirement. The box is isolated — no external force ever acts — so its center of energy must stay put. But energy $U$ moved a distance $\approx L$ to the right while the box (mass $M$ ) moved $\Delta x = vt = pL/Mc$ to the left. Balance the books:

M\,\Delta x = \frac{U}{c^2}\,L \qquad\Longrightarrow\qquad \frac{pL}{c} = \frac{UL}{c^2} \qquad\Longrightarrow\qquad p = \frac{U}{c}.

Why $U/c^2$ enters on the right: the "mass" that relocated with the pulse is its mass-equivalent $U/c^2$ — this argument is where Einstein first extracted $E = mc^2$ 's inertia-of-energy content. Requiring the center of energy to stay fixed for every $L$ and every $M$ leaves no freedom: light of energy $U$ must carry momentum exactly $U/c$ , or an isolated box could inch itself across the room by lamp-flashes.

Step 3. From pulse to density. The pulse's energy travelled at speed $c$ : an energy current. Localize the statement: wherever field energy flows at rate $\mathbf{S}$ (energy per area per time), the associated momentum per volume is that current divided by $c^2$ ,

\mathbf{g} = \frac{\mathbf{S}}{c^2} = \frac{1}{\mu_0 c^2}\,(\mathbf{E}\times\mathbf{B}) = \varepsilon_0\,(\mathbf{E}\times\mathbf{B}),

using $\mu_0\varepsilon_0 c^2 = 1$ in the last step. The Poynting vector of the sibling article The Poynting Vector — constructed there as an energy bookkeeping current — is thereby conscripted into a second job it was never designed for: momentum register.

Step 4. The reach and the fine print. Because Step 2 used only isolation and relativistic consistency, the conclusion binds every configuration, including static ones: a charge beside a magnet has circulating $\mathbf{S}$ , hence resident $\mathbf{g}$ , with nothing moving anywhere. And the same center-of-energy logic that fixed the coefficient will return with teeth in Hidden Momentum in Magnetic Systems: an isolated static system must have zero total momentum, so a nonzero $\int\mathbf{g}\,d^3r$ in static fields forces equal-and-opposite mechanical momentum inside the matter. The formal statement — $\mathbf{g} = \mathbf{S}/c^2$ as the symmetry of the stress–energy tensor, $T^{0i} = T^{i0}$ — is the closing article's business (Conservation Laws and Spacetime Symmetries). Why the box argument is exact: the treatment ignores $O(v/c)$ rigidity subtleties, but these cancel in the careful version, leaving the coefficient unchanged. Dimensional check: $\mathbf{S}/c^2$ carries units W·m⁻²/(m²·s⁻²) = kg·m⁻²·s⁻¹, consistent with the plane-wave result (3) and Derivation A's dynamical ratio.

Limiting cases

Case	Result	Meaning
plane wave	$g = u/c$ , along propagation	the photon relation $p = E/c$ , classical edition; radiation pressure follows
$\mathbf{E} \parallel \mathbf{B}$	$\mathbf{g} = 0$	both energy densities nonzero, yet no momentum — geometry, not intensity, decides
pure $\mathbf{E}$ or pure $\mathbf{B}$	$\mathbf{g} = 0$	a lone capacitor or lone solenoid stores energy but no momentum
static crossed fields	$\mathbf{g} \neq 0$ , constant	momentum in residence with nothing moving — the chapter's standing puzzle (hidden momentum)
$c \to \infty$	$\mathbf{g} = \mathbf{S}/c^2 \to 0$	in instantaneous-action physics the field carries no momentum — Newton's third law returns (next articles)
slow charges, $v \ll c$	field momentum $\sim (v/c)^2$ corrections to mechanics	why two centuries of mechanics never noticed the field's column in the ledger

Common trapS over c squared does not mean momentum lives only where energy flows

The proportionality $\mathbf{g} = \mathbf{S}/c^2$ invites two opposite mistakes. First: "no net energy transport, so no momentum." False — in static crossed fields $\mathbf{S}$ circulates in closed loops, delivering no energy anywhere, yet $\mathbf{g}$ is genuinely there, and its volume integral can be nonzero (charge beside a magnet). Momentum density is a state of the field, not a report of traffic. Second, the reverse: "nonzero total field momentum, so the system must drift." Also false — for static isolated systems the matter carries an exactly compensating hidden mechanical momentum, and nothing drifts. Both errors come from treating $\mathbf{S}/c^2$ as a description of motion rather than what it is: a density of stored momentum, as real when parked as when travelling. The audit that settles every such case is the total ledger — mechanical plus field plus flux through the boundary — never intuition about what "looks like" it is moving.

Worked examples

1 · Sunlight as a propulsion system: IKAROS (confidence builder)

In 2010 the spacecraft IKAROS deployed a $14\times14$ m ( $A = 196$ m²) reflective sail at roughly Earth's distance from the Sun ( $\langle S\rangle = 1361$ W/m²). Estimate the radiation force, and the momentum density in the sunlight doing the pushing.

Step 1. Momentum density on arrival, from equation (1): $g = S/c^2 = 1361/(2.998\times10^8)^2 = \mathbf{1.5\times10^{-14}}$ kg·m⁻²·s⁻¹. Absurdly dilute — but it is restocked at the speed of light.

Step 2. Pressure. A perfect mirror at normal incidence reverses the incoming momentum column: $P = 2\langle S\rangle/c = 2\times1361/2.998\times10^8 = \mathbf{9.1\ \mu Pa}$ . (A perfect absorber would feel half: $4.5\ \mu$ Pa.)

Step 3. Force on the sail: $F = PA = 9.1\times10^{-6}\times196 = \mathbf{1.8\ mN}$ for an ideal mirror. IKAROS's measured thrust was $1.12$ mN — between the absorber floor ( $0.89$ mN) and the mirror ceiling, exactly where a real aluminized membrane with imperfect reflectivity and some tilt should sit. The mission's accelerometers confirmed the force to within a few percent: equation (1), flight-tested.

Step 4. Calibrate the patience required. On IKAROS's $315$ kg, $1.12$ mN gives $a = 3.6\times10^{-6}$ m/s² — about $110$ m/s of velocity per year. Tiny, but continuous, free, and propellant-less; over months it measurably reshaped the orbit. That is field momentum as an engineering budget line.

2 · The momentum audit of a laser pulse (exam level)

A laboratory pulse laser delivers $E_{\text{pulse}} = 1.0$ J in $\tau = 10$ ns, focused onto a mirror of mass $m = 1.0$ g hanging as a pendulum. Find the pulse's momentum, the average force during reflection, the mirror's recoil velocity, and the peak pressure if the spot is $1.0$ mm².

Step 1. Pulse momentum, from Derivation A's exchange rate: $p = E_{\text{pulse}}/c = 1.0/2.998\times10^8 = \mathbf{3.3\times10^{-9}}$ kg·m/s — a mosquito at a tenth of a millimetre per second.

Step 2. The mirror reverses it: momentum transfer $\Delta p = 2p = 6.7\times10^{-9}$ kg·m/s, so the recoil velocity is $v = \Delta p/m = 6.7\times10^{-9}/10^{-3} = \mathbf{6.7\times10^{-6}\ m/s}$ . Measurable with interferometry — and this is precisely how modern "photon-pressure" force standards calibrate small thrusts.

Step 3. Average force during the pulse: $F = \Delta p/\tau = 6.7\times10^{-9}/10^{-8} = \mathbf{0.67\ N}$ . Note the lesson: the same momentum delivered in 10 ns exerts a very respectable force — radiation pressure is weak only because sources are usually dilute in time and space.

Step 4. Peak pressure on the $1.0$ mm² spot. Intensity $S = E_{\text{pulse}}/(\tau A) = 1.0/(10^{-8}\times10^{-6}) = 10^{14}$ W/m²; mirror pressure $P = 2S/c = 2\times10^{14}/2.998\times10^8 = \mathbf{6.7\times10^{5}\ Pa}$ — about 6.6 atmospheres of pure light. Push the same arithmetic to the National Ignition Facility's $10^{19}$ W/m² and light pressure reaches millions of atmospheres: the regime where radiation pressure implodes fusion capsules and levitates the interiors of massive stars.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §8.2.3 · Jackson, Classical Electrodynamics, 3rd ed., §6.7 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 9 · Zangwill, Modern Electrodynamics, §15.7 · Feynman Lectures, Vol. II, §27 The Lorentz-force density $\mathbf{f}=\rho\mathbf{E}+\mathbf{J}\times\mathbf{B}$ can be manipulated into the local momentum-balance form $\partial_t\mathbf{g}=\nabla\cdot\overleftrightarrow{T}-\mathbf{f}$ ; that full stress-tensor derivation is carried out in The Maxwell Stress Tensor and Conservation of Linear Momentum in Electrodynamics. Key numerical benchmarks: $g=u/c$ for a plane wave; radiation pressure from the solar constant $1361\,\text{W}\,\cdot\,\text{m}^{-2}$ gives $P_{\text{abs}}\approx4.5\times10^{-6}\,\text{Pa}$ . Worked values: sunlight $g = 1.51\times10^{-14}$ kg·m⁻²·s⁻¹, $P_{\text{abs}} = 4.54\ \mu$ Pa, $P_{\text{mirror}} = 9.08\ \mu$ Pa; IKAROS 196 m²: mirror ceiling 1.78 mN, absorber floor 0.89 mN, measured 1.12 mN; laser example: $p = 3.34\times10^{-9}$ kg·m/s, $F_{\text{avg}} = 0.67$ N, $P = 6.7\times10^{5}$ Pa.

Builds on: The Poynting Vector · Energy Density of the Electromagnetic Field · Work Done by Electromagnetic Fields on Charges · Charge Conservation and the Continuity Equation

Leads to: The Maxwell Stress Tensor · Conservation of Linear Momentum in Electrodynamics · Newton's Third Law in Electrodynamics · Hidden Momentum in Magnetic Systems · Angular Momentum of the Electromagnetic Field

If radiation pressure is only microopascals, how does light hold up a star?

By having astronomical amounts of it and a very long lever of time. Pressure scales with intensity, and intensity in a massive star's interior is beyond anything laboratory optics produces: the radiative flux threading the core of an O star corresponds to pressures competitive with the gas pressure itself. The scaling to remember is $P = \langle S\rangle/c$ for absorption-dominated transfer: at the Sun's surface that is microopascals, but deep inside a $50\,M_\odot$ star the photon field is so dense that radiation pressure supplies a large fraction of the hydrostatic support. Push further and you reach the Eddington limit — luminosity high enough that radiation pressure on electrons beats gravity's pull on the associated protons — beyond which the star sheds its own envelope. The same tiny-coefficient, huge-flux logic runs Nichols radiometers ( $10^{-6}$ Pa, needs a torsion fibre) up to inertial-confinement fusion ( $10^{11}$ Pa, needs the world's largest laser). Nothing about the physics changes; only $\langle S\rangle$ does.

Does the momentum belong to the photons or to the E and B fields? Which is the real story?

They are the same story told at different resolutions. Classically, momentum resides continuously in the field at density $\varepsilon_0\mathbf{E}\times\mathbf{B}$ ; quantum mechanically, the field's excitations are photons each carrying $p = \hbar k = E/c$ . The classical density is exactly what a dense stream of photons averages to: intensity $S$ means photon flux $S/\hbar\omega$ , each with momentum $\hbar\omega/c$ , giving momentum delivery $S/c$ — identical to equation (4). The classical formula is not an approximation being tolerated; for the coherent states a laser produces it is the exact expectation value. Where the quantum picture earns its keep is at low intensity (single-photon recoil in laser cooling, one $\hbar k$ at a time) and in fluctuations: the shot noise of photon momentum kicks is LIGO's radiation-pressure noise, which no smooth classical $\mathbf{g}$ predicts. Use whichever resolution the experiment sees.

Why did nobody notice field momentum for two hundred years of successful mechanics?

Because it is suppressed by $c^2$ in every slow-motion situation. For charges moving at $v \ll c$ , the field momentum of their interaction is smaller than their mechanical momentum by order $(v/c)^2$ — for laboratory currents ( $v \sim$ mm/s drift) or planetary orbits, that is beneath any classical instrument. Newton's third law and $\sum m\mathbf{v}$ conservation are the $c \to \infty$ limit of electrodynamics, and an excellent one: instantaneous action-at-a-distance is wrong in principle but right to parts in $10^{16}$ for everyday mechanics. The field's momentum column only becomes legible when something moves near $c$ (light itself, radiation pressure) or when precision demands it (the perpendicular-charges force imbalance, coming two articles hence, is a $(v/c)^2$ effect). This is the standard shape of a superseding theory: the old law survives as a controlled limit, and the new bookkeeping only bills you at high speed.

In static crossed fields, is the circulating Poynting vector actually carrying energy around in a loop?

The honest answer: the formalism declines to distinguish "energy circulating in closed loops" from "no local transport, with $\mathbf{S}$ as an idle gauge-like artifact," because only $\nabla\cdot\mathbf{S}$ (net deposit) and, as this article now adds, $\mathbf{S}/c^2$ (momentum density) enter measurable statements. What is not optional is the momentum: you can shrug at circulating energy, but $\varepsilon_0\int\mathbf{E}\times\mathbf{B}\,d^3r \neq 0$ in a charge-plus-magnet system has mechanical consequences — discharge the capacitor or quench the magnet and real impulses appear (the disk paradox, three articles ahead, is this made rotational). The chapter's working doctrine, inherited from the Poynting articles: treat $\mathbf{S}$ pointwise as bookkeeping, trust its integrals — and now, trust its $1/c^2$ rescaling as genuinely stored momentum. Static-field circulation is the one place the two jobs of $\mathbf{E}\times\mathbf{B}$ pull apart, and momentum is the job that bites.

Light has no mass. How can it have momentum — does p equal mv fail?

The definition $p = mv$ fails; momentum does not. Relativity's general relation is $E^2 = (pc)^2 + (mc^2)^2$ , of which $p = mv$ is the slow-massive corner and $p = E/c$ the massless edge. Light sits on the edge: energy $U$ , momentum $U/c$ , mass zero, always moving at $c$ . Derivation A shows the classical field version needs no photon: the Lorentz force delivers momentum to whatever absorbs the wave, in the ratio $1/c$ , from the fields alone. If it helps intuition, the wave has inertia-equivalent $U/c^2$ in transit (Derivation B literally weighs it), and "mass-equivalent times $c$ " gives back $U/c$ . But the cleaner habit is to let go of mass as momentum's prerequisite: momentum is the conserved quantity of spatial translation symmetry (the chapter closer makes this precise), and fields qualify on symmetry grounds, mass or no mass.

Do I add the field momentum of every charge in a wire when current flows — does a circuit carry huge hidden E cross B momentum?

No — for ordinary circuits it is fantastically small, which is why circuit theory never mentions it. Estimate: a wire with surface field $E \sim 100$ V/m (driving the current) and internal $B \sim$ mT has $g = \varepsilon_0 EB \sim 10^{-12}$ kg·m⁻²·s⁻¹, and integrating over the field volume of a bench circuit gives momenta of order $10^{-15}$ kg·m/s — several orders below the mechanical momentum of the drifting electrons themselves. Two real exceptions: coaxial transmission lines carrying serious power do have a well-defined field momentum flowing down the dielectric (the next-but-one article computes a coax's stored momentum explicitly — the fields, not the electrons, carry the delivered momentum), and any configuration pairing strong static $\mathbf{E}$ with strong static $\mathbf{B}$ (superconducting magnets beside charged electrodes) stores enough field momentum that its compensating hidden momentum becomes a live theoretical issue. The rule: negligible for signals-and-breadboards, structural for power transmission and for paradox-hunting.

Is epsilon-zero E cross B the unique correct momentum density, or just one convention?

For vacuum fields with localized sources, the total $\int\mathbf{g}\,d^3r$ is unambiguous — it is fixed by conservation against the mechanical momentum, and every consistent formulation agrees on it. The density has the same modest gauge-like freedom all densities do: you could add any divergence-free field that integrates to zero and no vacuum experiment would notice. The standard density earns its privileged status three ways: it is what the momentum-balance theorem produces naturally from the Lorentz force (next two articles); it sits in the symmetric stress-energy tensor that gravity couples to (general relativity does measure densities, and it votes for this one); and its $\mathbf{S}/c^2$ relation implements the relativity of energy transport pointwise. The one place a genuine, physical ambiguity opens is inside matter, where "field" versus "medium" momentum can be split two defensible ways — the celebrated Abraham–Minkowski problem, owned honestly by this chapter's Electromagnetic Stress, Momentum, and Energy in Media.