Intensity and Time-Averaged Quantities

The plane wave's books, opened two articles ago, pulse at $2\omega$ — for visible light, a thousand trillion beats per second. No photodiode, no bolometer, no eye, no antenna-plus-rectifier follows that. Every instrument that reports "how bright" integrates over many cycles and returns a mean. This article is about that act of averaging: innocent-looking, universally performed, and carrying exactly one number — the factor ½ — whose placement separates correct radiometry from a persistent family of factor-of-two errors. The centerpiece is the intensity formula $I = \tfrac12 c\varepsilon_0E_0^2$ : the time-averaged Poynting flux of a plane wave, the single most-used equation of applied electromagnetism. Around it this article builds the working toolkit — the rms amplitudes that absorb the ½ into the notation, the complex-notation shortcut $\tfrac12\mathrm{Re}(\tilde{\mathbf{E}}\times\tilde{\mathbf{H}}^*)$ that computes averages without ever writing a cosine, the inverse-square law as energy conservation rather than geometry folklore, and the averaged momentum $\langle g\rangle = I/c^2$ handed back to the pressure article. The payoff is fluency across an absurd dynamic range. One formula converts the solar constant into the kilovolt-per-metre field crest of sunlight, prices a light bulb's field at arm's length, and pulls the microvolts-per-metre whisper of a GPS satellite out of the sky — sixteen decades of intensity, one ½, no exceptions. Learn where the half lives and every brightness in physics becomes a two-line calculation.

Key takeaways

Detectors are slow: they average the $2\omega$ -pulsing books over many cycles. The average of $\cos^2$ is exactly ½ — the one new number in this article.
Intensity is the time-averaged Poynting flux: $I \equiv \langle S\rangle = \tfrac12 c\varepsilon_0E_0^2 = E_0^2/2Z_0$ , with $\langle u\rangle = \tfrac12\varepsilon_0E_0^2$ and $\langle g\rangle = I/c^2$ riding along.
Root-mean-square amplitudes absorb the ½: $E_{\text{rms}} = E_0/\sqrt2$ and $I = c\varepsilon_0E_{\text{rms}}^2$ — the convention AC engineering runs on.
In complex notation the average needs no integral: $\langle\mathbf{S}\rangle = \tfrac12\mathrm{Re}(\tilde{\mathbf{E}}\times\tilde{\mathbf{H}}^*)$ — the previous lemma, cashed.
The inverse-square law is conservation, not geometry trivia: $I(r) = P/4\pi r^2$ because spheres must pass the same power.
Key idea: every measured brightness is a cycle average, and for sinusoidal waves that average is half the peak — put the ½ in once, via rms or the lemma, and never lose it.

The core insight

Intensity is the cycle-averaged energy flux of the wave, $I = \langle S\rangle = \tfrac12c\varepsilon_0E_0^2$ : the ½ is $\langle\cos^2\rangle$ , absorbed either into rms amplitudes or into the complex-notation formula $\tfrac12\mathrm{Re}(\tilde{\mathbf{E}}\times\tilde{\mathbf{H}}^*)$ .

Where this lives today

Intensity bookkeeping is the quiet backbone of the connected world. Every wireless standard is a chain of this article's two derivations: transmit power times gain, divided by $4\pi r^2$ , converted to field by $\sqrt{2Z_0I}$ , compared against receiver noise — run the chain for your phone (half a watt, twenty kilometres of cell edge) and the microvolt-per-metre answers explain both why base stations are everywhere and why exposure levels sit orders of magnitude below safety limits (ICNIRP's limits are themselves written as time-averaged intensities and rms fields — regulatory text in this article's variables). Laser safety runs the same books in reverse: maximum permissible exposures are quoted as averaged intensities with duty-cycle corrections, because the eye is a slow detector focusing a fast comb. The precision frontier is radiometric. The solar constant is monitored to ~0.01% by spaceborne radiometers (its 0.1% solar-cycle wobble is a climate-model input); the candela — the SI unit of visible brightness — is defined through watts per steradian at 540 THz; and gravitational-wave interferometry closes the loop elegantly: LIGO reads out mirror positions as intensity changes at a photodiode, meaning the most delicate length measurement in history is, at its final stage, an exercise in exactly this article's ⟨S⟩.

The physical picture

Consider what a detector actually does. A silicon photodiode absorbs the wave's energy and delivers a photocurrent; its junction responds in nanoseconds at best — ten million optical cycles. A thermal sensor is slower by orders more. Your eye integrates for tens of milliseconds. Each instrument sits in the $2\omega$ -pulsing energy stream of article 0585 like a rain gauge in a storm: it cannot see individual drops, only the accumulated level. The reading it returns is the mean flux, and for a sinusoidal wave that mean is pinned by one small theorem: $\cos^2$ spends as much of its time above ½ as below, symmetrically, so its average over any whole number of cycles — indeed over any interval long compared to a cycle — is exactly ½. That single number restructures all the peak formulas of article 0585 into measured quantities. Peak flux $c\varepsilon_0E_0^2$ becomes measured intensity $\tfrac12c\varepsilon_0E_0^2$ ; peak density $\varepsilon_0E_0^2$ becomes mean density $\tfrac12\varepsilon_0E_0^2$ ; the equal partition survives averaging untouched (both accounts halve together). Nothing physical happened at the wave — the halving lives entirely in the act of reading.

Figure

What a slow detector sees — the 2ω-pulsing flux integrates to a steady mean at exactly half the peak; the rms amplitude is the DC-equivalent field

Two housekeeping matters attach to the mean, and both save future grief. First, vocabulary: this track, with Griffiths, calls $\langle S\rangle$ the intensity $I$ (W/m²). Radiometry proper calls the same quantity irradiance, reserving "intensity" for watts per steradian; astronomy layers on flux densities per hertz. The physics is identical; when reading across fields, check the units, not the word. Second, conventions for the ½: you can carry it explicitly (peak amplitudes, ½ in every formula), absorb it into rms amplitudes ( $E_{\text{rms}} = E_0/\sqrt2$ , no ½ anywhere — the universal AC-power convention: mains "230 V" is rms), or let complex notation supply it through the lemma. All three appear in the wild; mixing them without noticing is the standard source of silent factor-of-two errors in link budgets and laser-damage estimates.

The toolkit

The formal centerpiece takes one line to state and inherits everything from the articles before it:

I \equiv \langle S\rangle = \frac{1}{2}\,c\,\varepsilon_0 E_0^2 = \frac{E_0^2}{2Z_0} = c\,\varepsilon_0 E_{\text{rms}}^2, \qquad \langle u\rangle = \frac{I}{c}, \qquad \langle g\rangle = \frac{I}{c^2} \qquad (1)

where: $E_0$ — peak field amplitude; $Z_0 = \sqrt{\mu_0/\varepsilon_0} = 1/c\varepsilon_0 \approx 377\ \Omega$ — the impedance of free space, whose role as the E-to-H exchange rate the E7 Poynting article owns and this formula merely wears; $E_{\text{rms}} = E_0/\sqrt2$ . Read it physically: intensity is mean density in transit ( $I = c\langle u\rangle$ — the transport identity, averaged), and mean momentum flux rides along at one more factor of $c$ down, feeding the radiation-pressure catalogue of the previous article. Inverting for the field is the working direction of most problems: $E_0 = \sqrt{2I/c\varepsilon_0} = \sqrt{2Z_0 I}$ — with the mnemonic that $2Z_0 \approx 754\ \Omega$ turns W/m² directly into (V/m)².

Derivation AThe half — averaging the comb honestly

The goal is $\langle\cos^2\rangle = \tfrac12$ with its exact scope of validity, and the assembled averages (1). The derivation is deliberately short — the point is to do the one integral in this subject that everyone must do once, note the two independent routes to it, and mark precisely when the ½ is wrong (non-sinusoidal or unresolved-cycle cases at the end). The strategy: direct integration, then the complex-lemma cross-check, then assembly.

Step 1. The integral, once. Over one period $T = 2\pi/\omega$ , with $\varphi = kz - \omega t + \delta$ at fixed $z$ :

\langle\cos^2\varphi\rangle = \frac{1}{T}\int_0^T\cos^2\varphi\ \mathrm{d}t = \frac{1}{T}\int_0^T\frac{1 + \cos2\varphi}{2}\,\mathrm{d}t = \frac12 + 0.

The half-angle identity splits the integrand into the constant ½ and a pure oscillation at $2\omega$ , which completes exactly two full cycles in $T$ and integrates to zero. Why the answer is robust: any averaging window that is a whole number of half-periods gives exactly ½; a window of duration $\tau$ misaligned with the cycle errs only by $O(1/\omega\tau)$ — for a nanosecond measurement of green light, parts in $10^6$ . "Slow detector" is a superbly controlled approximation.

Step 2. The lemma route (cross-check and generalization). Complex notation's averaging lemma gives $\langle E^2\rangle = \tfrac12\mathrm{Re}(\tilde{E}\tilde{E}^*) = \tfrac12|\tilde{E}_0|^2$ — the same ½, no integral, and valid with complex amplitudes whose phases would make the direct integral tedious. Applied to the flux with $\tilde{\mathbf{H}} = \tilde{\mathbf{B}}/\mu_0$ :

\langle\mathbf{S}\rangle = \frac{1}{2}\,\mathrm{Re}\left(\tilde{\mathbf{E}}\times\tilde{\mathbf{H}}^*\right),

the workhorse form — for the vacuum plane wave it evaluates to $\tfrac12 E_0(E_0/\mu_0 c)\hat{\mathbf{k}} = \tfrac12c\varepsilon_0E_0^2\hat{\mathbf{k}}$ , and in later media problems, where $\tilde{\mathbf{E}}$ and $\tilde{\mathbf{H}}$ acquire a relative phase, the same expression automatically delivers the $\cos$ (phase-lag) power factor. Why this form is the one to memorize: it is basis-free, integral-free, correct in matter, and its conjugate placement encodes which field "works against" which — the entire content of average power flow in one expression.

Step 3. Assembly of (1). Averaging the comb of article 0585 term by term: $\langle u\rangle = \varepsilon_0E_0^2\langle\cos^2\rangle = \tfrac12\varepsilon_0E_0^2$ (equal E/B shares of $\tfrac14\varepsilon_0E_0^2$ each); $I = \langle S\rangle = c\langle u\rangle$ ; $\langle\mathbf{g}\rangle = \langle\mathbf{S}\rangle/c^2$ . The rms repackaging: defining $E_{\text{rms}}^2 \equiv \langle E^2\rangle$ gives $I = c\varepsilon_0E_{\text{rms}}^2$ with no visible ½ — the half now lives inside the amplitude definition, paid exactly once.

Step 4. Where the ½ is a lie. Three boundary cases patrol the result. Non-sinusoidal waveforms: $\langle f^2\rangle/f_{\text{peak}}^2$ is ½ only for sinusoids (a square-wave chop gives 1; pulsed lasers give the duty cycle) — for anything modulated, integrate the actual envelope or use rms honestly. Fast detectors: streak cameras and RF rectifiers resolve the cycle; their readings are the comb, not the mean, and applying (1) to them double-halves. Superpositions: the mean of a two-wave field carries interference cross terms ( $I \neq I_1 + I_2$ pointwise — the E7 ledger); (1) applies per coherent mode, after the modes are combined into one total amplitude. Verified: integral standard; lemma route consistent with the complex-notation article's Derivation B; $\tfrac12\mathrm{Re}(\tilde{\mathbf{E}}\times\tilde{\mathbf{H}}^*)$ per Jackson §6.9/Griffiths Eq. 9.73 context; assembled values against Griffiths Eqs. 9.61–9.64.

Go deeper · Derivation B · 2 minInverse square is conservation — intensity from a compact source

Derivation BInverse square is conservation — intensity from a compact source

The goal is $I(r) = P/4\pi r^2$ for an isotropic emitter of average power $P$ , derived as an energy audit rather than quoted as geometry, plus its anisotropic generalization — the two-line argument behind every link budget, stellar magnitude, and safety-distance calculation. The strategy: apply the E7 energy machinery (Poynting's theorem in steady state, recalled by name) to nested spheres, then localize with the plane-wave license of article 0583. Watch what the derivation does not assume: nothing about the wave's detailed shape — only conservation and steadiness.

Step 1. Steady state audits to zero storage. Over a full cycle, the field energy stored between two concentric spheres around the source is unchanged, and (in vacuum) nothing is dissipated there. Poynting's theorem in its integral, time-averaged form then says the mean power crossing every enclosing sphere is the same: $\oint\langle\mathbf{S}\rangle\cdot\mathrm{d}\mathbf{a} = P$ , independent of radius.

Step 2. Isotropy distributes it. If the source radiates equally in all directions, $\langle S\rangle$ is uniform over the sphere of area $4\pi r^2$ :

I(r) = \frac{P}{4\pi r^2}.

The $1/r^2$ is not a property of fields but of area: the same watts through ever-larger spheres. Consistently, the far field's amplitude must fall as $E_0 \propto 1/r$ (since $I \propto E_0^2$ ) — the spherical-wave amplitude law the wave-equation article exhibited, now derived from the books instead of the PDE.

Step 3. Real sources: gain. Antennas and lasers concentrate power into a solid angle; the practical form is $I = G\,P/4\pi r^2$ , with gain $G$ the concentration factor ( $G = 1$ isotropic; a dish's $G$ can be $10^4$ ; the product $GP$ is the "EIRP" of link budgets). Locally, at any $r$ large compared to the source, the wave is a plane wave (0583's license), so (1) converts $I$ to fields on the spot.

Step 4. The scope fence. The audit assumed nothing absorbed and nothing stored between spheres: in fog, plasma, or any lossy medium, $P(r)$ itself decays (exponential attenuation — the absorption article, 0595, by name) and inverse-square is the floor on dimming, not the whole law. Near the source (distances comparable to the antenna or within the reactive near field), storage is nonzero, the wave is not locally plane, and neither (1) nor inverse-square applies — the radiation chapter (E9) owns near fields. Verified: standard conservation argument (Griffiths §9.2.3 context, Purcell Ch. 9); consistent with the E7 Poynting article's coax audit method — same theorem, spherical bookkeeping; EIRP/gain form per any antenna text.

Figure

Inverse square is an audit, not a decree — the same mean power crosses every sphere, so intensity dilutes with area and the field falls as 1/r

Limiting cases

Case	Result	Meaning
sinusoid, slow detector	$\langle\cos^2\rangle = \tfrac12$ exactly	the article's one number; error $O(1/\omega\tau)$
rms convention	$I = c\varepsilon_0E_{\text{rms}}^2$ , no visible ½	the half paid once, inside the amplitude
complex convention	$\langle\mathbf{S}\rangle = \tfrac12\mathrm{Re}(\tilde{\mathbf{E}}\times\tilde{\mathbf{H}}^*)$	integral-free; survives phase lags in media
E, H in quadrature	$\langle S\rangle = 0$	reactive fields transport nothing on average
square-wave modulation	$\langle f^2\rangle = f_{\text{peak}}^2\times$ duty cycle	½ is sinusoid-specific
pulsed laser	$I_{\text{peak}} = I_{\text{avg}}/\text{duty}$	mJ/fs physics (0585's example) hides under mW averages
isotropic source	$I = P/4\pi r^2$ , $E_0 \propto 1/r$	conservation through nested spheres
lossy path	$I$ falls faster than $1/r^2$	absorption article 0595 owns the exponential

Common trapThe factor-of-two farm — peak vs rms vs mean is where intensity calculations go to die

Three amplitudes describe one wave — peak $E_0$ , rms $E_0/\sqrt2$ , and (in older engineering texts) peak-to-peak $2E_0$ — and three intensity conventions coexist: $\tfrac12c\varepsilon_0E_0^2$ , $c\varepsilon_0E_{\text{rms}}^2$ , and the complex $\tfrac12\mathrm{Re}(\tilde E\tilde H^*)$ whose ½ is already included. Every wrong-by-2 (or by-4) intensity in the wild is a crossing error among these: applying the ½ to an rms value (half of a half), quoting a peak field where a datasheet meant rms (mains: 230 V rms = 325 V peak), or adding the lemma's ½ on top of rms amplitudes inside $\tilde E$ . The discipline that ends it: label every amplitude at birth (peak or rms), convert to peak before using this article's (1), and when using complex notation feed it peak-convention amplitudes and let the lemma's ½ do all the halving. One further cousin: laser specifications quote average power; for pulsed systems the peak exceeds it by the inverse duty cycle — safety calculations done with the average are wrong by up to $10^{10}$ (femtosecond systems), which is not a rounding error.

Worked examples

1 · The fields of sunlight (confidence builder)

The solar constant — mean flux above the atmosphere — is $I = 1361$ W/m². Find $E_0$ , $E_{\text{rms}}$ , $B_0$ , the mean energy density, and the mean momentum flux; check via the 377 Ω route.

Step 1. Peak field: $E_0 = \sqrt{2I/c\varepsilon_0} = \sqrt{2\times1361/(2.998\times10^8\times8.854\times10^{-12})} = \sqrt{1.026\times10^6} = \mathbf{1013\ V/m}$ — a kilovolt per metre riding in ordinary daylight; nothing sparks because it reverses every $10^{-15}$ s and everyday matter responds only to the mean.

Step 2. Companions: $E_{\text{rms}} = E_0/\sqrt2 = \mathbf{716\ V/m}$ ; $B_0 = E_0/c = \mathbf{3.38\ \mu T}$ — a tenth of Earth's magnetic field, oscillating at $10^{15}$ Hz.

Step 3. Impedance check: $I = E_0^2/2Z_0 = (1013)^2/(2\times376.7) = 1361$ W/m² ✓ — the 377 Ω conversion (E7 Poynting's reading) closing exactly.

Step 4. The mean books: $\langle u\rangle = I/c = 4.54\ \mu$ J/m³ (the E7 shelf's sunlight anchor, re-derived); $\langle g\rangle = I/c^2 = 1.51\times10^{-14}$ kg·m⁻²·s⁻¹ — which, delivered to an absorber, is precisely the 4.54 μPa the pressure article charged. The averaged trio $I$ , $I/c$ , $I/c^2$ : one number, three prices, again.

2 · From a desk lamp to a GPS satellite — sixteen decades on one formula (exam level)

(a) Find $E_0$ two metres from a 60 W incandescent bulb (treat it as an isotropic radiator — most of its output is near-infrared, all of it counts as flux). (b) A GPS satellite radiates an effective $GP \approx 500$ W toward Earth from $r = 20{,}200$ km. Find $I$ and $E_0$ at the ground and comment on detectability.

Step 1. Bulb intensity by the conservation law: $I = P/4\pi r^2 = 60/(4\pi\times4) = \mathbf{1.19\ W/m^2}$ — about a thousandth of sunlight, matching everyday experience of warmth at arm's length.

Step 2. Bulb field: $E_0 = \sqrt{2Z_0I} = \sqrt{753.5\times1.19} = \mathbf{30.0\ V/m}$ ; $B_0 = 0.10\ \mu$ T. Thirty volts per metre from a desk lamp — fields of this size saturate nothing and are noticed by nothing but antennas and eyes.

Step 3. GPS intensity: $I = GP/4\pi r^2 = 500/(4\pi(2.02\times10^7)^2) = \mathbf{9.8\times10^{-14}\ W/m^2}$ — sixteen orders of magnitude below sunlight.

Step 4. GPS field: $E_0 = \sqrt{2Z_0 I} = \sqrt{753.5\times9.8\times10^{-14}} = \mathbf{8.6\ \mu V/m}$ . A few microvolts per metre: your receiver's antenna, a few centimetres of conductor, develops nanovolts of signal, buried ~20 dB below thermal noise — recovered only because the signal's phase structure is known (correlation against code replicas). The plane wave's coherence (article 0583) is what makes a $10^{-13}$ W/m² whisper usable for centimetre-level surveying. Answers: (a) 1.19 W/m², 30 V/m; (b) $9.8\times10^{-14}$ W/m², 8.6 μV/m.

Origins · 1 minLord Rayleigh and how this was discovered

Lord Rayleigh

1842–1919 · Terling and Cambridge — who made averages a precision instrument

John William Strutt, third Baron Rayleigh, is the physicist of the well-taken mean. His Theory of Sound (1877–78) built the modern machinery of time-averaged wave quantities — rms values, mean energy flux, the ½'s of sinusoidal averaging — that this article transplants to light; his name is on the statistical distribution governing the amplitude of many superposed random phasors (Rayleigh fading, in every wireless engineer's vocabulary) and on the resolution criterion that limits every telescope. Measuring averages carefully was his signature: his determination of electrical standards at Cambridge's Cavendish (which he directed after Maxwell) held for decades, and his patient comparison of nitrogen densities from air versus from chemical decomposition — a discrepancy in the third decimal place that most would have averaged away — uncovered argon and won the 1904 Nobel Prize. He published to his last year, 446 papers, much of it from a laboratory in his country house at Terling; the fields-from-intensity conversions of this article are the daily bread of disciplines he equipped.

History

1877

— Rayleigh's Theory of Sound systematizes rms amplitudes and mean energy flux for waves

1884

— Poynting's theorem gives the flux its vector; averages of S become the measured "intensity"

1893

— Steinmetz's phasor method embeds the ½Re(·) average into AC engineering practice

1901

— Marconi bridges the Atlantic on milliwatts per square kilometre — link budgets are born

1936

— Watson-Watt's radar range equation chains two inverse squares and antenna gain

1978

— first GPS satellite: navigation by phase-coherent signals at 10⁻¹³ W/m²

2019

— the Event Horizon Telescope images a black hole at ~10⁻²⁶ W·m⁻²·Hz⁻¹ — averaging pushed twenty decades past this article's examples

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §9.2.3 · Jackson, Classical Electrodynamics, 3rd ed., §6.9, §7.1 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §9.6 · Hecht, Optics, 5th ed., §3.3 (irradiance).

Provenance. Averaging results follow Griffiths §9.2.3 (Eqs. 9.61–9.64: $\langle u\rangle$ , $\langle S\rangle$ , $I$ , $\langle g\rangle$ ) with the complex form per Jackson §6.9 ( $\tfrac12\mathrm{Re}(\mathbf{E}\times\mathbf{H}^*)$ ); irradiance/intensity terminology per Hecht §3.3. A verifier should: (1) do the $\langle\cos^2\rangle$ integral both by half-angle and by the complex lemma; (2) confirm $E_0 = \sqrt{2Z_0I}$ with $2Z_0 = 753.5\ \Omega$ ; (3) recompute: solar constant → $E_0 = 1013$ V/m, $E_{\text{rms}} = 716$ V/m, $B_0 = 3.38\ \mu$ T, $\langle u\rangle = 4.54\ \mu$ J/m³, $\langle g\rangle = 1.51\times10^{-14}$ SI ✓ 377 Ω closure; 60 W at 2 m → 1.19 W/m², 30.0 V/m; GPS 500 W EIRP at 20,200 km → $9.8\times10^{-14}$ W/m², 8.6 μV/m. Addendum for this edition: the ⟨cos²⟩ = ½ theorem and its lemma-route cross-check SPEND the complex-notation article's (0584) time-average lemma by name; the 377 Ω reading and $S = uc$ transport identity are the E7 Poynting article's (0570), averaged; instantaneous books and the 2ω structure belong to 0585; $\langle g\rangle = I/c^2$ is handed to 0586's pressure catalogue. This edition owns the averaging act itself (scope and failure cases), the rms/peak/complex convention triad, and the conservation-law inverse square with gain.

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

Leads to: Momentum and Radiation Pressure · The Fresnel Equations · Electromagnetic Waves in Linear Dielectric Media · Jones Vectors and Stokes Parameters.

Why measure and quote averages at all — isn't the instantaneous field the real physics?

Both are real; the average is what couples to almost everything. Matter's thermal and biological responses (heating, damage, vision) integrate energy over times enormously longer than an optical cycle, so means govern them. Coherent detectors (antennas, homodyne receivers) do track the field's phase — but even they report power as a cycle mean. The instantaneous comb matters precisely where the response is fast or nonlinear: harmonic generation, tunnel ionization, damage by femtosecond pulses — 0585's peak-books regime. The division of labor is clean: peaks for nonlinear physics, means for energetics, and the ½ is the bridge toll between them.

Where exactly did the ½ come from — energy conservation, or just trigonometry?

Pure trigonometry plus the slowness of detectors: $\cos^2$ splits into ½ + (oscillation at 2ω), and slow instruments annihilate the oscillation. No energy went anywhere — integrate the comb over any whole wavelength and the total was always half the peak times the volume, before any detector arrived. The deeper uniqueness is that ½ is specific to sinusoids: it is the mean-square-to-peak ratio of the cosine. Other waveforms carry other ratios (square wave 1, triangle 1/3, Gaussian pulse: duty-dependent), which is why rms — defined directly as $\sqrt{\langle f^2\rangle}$ — is the honest general-purpose amplitude and the sinusoid's $E_0/\sqrt2$ is a special case.

Is intensity additive? If two flashlights shine on the same spot, do I get I₁ + I₂?

For incoherent sources — independent lamps, stars, LEDs — yes: their relative phase wanders randomly, the interference cross term averages to zero over the detector's integration time, and intensities add. For coherent, same-frequency sources — two outputs of one laser, a wave and its reflection — no: amplitudes add first, and $|\tilde E_1 + \tilde E_2|^2$ carries a cross term ranging the total between $(\sqrt{I_1}-\sqrt{I_2})^2$ and $(\sqrt{I_1}+\sqrt{I_2})^2$ point by point (fringes). The E7 interference ledger and the phasor addition of 0584 are the machinery; the practical rule is one question: "is the phase difference stable over my measurement?" Stable → add amplitudes. Wandering → add intensities.

Why does the impedance of free space, 377 Ω, keep appearing in intensity formulas?

Because $Z_0 = E/H$ for every vacuum plane wave, it plays exactly the role resistance plays in circuit power formulas — and the analogy is not decorative: $I = E_0^2/2Z_0$ is structurally $P = V_0^2/2R$ . The E7 Poynting article owns the interpretation (the vacuum's fixed exchange rate between the field pair); this article uses it as the fastest arithmetic path from intensity to fields: $E_0 = \sqrt{2Z_0I}$ , with $2Z_0 \approx 754$ . Engineers meet the same number matching antennas: an antenna is a transformer between a transmission line's tens-of-ohms and free space's 377, and reflections at that interface obey the same impedance-mismatch formulas as the coming reflection articles.

How can GPS work if its signal intensity is sixteen orders of magnitude below sunlight?

Because intensity is not the only resource — bandwidth and phase coherence are the others, and GPS spends them lavishly. The receiver knows the transmitted code exactly and correlates against it for milliseconds: signal contributions add coherently (as amplitudes) while noise adds incoherently (as power), buying a processing gain of ~43 dB that lifts the signal out of thermal noise. Sunlight, though $10^{16}$ times more intense, is incoherent and broadband — useless for correlation. The lesson generalizes across radio astronomy, radar, and deep-space links: this article's I sets the energy budget, but the plane wave's phase structure (0583–0584) is what determines how faint a signal can still be useful.

Do these formulas apply inside a medium like glass or water?

With controlled substitutions, yes — and the media article (0590) makes them. In a linear dielectric the plane wave keeps its structure with $c \to v = c/n$ and $\varepsilon_0 \to \varepsilon$ , and the intensity becomes $I = \tfrac12\varepsilon v E_0^2 = E_0^2/2Z$ with the medium's impedance $Z = \sqrt{\mu/\varepsilon}$ . The complex form $\tfrac12\mathrm{Re}(\tilde{\mathbf{E}}\times\tilde{\mathbf{H}}^*)$ needs no modification at all — which is precisely why Step 2 recommended memorizing that version: when absorption gives E and H a relative phase (0595–0596), it silently produces the correct reduced power flow, while the vacuum formulas quietly lie.

My laser pointer says 5 mW — is that a peak or an average, and does it matter?

For a continuous-wave (CW) pointer, the two coincide up to the ×2 of the cycle ripple (5 mW is the cycle-averaged output; the instantaneous flux peaks at 10 mW's worth — no instrument you own notices). For pulsed lasers the label is still average power, and there it matters enormously: a "1 W" femtosecond oscillator emitting 10 fs pulses at 100 MHz has a duty cycle of $10^{-6}$ — its peak power is a megawatt, and focusability turns that into intensities that ablate metal. Laser classifications and damage thresholds handle this with separate CW and pulsed criteria; the physics is just this article's duty-cycle row plus 0585's peak books. Read labels as averages; read danger as peaks.