Complex Notation for Electromagnetic Waves

The previous article built the lawful plane wave by hand, in real cosines — and the hand shows. Every derivative sprayed sine terms, every superposition threatened a page of trigonometric identities, and tracking the phase constant δ through a calculation meant carrying an extra symbol that mostly got in the way. Now multiply that friction by what is coming: adding reflected to incident waves at interfaces, waves with phase lags in conductors, polarization states built from components a quarter cycle apart. The chapter needs better machinery, and this article installs it. The machinery is a single move: write the field as the real part of a complex exponential, do all the algebra on the exponential, and take the real part at the end. Because $e^{i\varphi}$ turns differentiation into multiplication — $\nabla \to i\mathbf{k}$ , $\partial_t \to -i\omega$ — Maxwell's equations collapse from coupled partial differential equations into four lines of vector algebra. The triad that took a careful derivation last article falls out in three lines here. Amplitude and phase fuse into one complex number, so superposition becomes addition of arrows in the complex plane, and a phase lag becomes a complex factor instead of a trigonometric headache. The notation has exactly one danger zone, and this article owns it: the trick is licensed by linearity, and energies are quadratic. Squaring a complex field and taking the real part gives the wrong answer — sometimes catastrophically wrong. The repair is a small theorem, the time-average lemma, that converts products of complex fields into physical averages with one conjugate and one factor of ½. Every intensity formula in the rest of electromagnetism runs through that lemma, and it is proved, tested, and stress-tested here.

Key takeaways

Write $\tilde{\mathbf{E}} = \tilde{\mathbf{E}}_0\,e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ with complex amplitude $\tilde{\mathbf{E}}_0 = \mathbf{E}_0 e^{i\delta}$ ; the physical field is the real part. Amplitude and phase travel as one complex number.
On plane-wave exponentials, derivatives become algebra: $\nabla\cdot \to i\mathbf{k}\cdot$ , $\nabla\times \to i\mathbf{k}\times$ , $\partial_t \to -i\omega$ . Maxwell's equations become four algebraic constraints and deliver the transverse locked triad in three lines.
The license is linearity: adding, differentiating, and integrating commute with "take the real part". Superposition of waves = addition of complex arrows.
Products do NOT commute with taking real parts: $[\mathrm{Re}\,\tilde{f}\,]^2 \neq \mathrm{Re}(\tilde{f}^2)$ . Never square a complex field naively.
The repair for quadratics is the time-average lemma: $\langle fg\rangle = \tfrac12\mathrm{Re}(\tilde{f}\tilde{g}^*)$ for same-frequency fields — the engine behind every intensity formula ahead.
Key idea: complex exponentials diagonalize Maxwell's equations — do all linear algebra on $\tilde{\mathbf{E}}$ , and cross the quadratic bridge only via ½ Re(f̃ g̃*).

The core insight

Complex exponentials diagonalize Maxwell's equations — derivatives become $i\mathbf{k}$ and $-i\omega$ , so wave problems become vector algebra; quadratic quantities cross back to physics only through $\langle fg\rangle = \tfrac12\mathrm{Re}(\tilde f\tilde g^*)$ .

Where this lives today

Complex field notation is the native language of every wave-engineering stack on Earth. RF and photonics tools hand you S-parameters, impedances, and refractive indices as complex numbers because the notation of this article is baked into the instruments themselves — a vector network analyzer literally measures $\tilde g_0/\tilde f_0$ ratios, amplitude and phase together, at up to terahertz rates. Optical coherence theory, laser linewidth specs, and fiber dispersion budgets are all statements about complex amplitudes; the "I/Q data" streaming through every phone's radio (in-phase and quadrature samples) is the real and imaginary part of the local field's complex envelope, sampled — your handset performs the real-part projection of equation (1) billions of times per second. The deeper afterlife is conceptual. In classical electromagnetism the complex field is a shadow-caster — only $\mathrm{Re}$ is physical. Quantum mechanics promoted the complex amplitude to reality: the wavefunction's phase is the physics (interference of alternatives), not a notational convenience, and the classical lemma $\tfrac12\mathrm{Re}(\tilde f\tilde g^*)$ reappears as the structure of expectation values. The boundary between "useful fiction" and "the actual state" is one of the cleaner doorways from this chapter into quantum field theory — where the electromagnetic field itself becomes an operator-valued version of the mode expansion this article set up.

The physical picture

Picture the field at one point as a clock hand. A real cosine $E_0\cos(\omega t - \delta)$ is what you see if you watch only the horizontal shadow of a hand of length $E_0$ rotating clockwise at $\omega$ rad/s, having started at angle δ. The complex field $\tilde{E} = E_0 e^{i\delta}e^{-i\omega t}$ is the whole hand; the physical field is its shadow. Nothing about the shadow is lost — but the hand is a far better object to compute with, because rotation is multiplication: advancing time multiplies by $e^{-i\omega\Delta t}$ , a phase lag is a factor $e^{i\phi}$ , and the derivative of rotation is just $-i\omega$ times the hand itself. The real power appears when waves combine. Add two same-frequency cosines with different phases and the trigonometric identities arrive like weather. Add the two hands instead: arrows in the complex plane, tip to tail, one resultant arrow — whose length is the combined amplitude and whose angle is the combined phase, read off in one line. Interference, in this picture, is nothing but the geometry of arrow addition: aligned arrows reinforce, opposed arrows cancel, and the general case is the law of cosines. Every interferometer trace and every Fresnel coefficient in the coming articles is arrows being added.

Figure

The phasor picture — the complex field is a rotating arrow, the physical field its horizontal shadow, and superposition is arrow addition

Why is the shadow trick legitimate at all? Because everything Maxwell's equations do to fields is linear — differentiation, addition, integration — and linear operations commute with taking real parts: the shadow of the sum of two hands is the sum of the shadows, the shadow of a rotated hand's velocity is the velocity of the shadow. So you may carry the complex field through an entire linear calculation and project at the very end, once, and the answer is identical to grinding through cosines. The one place the license expires — products of fields, which is to say every energy, intensity, force, and pressure — is exactly where students get hurt, and the second half of this article builds the bridge for crossing it safely.

The machinery

The notational contract, stated once and used forever: physical fields are real; tilded fields are complex; and they are related by

\mathbf{E}(\mathbf{r},t) = \mathrm{Re}\left[\tilde{\mathbf{E}}(\mathbf{r},t)\right], \qquad \tilde{\mathbf{E}}(\mathbf{r},t) = \tilde{\mathbf{E}}_0\,e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}, \qquad \tilde{\mathbf{E}}_0 = \mathbf{E}_0\,e^{i\delta} \qquad (1)

where: $\tilde{\mathbf{E}}_0$ — the complex amplitude, a constant vector with complex components, packaging the real amplitude $E_0$ and the phase constant δ of the previous article into one symbol. Read it physically: last article's birth certificate listed $E_0$ and δ as separate freedoms; the complex amplitude fuses them, because in every calculation they enter together. Expanding $\mathrm{Re}[E_0e^{i\delta}e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}] = E_0\cos(\mathbf{k}\cdot\mathbf{r}-\omega t+\delta)$ recovers exactly the real wave of article 0583 — the two descriptions are the same wave in different scripts. (Convention note: physics writes $e^{-i\omega t}$ for time evolution; electrical engineering writes $e^{+j\omega t}$ . Same mathematics, conjugate conventions — check the sign before importing any formula across the border, or every phase lag becomes a lead.) The entire computational advantage is contained in what derivatives do to the exponential. Each spatial derivative brings down a component of $i\mathbf{k}$ ; each time derivative brings down $-i\omega$ . Assembled into the three vector operators, acting on any plane-wave field:

\nabla\cdot\tilde{\mathbf{E}} = i\mathbf{k}\cdot\tilde{\mathbf{E}}, \qquad \nabla\times\tilde{\mathbf{E}} = i\mathbf{k}\times\tilde{\mathbf{E}}, \qquad \frac{\partial\tilde{\mathbf{E}}}{\partial t} = -i\omega\,\tilde{\mathbf{E}} \qquad (2)

where: each rule replaces an analysis operation by an algebra operation. This is the precise sense in which plane waves "diagonalize" the derivatives — the exponential is an eigenfunction of every one of these operators, with eigenvalues built from $\mathbf{k}$ and $\omega$ . Note what the rules quietly assume: a single exponential of fixed $\mathbf{k}$ and ω. A superposition must be differentiated term by term, each with its own $i\mathbf{k}$ — the rules diagonalize modes, not arbitrary fields.

Derivation AMaxwell's equations as algebra — the locked triad in three lines

The goal is to reproduce the entire anatomy of the lawful plane wave — transversality, the constructed magnetic partner, the amplitude lock, the dispersion relation — using the substitution rules (2), and to display the contrast with the previous article's real-cosine derivation as the sales pitch it is. The strategy: insert $\tilde{\mathbf{E}} = \tilde{\mathbf{E}}_0e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ and $\tilde{\mathbf{B}} = \tilde{\mathbf{B}}_0e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ into the four vacuum Maxwell equations and cancel the common exponential everywhere. Watch how each equation becomes one line of vector algebra, and note that the results are not merely the same as before — they are the same with the phase information handled automatically by the complex amplitudes.

Step 1. The two divergences. $\nabla\cdot\tilde{\mathbf{E}} = 0$ and $\nabla\cdot\tilde{\mathbf{B}} = 0$ become, by (2),

\mathbf{k}\cdot\tilde{\mathbf{E}}_0 = 0, \qquad \mathbf{k}\cdot\tilde{\mathbf{B}}_0 = 0

— transversality of both fields, one line each. Why the complex version is stronger than it looks: $\tilde{\mathbf{E}}_0$ may have complex components (two transverse components with different phases — the elliptic polarization states of articles 0588–0589), and the constraint kills the longitudinal component of both the real and imaginary parts at once.

Step 2. Faraday's law builds the partner. $\nabla\times\tilde{\mathbf{E}} = -\partial_t\tilde{\mathbf{B}}$ becomes $i\mathbf{k}\times\tilde{\mathbf{E}}_0 = i\omega\tilde{\mathbf{B}}_0$ , hence

\tilde{\mathbf{B}}_0 = \frac{\mathbf{k}\times\tilde{\mathbf{E}}_0}{\omega} = \frac{1}{c}\,\hat{\mathbf{k}}\times\tilde{\mathbf{E}}_0.

Direction ( $\perp$ both), magnitude ( $B_0 = E_0/c$ ), and relative phase (the factor multiplying $\tilde{\mathbf{E}}_0$ is real, so $\tilde{\mathbf{B}}_0$ inherits δ unchanged — in phase) delivered in one line. Why the phase statement is now automatic: "in phase" is the statement that the complex ratio $\tilde{B}_0/\tilde{E}_0$ is real and positive. The notation carries phase relationships as the arguments of complex numbers, so they never need separate tracking — this is precisely the bookkeeping that cosines force you to do by hand.

Step 3. Ampère–Maxwell audits. $\nabla\times\tilde{\mathbf{B}} = \mu_0\varepsilon_0\,\partial_t\tilde{\mathbf{E}}$ becomes $i\mathbf{k}\times\tilde{\mathbf{B}}_0 = -i(\omega/c^2)\tilde{\mathbf{E}}_0$ . Substituting Step 2's $\tilde{\mathbf{B}}_0$ and using $\mathbf{k}\times(\mathbf{k}\times\tilde{\mathbf{E}}_0) = -k^2\tilde{\mathbf{E}}_0$ (transversality):

-\frac{k^2}{\omega}\tilde{\mathbf{E}}_0 = -\frac{\omega}{c^2}\tilde{\mathbf{E}}_0 \quad\Longrightarrow\quad \omega = ck.

The audit closes by re-deriving the dispersion relation, exactly as in the real-form derivation — same physics, a quarter of the ink. The comparison is the lesson: the cosine derivation of article 0583 needed a preparatory lemma about derivatives of $\cos\varphi$ , careful sign tracking through an integration, and a separate argument for the phase lock. Here, every one of those is absorbed into the algebra of $i$ . Verified: constraints identical to Griffiths §9.2.2 Eqs. 9.47–9.49 and to article 0583's Derivation A; the real-part projection of each line reproduces the corresponding real-form statement (checked by expanding $\mathrm{Re}$ explicitly for Step 2); dispersion consistent with WEQ 0582.

The danger zone: quadratic quantities

Everything so far ran on linearity, and linearity has a boundary. The chapter's next business — energy density, Poynting flux, intensity, radiation pressure — is built from products of fields, and products break the shadow trick. The failure is worth exhibiting brutally. Take $\tilde{E} = E_0e^{-i\omega t}$ (a field at one point, δ = 0) at the moment $\omega t = 45°$ . The physical field is $E = E_0\cos45° = 0.707\,E_0$ , so the true $E^2 = 0.500\,E_0^2$ . But the "complex square" $\mathrm{Re}(\tilde{E}^2) = E_0^2\cos90° = 0$ . Not close — zero, where the right answer is half the peak. Squaring first and projecting after double-counts the rotation: $\tilde{E}^2$ spins at $2\omega$ , and its real part is a different (and wrong) oscillation.

\left[\mathrm{Re}\,\tilde{f}\,\right]^2 \neq \mathrm{Re}\left(\tilde{f}^2\right) \qquad (3)

where: the inequality is generic — equality holds only at isolated instants. The rule of practice: before multiplying two fields, take real parts — or use the lemma below, which is almost always what you actually want, because every detector and every steady force in this chapter responds to the time average of the product rather than its instantaneous value (the intensity article, 0587, owns why).

Go deeper · Derivation B · 2 minThe time-average lemma — products of complex fields, done once and for all

Derivation BThe time-average lemma — products of complex fields, done once and for all

The goal is the workhorse identity $\langle f(t)\,g(t)\rangle = \tfrac12\mathrm{Re}(\tilde{f}\tilde{g}^*)$ for two same-frequency quantities $f = \mathrm{Re}(\tilde{f}_0e^{-i\omega t})$ , $g = \mathrm{Re}(\tilde{g}_0e^{-i\omega t})$ , where $\langle\cdot\rangle$ denotes the average over one period. This one identity powers every time-averaged energy, flux, and force formula in the rest of the track — the intensity article spends it immediately, the media and conductor articles spend it with complex amplitudes whose relative phases matter, which is where its conjugate earns its keep. The strategy: write each real part as half the sum of the quantity and its conjugate, multiply, and watch which terms survive averaging. Watch the split into a DC term and a $2\omega$ term — both matter: the DC term is the lemma, and the $2\omega$ term is physics too (the energy article's oscillating ledger rides on it).

Step 1. Write real parts symmetrically: $f = \tfrac12(\tilde{f}_0e^{-i\omega t} + \tilde{f}_0^*e^{+i\omega t})$ and likewise $g$ . Multiply:

fg = \tfrac14\left(\tilde{f}_0\tilde{g}_0\,e^{-2i\omega t} + \tilde{f}_0^*\tilde{g}_0^*\,e^{+2i\omega t} + \tilde{f}_0\tilde{g}_0^* + \tilde{f}_0^*\tilde{g}_0\right).

Step 2. Read the four terms. The first two rotate at $2\omega$ and are conjugates of each other: together they form $\tfrac12\mathrm{Re}(\tilde{f}_0\tilde{g}_0e^{-2i\omega t})$ — a pure oscillation at twice the wave frequency, averaging to zero over a period. The last two are time-independent conjugates summing to $\tfrac12\mathrm{Re}(\tilde{f}_0\tilde{g}_0^*)$ . Hence, exactly:

f(t)\,g(t) = \underbrace{\tfrac12\mathrm{Re}\left(\tilde{f}_0\,\tilde{g}_0^*\right)}_{\text{DC: the average}} + \underbrace{\tfrac12\mathrm{Re}\left(\tilde{f}_0\,\tilde{g}_0\,e^{-2i\omega t}\right)}_{2\omega\text{ ripple, averages to }0}

and averaging over a period leaves

\boxed{\ \langle fg\rangle = \tfrac12\,\mathrm{Re}\left(\tilde{f}\,\tilde{g}^*\right)\ }

(the tildes may carry the full exponentials: the $e^{-i\omega t}$ factors cancel between $\tilde f$ and $\tilde g^*$ , which is the whole point of the conjugate).

Step 3. Sanity checks that calibrate the lemma. Same quantity, $g = f$ , real amplitude: $\langle f^2\rangle = \tfrac12|\tilde{f}_0|^2 = \tfrac12 f_0^2$ — the familiar $\langle\cos^2\rangle = \tfrac12$ . Quadrature pair, $\tilde{g}_0 = i\tilde{f}_0$ (g lags f by 90°): $\langle fg\rangle = \tfrac12\mathrm{Re}(-i|\tilde f_0|^2) = 0$ — orthogonal-phase quantities average to nothing, the mathematical seed of "reactive power does no net work." Why the conjugate is the load-bearing part: without it, the product would retain a rotating factor and the relative phase between $\tilde f_0$ and $\tilde g_0$ would be double-counted; with it, exactly the phase difference survives, inside the Re. When later articles compute $\tfrac12\mathrm{Re}(\tilde{\mathbf{E}}\times\tilde{\mathbf{H}}^*)$ in a conductor and find it smaller than $\tfrac12 E_0H_0$ , the deficit is $\cos(\text{phase lag})$ — physics read directly off the lemma. *Verified: expansion algebra standard (Griffiths Eq. 9.30 footnote route; Zangwill §1.6.3); the ωt = 45° counterexample above is reproduced by the exact decomposition (DC term $\tfrac12E_0^2$ plus ripple $\tfrac12E_0^2\cos2\omega t$ , summing to $0.5E_0^2$ at 45°, while $\mathrm{Re}(\tilde f^2)$ contains only the ripple); quadrature and equal-field limits check.*

Figure

The quadratic bridge — a product of same-frequency oscillations is a DC level plus a 2ω ripple; the lemma extracts the level, the ripple is physics too

The exact decomposition in Step 2 — a DC part plus a $2\omega$ ripple — is worth keeping visible rather than averaging away immediately: the next article shows the plane wave's energy density really does pulse at $2\omega$ about its mean, and only the article after that (intensity) commits to the average. The lemma hands both terms over in one computation. One more piece of foreshadowing that the notation makes nearly free. Nothing stops the complex amplitude $\tilde{\mathbf{E}}_0$ from having components with different phases — say $\tilde{\mathbf{E}}_0 = E_0(\hat{\mathbf{x}} + i\hat{\mathbf{y}})/\sqrt2$ , whose real field rotates rather than oscillates. Such complex polarization vectors are perfectly lawful (Step 1 of the derivation above never assumed $\tilde{\mathbf{E}}_0$ real up to one overall phase), and they are the natural language for circular and elliptical polarization — articles 0588 and 0589 build the whole taxonomy on them. Likewise, nothing stops $k$ or the wave impedance from going complex when matter arrives — absorption and conductors (0595–0596) will be phase lags wearing complex constants. The notation installed here is the chapter's load-bearing infrastructure.

Limiting cases

Case	Result	Meaning
δ = 0, real $\tilde{\mathbf{E}}_0$	$\tilde{\mathbf{E}}$ reduces to the cosine wave of 0583	complex notation is a superset, not a new theory
linear operations	$\mathrm{Re}$ commutes: compute complex, project once at the end	the license, used silently everywhere
products	$\mathrm{Re}$ does not commute; use the lemma	the boundary of the license
$g = f$	$\langle f^2\rangle = \tfrac12\|\tilde f_0\|^2$	the ubiquitous factor ½ in every intensity formula
quadrature ( $\tilde g_0 = i\tilde f_0$ )	$\langle fg\rangle = 0$	90°-shifted quantities do no average work
$\tilde{\mathbf{E}}_0$ complex-direction	rotating field vectors	polarization states of 0588–0589, already lawful
$k$ or Z complex	decaying/phase-lagged waves	matter articles 0595–0596, same machinery

Common trapNever square the complex field — the shadow trick expires at products

The single most common error in wave calculations: computing an energy or intensity as $\mathrm{Re}(\tilde{E}^2)/\text{something}$ , or worse, $|\tilde E|^2$ presented as the instantaneous $E^2$ . Concretely: at $\omega t = 45°$ the true $E^2$ is $0.5E_0^2$ , while $\mathrm{Re}(\tilde E^2) = 0$ — the naive square is not approximately wrong, it is structurally wrong (it spins at 2ω and has lost the DC part entirely). Meanwhile $|\tilde E|^2 = E_0^2$ is time-independent — it is the peak squared, exactly double the average $\langle E^2\rangle = \tfrac12E_0^2$ . The discipline: linear steps in complex, then either (a) take real parts before any product, or (b) for time averages, use $\langle fg\rangle = \tfrac12\mathrm{Re}(\tilde f\tilde g^*)$ — with the conjugate, with the ½. A missing conjugate silently discards phase-lag physics (it costs you the $\cos\phi$ power factor in any medium with loss); a missing ½ inflates every intensity by 2. Both mistakes produce plausible-looking formulas that fail dimensional common sense only when checked against a known case — so check against $\langle\cos^2\rangle = \tfrac12$ every time.

Worked examples

1 · Two transmitters, one resultant (confidence builder)

Two co-located transmitters drive same-frequency, same-polarization plane waves toward a receiver, arriving with amplitudes $E_1 = 2.0$ V/m and $E_2 = 3.0$ V/m, the second leading the first by 60°. Find the resultant amplitude and phase, and the ratio of received power to the sum of individual powers.

Step 1. Set up the phasors: $\tilde{E}_1 = 2.0$ , $\tilde{E}_2 = 3.0\,e^{i60°}$ (choosing wave 1's phase as reference).

Step 2. Add arrows: $\tilde{E} = 2.0 + 3.0(\cos60° + i\sin60°) = 3.50 + 2.598\,i$ . Amplitude $|\tilde E| = \sqrt{3.50^2 + 2.598^2} = \sqrt{19.0} = \mathbf{4.36\ V/m}$ ; phase $\arctan(2.598/3.50) = \mathbf{36.6°}$ ahead of wave 1.

Step 3. Power bookkeeping via the lemma. Received power $\propto \langle E^2\rangle = \tfrac12|\tilde E|^2 = \tfrac12(19.0)$ ; individual powers $\propto \tfrac12(4.0) + \tfrac12(9.0) = \tfrac12(13.0)$ . Ratio $19.0/13.0 = \mathbf{1.46}$ — constructive interference has manufactured 46% "extra" power at this receiver. (No conservation crisis: other directions, where the arrival phases oppose, receive correspondingly less — the energy article's interference ledger, recalled by name, balances the books globally.)

Step 4. The cosine-free check: $|\tilde E|^2 = E_1^2 + E_2^2 + 2E_1E_2\cos60° = 4 + 9 + 6 = 19$ ✓ — the law of cosines, delivered by arrow geometry with no trigonometric identities invoked.

2 · The lemma under load — average energy and flux of a plane wave (exam level)

A plane wave $\tilde{\mathbf{E}} = 100\,e^{i(kz-\omega t)}\,\hat{\mathbf{x}}$ V/m propagates in vacuum. Compute $\langle E^2\rangle$ , the time-averaged energy density $\langle u\rangle$ , and the time-averaged Poynting magnitude $\langle S\rangle$ , using the lemma; then verify one of them by brute-force cosine integration.

Step 1. The partner field, by the algebraic lock: $\tilde{\mathbf{B}} = \tfrac{100}{c}e^{i(kz-\omega t)}\hat{\mathbf{y}}$ T, with $100/c = 3.34\times10^{-7}$ .

Step 2. Lemma applications (conjugate + half, every time): $\langle E^2\rangle = \tfrac12\mathrm{Re}(\tilde E\tilde E^*) = \tfrac12(100)^2 = 5.00\times10^3$ V²/m². Then, using the E7 chapter's density formula (recalled by name, owned there): $\langle u\rangle = \varepsilon_0\langle E^2\rangle = 8.854\times10^{-12}\times5.00\times10^3 = \mathbf{4.43\times10^{-8}\ J/m^3}$ (electric and magnetic halves equal — next article's business).

Step 3. Average flux: $\langle S\rangle = \tfrac{1}{\mu_0}\langle E B\rangle = \tfrac{1}{2\mu_0}\mathrm{Re}(\tilde E\tilde B^*) = \tfrac{100\times3.34\times10^{-7}}{2\times1.257\times10^{-6}} = \mathbf{13.3\ W/m^2}$ . Cross-check the identity $\langle S\rangle = c\langle u\rangle = 2.998\times10^8\times4.43\times10^{-8} = 13.3$ ✓ — the transport theorem of the E7 Poynting article, surviving the averaging intact.

Step 4. Brute-force audit of Step 2: $\langle E^2\rangle = E_0^2\langle\cos^2(kz-\omega t)\rangle = E_0^2\cdot\tfrac{1}{T}\int_0^T\cos^2\ \mathrm{d}t = E_0^2/2$ ✓. The lemma is the last time this integral ever needs doing — from here on, one conjugate and one ½ replace it. Answers: $\langle u\rangle = 4.43\times10^{-8}$ J/m³, $\langle S\rangle = 13.3$ W/m².

Origins · 1 minLeonhard Euler and how this was discovered

Leonhard Euler

1707–1783 · Basel, St. Petersburg, Berlin — whose identity turned oscillation into rotation

Euler never saw an electromagnetic wave — he died a century before Hertz — but every calculation in this article runs on his 1748 identity $e^{i\theta} = \cos\theta + i\sin\theta$ , published in the Introductio in analysin infinitorum. The identity's gift to physics is precisely the phasor trick: it embeds every oscillation as the shadow of a rotation, where calculus is trivial ( $e$ 's derivative is itself) and composition is multiplication. Euler produced roughly half of all mathematics of his century — 866 works, much of it dictated after he went almost fully blind in 1771, computed in a memory that famously held the first six powers of every integer up to a hundred. He also, fittingly for this chapter, spent decades on the wave theory of light, arguing against Newton's corpuscles in his Nova theoria lucis et colorum (1746) — wrong in details, right in spirit, and a century early. The engineering world completed his conquest when Steinmetz introduced complex phasors into AC circuit analysis in the 1890s; every impedance calculation since has been applied Euler.

History

1748

— Euler's Introductio publishes

e^{i\theta} = \cos\theta + i\sin\theta

: oscillation becomes the shadow of rotation

1807

— Fourier's heat memoir: arbitrary functions as sums of sinusoids, the superposition side of the machinery

1865

— Maxwell's waves arrive as real cosines; the Treatise computes without complex fields

1893

— Steinmetz's phasor method makes complex amplitudes the working language of AC engineering

1928

— Dirac's quantum electrodynamics elevates the complex field from bookkeeping to substance:

\psi

is not a shadow

1966

— Yee's FDTD algorithm returns to real fields on a grid — and modern solvers still validate against complex-notation plane-wave benchmarks

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §9.1.2, §9.2.1 · Jackson, Classical Electrodynamics, 3rd ed., §7.1 · Zangwill, Modern Electrodynamics, §1.6 (complex fields and time averages) · Hecht, Optics, 5th ed., §2.5.

Provenance. Standard complex-representation material following Griffiths §9.1.2 and Jackson §7.1; the time-average theorem is Griffiths's footnoted identity (his Eq. 9.30 context) and Zangwill §1.6.3. A verifier should: (1) confirm the substitution rules by differentiating $e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ directly; (2) re-derive the four algebraic Maxwell constraints and check them against the real-form results of the previous article; (3) prove $\langle fg\rangle = \tfrac12\mathrm{Re}(\tilde f\tilde g^*)$ by the conjugate-decomposition route and check the special cases $g=f$ and quadrature; (4) recompute the worked numbers. Addendum for this edition: the algebraic re-derivation of the triad (Derivation A) deliberately mirrors 0583's real-form Derivation A as its three-line contrast — division of labor: 0583 owns the real-form anatomy and parameter count, this article owns the machinery and the quadratic bridge; the lemma is owned HERE and spent by Intensity and Time-Averaged Quantities (0587) — noted for the ledger. Worked numbers recomputed: phasor sum $2 + 3e^{i60°}$ : amplitude $\sqrt{19} = 4.36$ , phase 36.59°, power ratio 19/13 = 1.46; plane wave 100 V/m: $\langle E^2\rangle = 5.00\times10^3$ V²/m², $\langle u\rangle = 4.43\times10^{-8}$ J/m³, $\langle S\rangle = 13.3$ W/m² $= c\langle u\rangle$ ✓; counterexample at $\omega t = 45°$ : $[\mathrm{Re}\tilde E]^2 = 0.5E_0^2$ , $\mathrm{Re}(\tilde E^2) = 0$ .

Builds on: Monochromatic Plane Waves in Vacuum · The Electromagnetic Wave Equation · Complex numbers and Euler's formula.

Leads to: Energy Density and the Poynting Vector · Intensity and Time-Averaged Quantities · Polarization of Electromagnetic Waves · Jones Vectors and Stokes Parameters · Absorption and Dispersion.

Is the complex field physically real, or just a calculational trick?

In classical electromagnetism, strictly a trick: the measurable field is $\mathrm{Re}\,\tilde{\mathbf{E}}$ , and the imaginary part is redundant scaffolding (it is the same oscillation shifted a quarter cycle — no new information). The trick is canonical because linear operations commute with Re, so the scaffolding never contaminates an answer. But hold the question — it has a better answer one theory over. In quantum mechanics the complex phase is physical (interference of amplitudes depends on it, and you cannot discard an imaginary part without changing predictions). Classical complex notation is, historically and pedagogically, the training ground for that promotion.

Why does the time-average formula need the complex conjugate?

Because the conjugate cancels the rotation and preserves the relative phase — the two things an average should do. $\tilde f\tilde g$ rotates at $2\omega$ (its average is genuinely zero — that term is the ripple, not the mean), while $\tilde f\tilde g^*$ is time-independent, and its argument is exactly the phase difference $\delta_f - \delta_g$ . The Re then converts that phase difference into the $\cos\Delta\delta$ factor you'd get from integrating cosines by hand. Physically: only the component of $g$ oscillating in phase with $f$ contributes to their average product; the conjugate performs that projection automatically.

When exactly may I use the ∇ → ik replacement — does it work for any field?

Only on a single plane-wave mode $e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ — the replacement is the statement that this exponential is an eigenfunction of the derivative operators. For a superposition, apply it term by term, each mode with its own $\mathbf{k}$ and ω; for a general field, that term-by-term application is the Fourier transform, and the replacement rules become the statement "differentiation in real space is multiplication in Fourier space." What you may never do is apply $i\mathbf{k}$ with one fixed $\mathbf{k}$ to a field that is not a single mode — a Gaussian beam, a pulse, a standing wave (two modes!) — that error assigns one plane wave's geometry to an object that has several.

What happens if I accidentally use e^{+iωt} instead of e^{−iωt}?

Internally, nothing — externally, everything. The choice is a convention; physics is invariant if you are consistent, because the physical field is the real part either way and $\mathrm{Re}(\tilde z) = \mathrm{Re}(\tilde z^*)$ . The damage comes from mixing: import an engineering formula (which uses $e^{+j\omega t}$ ) into a physics calculation and every $i$ flips sign — impedances conjugate, phase lags become phase leads, decaying waves grow. The tell-tale symptom is an absorption coefficient or an imaginary part of a refractive index with the wrong sign (a medium that amplifies instead of absorbs). When the conductor and dispersion articles introduce complex $n$ and $Z$ , their sign conventions trace directly back to $e^{-i\omega t}$ chosen here.

Why is |Ẽ| not the field's instantaneous magnitude?

$|\tilde E|$ strips the oscillation entirely: $|\tilde E_0e^{i\varphi}| = |\tilde E_0| = E_0$ , a constant — it is the envelope, the peak the oscillation touches twice per cycle, not the value now. The instantaneous magnitude is $|E_0\cos\varphi|$ , which spends most of its time below the envelope. The distinction earns money in two places: mean-square quantities ( $\langle E^2\rangle = \tfrac12E_0^2$ , not $E_0^2$ — the ½ is the whole difference), and modulated signals, where a slowly varying $|\tilde E_0(t)|$ is the transmitted information (AM radio, optical communications) riding on the fast carrier — the "complex envelope" of signal processing is precisely this object.

Do I lose anything by working with the complex field instead of real cosines?

Nothing, provided the endpoints are handled by contract. The map cosine-form ↔ complex-form is a bijection ( $E_0$ , δ ↔ $\tilde E_0$ ), all linear physics transfers exactly, and the quadratic bridge (the lemma) is exact, not approximate — including the 2ω ripple term if you keep it. What you can lose is physical alertness: the complex field makes some illegal objects easy to write (a "wave" with $\mathbf{k}\cdot\tilde{\mathbf{E}}_0 \neq 0$ , an energy computed from $\tilde E^2$ ), and the formalism will not stop you — it is notation, not law. The completion-routine discipline from the wave-equation article applies with the same force in either script.

How do complex polarization vectors manage to describe a rotating field with no rotation written anywhere?

Take $\tilde{\mathbf{E}}_0 = E_0(\hat{\mathbf{x}} + i\hat{\mathbf{y}})/\sqrt2$ and project: $\mathbf{E}(t) = \tfrac{E_0}{\sqrt2}[\hat{\mathbf{x}}\cos\omega t + \hat{\mathbf{y}}\sin\omega t]$ (at $\mathbf{r} = 0$ ) — the tip traces a circle. The $i$ is the rotation: multiplying by $i$ shifts phase by 90°, so the $\hat{\mathbf{y}}$ -component oscillates a quarter cycle behind the $\hat{\mathbf{x}}$ -component, and two orthogonal oscillations in quadrature compose into circular motion (Lissajous with equal amplitudes, 90° offset). One complex vector thus encodes amplitude, phase, and the shape of the polarization ellipse. The polarization articles (0588–0589) promote this observation into a two-complex-component formalism — the Jones vector — and the quantum reader will recognize a qubit.