Monochromatic Plane Waves in Vacuum

The previous article ended with a promise and a warning. The promise: Maxwell's equations support waves that carry themselves through empty space at $c$ , and the monochromatic family $e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ spans every solution by Fourier's theorem. The warning: the wave equation alone is a forger's charter — it admits fields Maxwell forbids, and only the full first-order system decides what a lawful electromagnetic wave looks like. This article runs that decision to completion for the single most important solution in classical field theory: one frequency, one direction, flat wavefronts filling all space. What comes out is astonishingly rigid. You get to choose the frequency, the propagation direction, the amplitude, a phase, and one angle of polarization — and then Maxwell's equations confiscate every other freedom. Both fields must lie in the wavefront plane (no longitudinal component survives Gauss's law); the magnetic field's direction, magnitude, and phase are all outputs, locked to the electric field by Faraday's law; and $\mathbf{E}$ , $\mathbf{B}$ , $\hat{\mathbf{k}}$ form a right-handed triad with $E = cB$ , peaking together, everywhere, forever. Mastering this one object pays for the rest of the chapter. Energy, momentum, intensity, and polarization (the next five articles) are all bookkeeping performed on the plane wave; media, interfaces, and waveguides (the articles after) are the plane wave meeting matter. And because the monochromatic family is a complete basis, nothing is lost by the idealization: every laser pulse, radar chirp, and starlight photon wavepacket is a superposition of exactly the object built here.

Key takeaways

A monochromatic plane wave oscillates at a single angular frequency $\omega$ and depends on position only through $\mathbf{k}\cdot\mathbf{r}$ : the fields are uniform over every plane perpendicular to the wavevector $\mathbf{k}$ , and the whole pattern slides along $\hat{\mathbf{k}}$ at $c = \omega/k$ .
Gauss's laws force transversality: $\mathbf{k}\cdot\mathbf{E}_0 = 0$ and $\mathbf{k}\cdot\mathbf{B}_0 = 0$ . A longitudinal electromagnetic wave in vacuum is illegal, not merely unusual.
Faraday's law constructs the magnetic field: $\mathbf{B} = \tfrac{1}{c}\,\hat{\mathbf{k}}\times\mathbf{E}$ — direction, magnitude ( $B_0 = E_0/c$ ), and phase (in step with $\mathbf{E}$ ) are all outputs, not choices.
The free parameters are exactly: frequency, propagation direction (two angles), amplitude, overall phase, and one polarization angle. Everything else is locked.
The pure plane wave is an idealization — infinite extent, infinite energy, perfectly sharp frequency — but it is the complete Fourier scaffolding out of which every real, finite field is built.
Key idea: Maxwell's equations promote the generic wave-equation solution to something far more constrained — $\mathbf{E}$ , $\mathbf{B}$ , $\mathbf{k}$ mutually orthogonal, in phase, and tied by $\mathbf{B} = \tfrac{1}{c}\,\hat{\mathbf{k}}\times\mathbf{E}$ .

The core insight

Maxwell's equations promote the generic wave-equation solution to something far more constrained: $\mathbf{E}$ , $\mathbf{B}$ , $\mathbf{k}$ are mutually orthogonal, in phase, and tied by $\mathbf{B} = \tfrac{1}{c}\,\hat{\mathbf{k}}\times\mathbf{E}$ .

Where this lives today

The plane wave is the working coordinate system of the electromagnetic industries. Antenna engineers characterize every device by its far-field plane-wave response and synthesize arbitrary beams as sums over $\hat{\mathbf{k}}$ — a 5G massive-MIMO panel is a machine for sculpting which plane-wave superposition leaves the tower, steered electronically hundreds of times a second. Optical engineers run the same decomposition at $10^{14}$ Hz: the angular-spectrum method propagates laser fields through lens systems as integrals over plane waves, each component transported by exactly this article's rigid anatomy. And metrology has pushed the monochromatic idealization astonishingly close to reality: the best cavity-stabilized lasers hold linewidths below 10 mHz at optical frequencies — a fractional frequency purity of $10^{-17}$ , a wave that stays in phase with itself for over a hundred million kilometres of propagation. The anatomy itself remains a live test target. The two transverse polarizations per $(\omega, \hat{\mathbf{k}})$ become, upon quantization, the photon's two helicity states — the observed absence of a third (longitudinal) mode is Gauss's law counted in particle language, and it is equivalent to the photon's masslessness. Current bounds put the photon mass below $10^{-54}$ kg: to that precision, transversality — Step 2 of this article's first derivation — is exact.

The physical picture

Stand somewhere in empty space and let a monochromatic plane wave from a distant transmitter wash over you. What does the field actually do? At your location, $\mathbf{E}$ points in one fixed transverse direction and oscillates sinusoidally — up to full strength, down through zero, to full strength the other way — $5\times10^{14}$ times per second if the wave is yellow light. The magnetic field does exactly the same thing at right angles to $\mathbf{E}$ , in perfect step: both peak at the same instant, both vanish at the same instant. And your neighbor one wavelength ahead of you along $\hat{\mathbf{k}}$ sees precisely the same show, delayed by one period. Now zoom out. At any frozen instant the field forms a comb of flat sheets — the wavefronts — stacked perpendicular to $\hat{\mathbf{k}}$ with spacing $\lambda$ , each sheet a plane on which the field is the same everywhere. Yellow light packs about 1,700 of these sheets into a millimetre. Unfreeze time and the entire comb glides rigidly along $\hat{\mathbf{k}}$ at $c$ , no sheet changing its value as it moves — that is what "the phase propagates" means, and the d'Alembert rigidity of the previous article guarantees the shape is preserved exactly.

Figure

The monochromatic plane wave at one instant — flat wavefronts stacked along k, and the locked right-handed triad riding on them

The deepest feature of the picture is what you do not get to draw. You cannot tilt $\mathbf{E}$ out of the wavefront plane — Gauss's law forbids it. You cannot choose $\mathbf{B}$ 's direction, or make it lag $\mathbf{E}$ by a quarter cycle the way current lags voltage in an inductor — Faraday's law has already decided both. The one genuine freedom left in the transverse plane, the direction of $\mathbf{E}$ within its sheet, has a name — polarization — and enough physics in it to fill two later articles. Everything else about the wave's internal anatomy is fixed by the field equations, which is why every plane wave in the universe, from a 50 Hz power-line hum with a 6,000 km wavelength to a gamma ray with a wavelength smaller than a nucleus, has exactly the same internal architecture. One honesty note before the mathematics. A true monochromatic plane wave fills all space and all time, carries infinite energy, and has never existed. Its status is that of the perfect basis element: real fields are finite superpositions of these modes, and because Maxwell's equations are linear, every theorem proved here transfers to real fields term by term. The previous article proved the completeness; this article builds the element.

The anatomy of the wave

Every statement about the wave hangs on a single scalar function, the phase. Before writing any field, it pays to fix the dictionary that converts between the five equivalent ways of quoting "how fast it wiggles" — this dictionary is used silently in every article and every lab notebook for the rest of the subject. The wave itself is then a fixed amplitude vector times a cosine of the phase:

\mathbf{E}(\mathbf{r},t) = \mathbf{E}_0\cos(\mathbf{k}\cdot\mathbf{r} - \omega t + \delta) \qquad (1)

where: $\mathbf{E}_0$ — constant amplitude vector, whose direction is the polarization; $\mathbf{k}$ — the wavevector, pointing along propagation, with magnitude $k = 2\pi/\lambda$ (the wavenumber); $\omega = 2\pi f$ — angular frequency; $\delta$ — phase constant fixing where in its cycle the wave sits at the origin at $t=0$ . Read it physically: the argument $\varphi = \mathbf{k}\cdot\mathbf{r} - \omega t + \delta$ is a number that counts cycles — advance $\mathbf{r}$ by one wavelength along $\hat{\mathbf{k}}$ and $\varphi$ grows by $2\pi$ ; wait one period $T = 1/f$ and it drops by $2\pi$ ; the field cares only about $\varphi$ , never about $\mathbf{r}$ or $t$ separately. The geometry of $\varphi$ is the geometry of the wave. A surface of constant phase satisfies $\mathbf{k}\cdot\mathbf{r} = \text{const}$ at each instant — the equation of a plane perpendicular to $\mathbf{k}$ (that is the "plane" in plane wave). As $t$ advances, keeping $\varphi$ fixed requires $\mathbf{k}\cdot\mathbf{r}$ to grow at rate $\omega$ , so each phase plane advances along $\hat{\mathbf{k}}$ at the phase velocity

v_{\text{phase}} = \frac{\omega}{k} = c \qquad (2)

where: the last equality is vacuum's dispersion relation $\omega = ck$ , inherited from the previous article's trial-exponential computation. The equality of $\omega/k$ with the same $c$ for every frequency is precisely what "vacuum is non-dispersive" means — the property that lets pulses hold their shape, and the property that dies first when matter enters (article 0590 and, fully, 0595 own that story). For orientation across the trade's units: quoting $\lambda$ , $f$ , $\omega$ , $k$ , or the photon energy are five dialects for one number. Yellow sodium light has $\lambda = 589$ nm, $f = 5.09\times10^{14}$ Hz, $\omega = 3.20\times10^{15}$ rad/s, $k = 1.07\times10^7$ rad/m; a 10 GHz radar has $\lambda = 3.0$ cm; the CMB's peak sits near 160 GHz, $\lambda \approx 1.9$ mm. The electromagnetic spectrum is this single dictionary run across twenty decades:

Figure

One structure, twenty decades — the electromagnetic spectrum as a single plane-wave dictionary from power hum to gamma rays

Equation (1) is, so far, only a candidate — a solution of the wave equation, which the previous article warned is a necessary condition and nothing more. The real content of this article is what happens when the candidate faces all four of Maxwell's equations. That confrontation is the first derivation, and it is the monochromatic, arbitrary-direction instantiation of the completion routine the wave-equation article established for $\hat{\mathbf{z}}$ -directed profiles: divergences first to kill impostors, one curl to build the partner, the other curl as audit.

Derivation ATransversality and the locked triad — Maxwell tightens the wave equation

The goal is the full anatomy of the lawful monochromatic plane wave: transversality of both fields, the construction of $\mathbf{B}$ from $\mathbf{E}$ , the amplitude lock $B_0 = E_0/c$ , and the in-phase right-handed triad — for a wave running in an arbitrary direction $\hat{\mathbf{k}}$ , not just down a coordinate axis. The strategy is the completion routine of the previous article, executed on the real cosine field (1): impose the two divergence equations, then use Faraday's law to construct the magnetic partner, then let Ampère–Maxwell audit the result. Everything reduces to one repeated observation — every derivative of $\cos\varphi$ acts through the phase — so watch how each vector operation converts into an algebraic operation with $\mathbf{k}$ , a conversion the next article will industrialize.

Step 1. Derivatives act through the phase. With $\varphi = \mathbf{k}\cdot\mathbf{r} - \omega t + \delta$ , each spatial derivative of $\cos\varphi$ pulls out the corresponding component of $\mathbf{k}$ : $\partial_x\cos\varphi = -k_x\sin\varphi$ , and likewise for $y$ , $z$ , while $\partial_t\cos\varphi = +\omega\sin\varphi$ . Assembled into vector operators, for any constant vector $\mathbf{a}$ :

\nabla\cdot\left[\mathbf{a}\cos\varphi\right] = -(\mathbf{k}\cdot\mathbf{a})\sin\varphi, \qquad \nabla\times\left[\mathbf{a}\cos\varphi\right] = -(\mathbf{k}\times\mathbf{a})\sin\varphi.

Why this is the whole calculation: the fields' only space-time dependence is through the scalar $\varphi$ , so the machinery of vector calculus collapses to dot and cross products with $\mathbf{k}$ . This is the structural reason plane waves are the preferred basis — they diagonalize the derivatives.

Step 2. Gauss's law kills the longitudinal component. Applying the divergence formula to (1), $\nabla\cdot\mathbf{E} = 0$ requires $(\mathbf{k}\cdot\mathbf{E}_0)\sin\varphi = 0$ at every point and time, hence

\mathbf{k}\cdot\mathbf{E}_0 = 0.

The amplitude vector must lie in the wavefront plane. Why this matters beyond bookkeeping: the wave equation happily transports a longitudinal $\mathbf{E}\parallel\hat{\mathbf{k}}$ — it is Gauss's law, and only Gauss's law, that outlaws it. Sound waves (longitudinal, no divergence constraint) and quantum wavefunctions (scalar) train the wrong reflex here; electromagnetism is transverse by law.

Step 3. Faraday's law constructs the partner. $\nabla\times\mathbf{E} = -\partial_t\mathbf{B}$ becomes $-(\mathbf{k}\times\mathbf{E}_0)\sin\varphi = -\partial_t\mathbf{B}$ . Integrating in time (and discarding the static integration constant, which is a superposed static field, not part of the wave):

\mathbf{B}(\mathbf{r},t) = \frac{\mathbf{k}\times\mathbf{E}_0}{\omega}\,\cos\varphi = \frac{1}{c}\,\hat{\mathbf{k}}\times\mathbf{E},

using $\omega = ck$ in the last step. Three facts arrive at once. Direction: $\mathbf{B}$ is perpendicular to both $\hat{\mathbf{k}}$ and $\mathbf{E}$ — automatically transverse (Gauss-for-B is satisfied without being invoked: $\mathbf{k}\cdot(\mathbf{k}\times\mathbf{E}_0) = 0$ identically). Magnitude: $B_0 = kE_0/\omega = E_0/c$ . Phase: $\mathbf{B}$ carries the same $\cos\varphi$ — the fields oscillate in step, peaking and vanishing together. None of these was assumed.

Step 4. Ampère–Maxwell audits. The remaining equation, $\nabla\times\mathbf{B} = \mu_0\varepsilon_0\,\partial_t\mathbf{E}$ , must hold with everything already fixed. The left side is $-(\mathbf{k}\times\mathbf{B}_0)\sin\varphi$ with $\mathbf{B}_0 = (\mathbf{k}\times\mathbf{E}_0)/\omega$ ; the double cross product with transversality gives $\mathbf{k}\times(\mathbf{k}\times\mathbf{E}_0) = -k^2\mathbf{E}_0$ . The right side is $(\omega/c^2)\mathbf{E}_0\sin\varphi$ . Equality holds if and only if $k^2/\omega = \omega/c^2$ , i.e.

\omega = ck

— the audit does not impose a new constraint on the wave's shape; it re-derives the dispersion relation, exactly the closing-consistency signature the wave-equation article's routine promised. The system is complete: the lawful wave is (1) together with $\mathbf{B} = \tfrac1c\hat{\mathbf{k}}\times\mathbf{E}$ , and the triad $\{\hat{\mathbf{E}}, \hat{\mathbf{B}}, \hat{\mathbf{k}}\}$ is right-handed and orthonormal, with $\mathbf{E}\times\mathbf{B}$ pointing along propagation — the geometric seed of the energy-flow story two articles ahead. Verified: reproduces Griffiths §9.2.2 Eqs. 9.46–9.49 (with the general- $\hat{\mathbf{k}}$ form 9.49) and Jackson §7.1; the complex-exponential route of the next article gives identical constraints in three lines; the count of surviving freedoms (ω, two angles of $\hat{\mathbf{k}}$ , $E_0$ , δ, one polarization angle) matches the mode-counting of Derivation B.

The derivation leaves a small list of genuine choices, and it is worth making that list explicit because the rest of the chapter is organized around it. This is the plane wave's complete birth certificate:

\left\{\ \omega,\ \hat{\mathbf{k}}\ (\text{two angles}),\ E_0,\ \delta,\ \hat{\mathbf{n}}\ (\text{one angle in the wavefront plane})\ \right\} \qquad (3)

where: $\hat{\mathbf{n}} = \hat{\mathbf{E}}_0$ is the polarization vector, the one internal freedom Maxwell leaves open. Six numbers, and every one of them is put to work later: $\omega$ drives dispersion in media, $\hat{\mathbf{k}}$ bends at interfaces, $E_0$ and $\delta$ become the complex amplitude of the next article, and $\hat{\mathbf{n}}$ — allowed to go complex — generates the entire polarization zoo of articles 0588–0589. A second polarization direction orthogonal to the first exists for every $(\omega, \hat{\mathbf{k}})$ , so each propagation mode is doubly degenerate: the two transverse polarizations, the classical ancestor of the photon's two helicity states.

Go deeper · Derivation B · 2 minThe geometry of phase — why wavefronts are planes and what the wavevector really is

Derivation BThe geometry of phase — why wavefronts are planes and what the wavevector really is

The goal is to earn the geometric vocabulary used loosely so far: to show that the surfaces of constant phase are planes perpendicular to $\mathbf{k}$ , that they march at $\omega/k$ , and that $\mathbf{k}$ is best understood as the spatial frequency vector — the gradient of phase — a reading that survives intact when wavefronts stop being flat (spherical waves, lenses, general optics) and when quantum mechanics reinterprets $\hbar\mathbf{k}$ as momentum. The strategy is elementary vector geometry on the phase function $\varphi(\mathbf{r},t)$ ; the payoff is a definition of $\mathbf{k}$ that does not mention planes at all.

Step 1. Constant phase means constant $\mathbf{k}\cdot\mathbf{r}$ . At fixed time $t$ , the set $\varphi = \varphi_0$ is $\{\mathbf{r} : \mathbf{k}\cdot\mathbf{r} = \varphi_0 + \omega t - \delta\}$ . For a fixed right-hand side this is the standard equation of a plane with unit normal $\hat{\mathbf{k}}$ , at perpendicular distance $(\varphi_0 + \omega t - \delta)/k$ from the origin. Successive crests ( $\varphi_0$ differing by $2\pi$ ) are parallel planes separated by $2\pi/k = \lambda$ .

Step 2. The march. Differentiate the plane's distance from the origin with respect to $t$ at fixed $\varphi_0$ : it grows at $\omega/k$ . Every phase plane — crest, node, trough — advances along $\hat{\mathbf{k}}$ at the same speed $c$ , which is why the pattern translates rigidly rather than churning.

Step 3. $\mathbf{k}$ as the phase gradient. Compute $\nabla\varphi = \mathbf{k}$ and $-\partial_t\varphi = \omega$ . This pair of statements is the portable definition: the wavevector is the spatial rate of phase accumulation, the angular frequency is the temporal rate, and for any wave — plane or not — the local wavevector is $\nabla\varphi$ and the local wavefronts are $\varphi$ 's level surfaces, marching along $\nabla\varphi$ at $\omega/|\nabla\varphi|$ . Why this framing is the one to keep: a spherical wave from a small antenna has $\varphi = kr - \omega t$ , hence local wavevector $k\hat{\mathbf{r}}$ — radially outward, magnitude unchanged — and at distances where the sphere's curvature is negligible across your detector, the wave is locally indistinguishable from a plane wave. That single observation is the license for using this article's rigid anatomy on real fields: every far-field, every collimated beam, every starlight arrival is a local plane wave.

Step 4. How local is "locally plane"? A beam of waist $w_0$ stays effectively collimated over the Rayleigh range $z_R = \pi w_0^2/\lambda$ — for a 1 mm-waist helium–neon beam ( $\lambda = 633$ nm), $z_R \approx 5.0$ m: metres of laboratory over which the plane-wave idealization holds to high accuracy, purchased with a millimetre of transverse extent. The idealization fails gracefully and predictably, through diffraction — the optics track's opening subject. Verified: plane geometry standard; $\nabla\varphi = \mathbf{k}$ , $\partial_t\varphi = -\omega$ by direct differentiation of $\varphi = \mathbf{k}\cdot\mathbf{r} - \omega t + \delta$ ; Rayleigh range $\pi w_0^2/\lambda = \pi(10^{-3})^2/6.33\times10^{-7} = 4.96$ m; spherical-wave $1/r$ amplitude and local-plane-wave limit consistent with the wave-equation article's spherical solution.

Limiting cases

Case	Result	Meaning
$\hat{\mathbf{k}} = \hat{\mathbf{z}}$ , $\delta = 0$	$\mathbf{E} = E_0\cos(kz-\omega t)\hat{\mathbf{x}}$ , $\mathbf{B} = \tfrac{E_0}{c}\cos(kz-\omega t)\hat{\mathbf{y}}$	the textbook special case; all generality lives in rotating this picture
$\omega \to 0$	wavelength and period diverge; fields locally uniform	statics recovered as the zero-frequency limit of the dictionary
superposition over $\omega$	wave packet, moving rigidly at $c$ in vacuum	Fourier synthesis; the plane wave as complete basis (previous article's theorem)
superposition over $\hat{\mathbf{k}}$	standing waves, beams, diffraction patterns	same anatomy per mode; geometry from interference
vacuum → linear medium	$c \to v = c/n$ , $B_0 = E_0/v$ , triad intact	article 0590; structure survives, speed and lock rescale
distant spherical wave	locally plane: $\mathbf{k}_{\text{loc}} = k\hat{\mathbf{r}}$ , amplitude $\propto 1/r$	the license for applying this article to real sources
add the second polarization	two independent transverse modes per $(\omega,\hat{\mathbf{k}})$	the degeneracy that polarization articles 0588–0589 organize

Common trapE and B peak together — the inductor intuition that they should be a quarter cycle apart is a transplant error

Circuit training installs a reflex: energy sloshes between electric and magnetic forms, so when one is maximal the other should vanish — capacitor full when inductor empty, $90°$ apart. Standing waves in cavities do behave that way. But in a traveling plane wave, Faraday's law locks $\mathbf{B} = \tfrac1c\hat{\mathbf{k}}\times\mathbf{E}$ with the same cosine: the fields are in phase, peaking together and vanishing together, and nothing sloshes between them — the pair rides forward as one object, with equal energy in each field at every single instant (the energy article makes this quantitative). The quarter-cycle intuition belongs to resonators, where two counter-propagating waves superpose and the in-phase locks of the two directions interfere into a $90°$ offset. Diagnose before you assume: traveling wave → in phase; standing wave → in quadrature. Getting this wrong flips predictions for field maxima in cavities, antenna near-fields, and every "where is the energy at this instant" question.

Worked examples

1 · The sodium street lamp, in all five dialects (confidence builder)

A low-pressure sodium lamp radiates at $\lambda = 589.0$ nm. At a point in the far field the electric amplitude is $E_0 = 100$ V/m. Express the wave in every dialect of the dictionary and construct the full field.

Step 1. The dictionary. $k = 2\pi/\lambda = 2\pi/(5.890\times10^{-7}) = 1.067\times10^7$ rad/m; $f = c/\lambda = 2.998\times10^8/5.890\times10^{-7} = 5.09\times10^{14}$ Hz; $\omega = 2\pi f = 3.20\times10^{15}$ rad/s; $T = 1/f = 1.96$ fs. Five numbers, one fact.

Step 2. The magnetic partner. $B_0 = E_0/c = 100/2.998\times10^8 = \mathbf{3.34\times10^{-7}\ T}$ — a third of a microtesla, over a hundred times weaker than Earth's field, yet locked in a perfectly rigid ratio to $E_0$ for every plane wave ever measured.

Step 3. The full field, taking $\hat{\mathbf{k}} = \hat{\mathbf{z}}$ and polarization $\hat{\mathbf{x}}$ : $\mathbf{E} = 100\cos(1.067\times10^7 z - 3.20\times10^{15}t)\,\hat{\mathbf{x}}$ V/m, $\mathbf{B} = 3.34\times10^{-7}\cos(\text{same phase})\,\hat{\mathbf{y}}$ T. Check the triad: $\hat{\mathbf{x}}\times\hat{\mathbf{y}} = \hat{\mathbf{z}}$ ✓ — energy will flow along $+\hat{\mathbf{z}}$ , as the Poynting machinery will confirm.

Step 4. Scale reading. In one millimetre of beam there are $10^{-3}/\lambda = \mathbf{1{,}700}$ wavefront sheets; in one nanosecond the comb slides $30$ cm. The enormous $k$ and $\omega$ of optical waves are why light behaves like rays in daily life: the wave's internal structure is four orders of magnitude finer than anything on a desk.

2 · Building a lawful wave in an arbitrary direction (exam level)

A 10 GHz radar wave propagates in the $x$ – $y$ plane at $30°$ from the $x$ -axis, polarized along $\hat{\mathbf{z}}$ , with $E_0 = 50$ V/m and $\delta = 0$ . Construct $\mathbf{E}$ and $\mathbf{B}$ explicitly, verify the whole triad, and count the choices made.

Step 1. Kinematics. $\lambda = c/f = 3.00$ cm; $k = 2\pi f/c = 209.6$ rad/m; $\hat{\mathbf{k}} = (\cos30°, \sin30°, 0) = (0.866,\ 0.500,\ 0)$ , so $\mathbf{k} = (181.5,\ 104.8,\ 0)$ rad/m and the phase is $\varphi = 181.5\,x + 104.8\,y - 6.28\times10^{10}\,t$ .

Step 2. Check the polarization is legal. $\hat{\mathbf{k}}\cdot\hat{\mathbf{z}} = 0$ ✓ — $\hat{\mathbf{z}}$ lies in the wavefront plane, so $\mathbf{E} = 50\cos\varphi\ \hat{\mathbf{z}}$ V/m passes Gauss's law. (Had we requested polarization along $\hat{\mathbf{x}}$ , the check fails: $\hat{\mathbf{k}}\cdot\hat{\mathbf{x}} = 0.866 \neq 0$ ; only the component of $\hat{\mathbf{x}}$ perpendicular to $\hat{\mathbf{k}}$ could survive.)

Step 3. Construct $\mathbf{B}$ by the Faraday lock. $\hat{\mathbf{k}}\times\hat{\mathbf{z}} = (0.866, 0.500, 0)\times(0,0,1) = (0.500,\ -0.866,\ 0)$ , hence

\mathbf{B} = \frac{50}{c}\cos\varphi\ (0.500,\ -0.866,\ 0) = 1.67\times10^{-7}\cos\varphi\ (0.500,\ -0.866,\ 0)\ \text{T}.

Step 4. Audit the triad. Transversality of $\mathbf{B}$ : $\hat{\mathbf{k}}\cdot\hat{\mathbf{B}} = 0.866\times0.500 + 0.500\times(-0.866) = 0$ ✓. Right-handedness: $\hat{\mathbf{E}}\times\hat{\mathbf{B}} = \hat{\mathbf{z}}\times(0.500, -0.866, 0) = (0.866,\ 0.500,\ 0) = \hat{\mathbf{k}}$ ✓. Amplitude lock: $B_0 = E_0/c = \mathbf{0.167\ \mu T}$ ✓. In-phase: same $\cos\varphi$ ✓.

Step 5. The birth-certificate count. Choices made: $\omega$ (1), direction (one angle sufficed here since $\hat{\mathbf{k}}$ was confined to a plane; two in general), $E_0$ (1), $\delta$ (1), polarization $\hat{\mathbf{z}}$ (1 angle). Everything else — $\mathbf{B}$ 's direction, its $0.167\ \mu$ T magnitude, its phase — was computed, not chosen. An exam answer that "chooses" any property of $\mathbf{B}$ has already failed the completion routine.

Origins · 1 minHeinrich Hertz and how this was discovered

Heinrich Hertz

1857–1894 · Karlsruhe — who built the wave Maxwell predicted, then measured its anatomy

Hertz was Helmholtz's prize student, set the problem his teacher had put on a prize list: decide experimentally whether Maxwell's displacement current — and hence the waves — was real. In 1886–88 at Karlsruhe, with a spark-gap dipole as transmitter and a wire loop with a micrometer gap as receiver, he generated electromagnetic waves of roughly 60 cm to several metres wavelength, and then did the thing that makes him this article's patron: he measured the anatomy. Standing waves against a zinc sheet gave him the wavelength; the known circuit frequency then gave the speed — Maxwell's $c$ . He verified the polarization by rotating his receiver, the transversality, reflection, refraction in a prism of pitch, and focusing with cylindrical mirrors: every structural property derived in this article, checked on a laboratory bench within two years. His summary was characteristically modest — the experiments merely proved "the correctness of the Maxwellian theory". Asked about applications, he reportedly saw none: "it is of no use whatsoever." Marconi's transatlantic signal came seven years after Hertz's death at 36, and the unit of the dictionary's $f$ carries his name into every spec sheet on Earth.

History

1865

— Maxwell's "A Dynamical Theory of the Electromagnetic Field" predicts transverse waves at

c

and identifies light with them

1873

— the Treatise systematizes the prediction; polarization and pressure of waves discussed

1887

— Hertz generates and detects electromagnetic waves from a spark-gap dipole at Karlsruhe

1888

— Hertz measures wavelength via standing waves, confirms speed

c

, transversality, and polarization

1895

— Marconi turns the laboratory wave into wireless telegraphy; Röntgen's X-rays extend the spectrum upward

1960

— Maiman's ruby laser: a source approaching the monochromatic plane wave more closely than anything before it

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §9.2 · Jackson, Classical Electrodynamics, 3rd ed., §7.1 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 9 · Zangwill, Modern Electrodynamics, §16.2

Provenance. Equations and structure follow Griffiths §9.2 and Jackson §7.1. A verifier should: (1) substitute the plane-wave ansatz into the four vacuum Maxwell equations to confirm $\mathbf{k}\cdot\mathbf{E}_0=0$ , $\mathbf{B}_0=\tfrac1c\hat{\mathbf{k}}\times\mathbf{E}_0$ , and $\omega=ck$ ; (2) check $B_0=E_0/c$ numerically with $c=2.998\times10^8\,\mathrm{m/s}$ (the $300\,\mathrm{V/m}\to1\,\mu\mathrm{T}$ example); (3) confirm $I=\tfrac12 c\varepsilon_0 E_0^2$ against the time-averaged Poynting vector; (4) verify the solar constant $\approx1.36\times10^3\,\mathrm{W/m^2}$ (standard astronomical value) and the implied radiation pressure $I/c\sim10^{-6}\,\mathrm{Pa}$ . All numbers are standard textbook values. Addendum for this edition: the base's intensity and radiation-pressure results are owned by the dedicated articles ahead (0585–0587 energy/momentum/intensity — deferred BY NAME throughout); the wave-equation derivation and d'Alembert/completion-routine material is owned by The Electromagnetic Wave Equation (0582) and recalled, never re-derived. This edition owns the real-form general- $\hat{\mathbf{k}}$ anatomy (Derivation A), the phase-geometry/mode dictionary (Derivation B), and the parameter count (3). Worked numbers recomputed: Na 589.0 nm → $k=1.067\times10^7$ rad/m, $f=5.09\times10^{14}$ Hz, $\omega=3.20\times10^{15}$ rad/s, $T=1.96$ fs, $B_0(100\ \text{V/m})=0.334\ \mu$ T, 1,700 fronts/mm; 10 GHz → $\lambda=3.00$ cm, $k=209.6$ rad/m, $B_0(50\ \text{V/m})=0.167\ \mu$ T, triad audited componentwise; $z_R(\text{1 mm, 633 nm})=4.96$ m.

Builds on: The Electromagnetic Wave Equation · Maxwell's Equations in Vacuum · Vector calculus and complex exponentials.

Leads to: Complex Notation for Electromagnetic Waves · Energy Density and the Poynting Vector · Intensity and Time-Averaged Quantities · Polarization of Electromagnetic Waves · Electromagnetic Waves in Linear Dielectric Media.

If the wave equation permits longitudinal waves, why doesn't nature find some way to excite them?

Because the wave equation is not the law — it is a consequence of the law, and a lossy one. The curl-of-curl maneuver that produces it discards the divergence information; Maxwell's actual system carries $\nabla\cdot\mathbf{E} = 0$ at all times, and no source, boundary, or cleverness can generate a vacuum field violating it. In particle language: exciting a longitudinal photon would require the photon to have mass, and the measured bound is below $10^{-54}$ kg. Inside matter the story loosens — plasmas support longitudinal plasma oscillations (charge density waves, not electromagnetic waves proper), which is exactly the loophole: there $\nabla\cdot\mathbf{E} = \rho/\varepsilon_0 \neq 0$ because real charges participate.

E and B both vanish twice per cycle at any fixed point. Where is the wave's energy at those instants?

One wavelength ahead and behind of you — not "hidden" anywhere locally. The zeros of a traveling wave are zeros of the whole local field, and the energy density there is genuinely zero at that instant; the energy is concentrated in the moving sheets where $\cos^2\varphi$ is large, and those sheets are sliding forward at $c$ . A quarter period later a crest has arrived at your location, bringing its energy with it. Nothing oscillates between E-form and B-form at a point (that is the standing-wave picture); in a traveling wave the two forms are equal at every instant and the whole inventory travels. The energy article (0585) draws this comb explicitly.

Is a laser beam actually a plane wave?

Locally and temporarily, to excellent approximation — globally, no. A real beam has finite width, so it must be a superposition of plane waves with slightly spread directions (that spread is diffraction), and finite duration, so it carries a spread of frequencies (linewidth). The plane-wave description is licensed over the Rayleigh range $z_R = \pi w_0^2/\lambda$ — metres for a millimetric visible beam — and over the coherence time set by the linewidth, which for stabilized lasers can exceed seconds. Within that window, every formula of this article applies with negligible error. The idealization is not sloppy; it is controlled, and the corrections (Gaussian beam optics) are themselves built as plane-wave superpositions.

What does the phase constant δ actually do, and when does it matter?

δ records where in its cycle the wave sits at the origin of your coordinates at $t = 0$ — nothing more. For a single wave it is pure bookkeeping: shift your clock or your origin and you can set δ to zero, and no measurement on one wave alone can detect it. It becomes physical the moment two waves meet: interference, standing waves, and polarization states built from two components all depend on relative phases, which no re-zeroing can remove. The next article packages $E_0$ and δ into a single complex amplitude precisely because they always travel together — one knob for "how big", one for "when", fused into one complex number.

Could a left-handed triad wave exist — with E × B pointing against k?

No, and the question hides a sign confusion worth untangling. $\mathbf{E}\times\mathbf{B}$ for a lawful plane wave always points along the propagation direction — that is Faraday's law's doing, and it is the statement that energy flows the way the wave moves. If you write down a field with $\mathbf{E}\times\mathbf{B}$ along $-\hat{\mathbf{k}}$ while the phase moves along $+\hat{\mathbf{k}}$ , the Ampère audit fails: it is an impostor. What you can have is a wave propagating along $-\hat{\mathbf{k}}$ — flip the sign of $\mathbf{k}$ in the phase and the lock $\mathbf{B} = \tfrac1c\hat{\mathbf{k}}\times\mathbf{E}$ flips $\mathbf{B}$ accordingly, keeping the triad right-handed about the new direction. (Engineered "left-handed" metamaterials realize negative refraction by making the phase velocity oppose the energy flow inside a structured medium — a fascinating cousin, but it requires matter with tailored ε and μ, never vacuum.)

Since B₀ = E₀/c is so tiny, can I just ignore the magnetic field of a light wave?

Ignore it for forces on slow charges, never for the wave's own existence or its momentum. The magnetic force on a charge moving at $v$ is smaller than the electric force by $vB/E = v/c$ — for electrons in matter ( $v/c \sim 10^{-2}$ or less) the electric force dominates, which is why "the optical field" in spectroscopy usually means $\mathbf{E}$ alone. But the wave propagates because the fields regenerate each other — kill $\mathbf{B}$ and Faraday/Ampère stop the show — and the small magnetic force acting on the charge's E-driven motion is exactly what transfers the wave's momentum (radiation pressure, article 0586). Small ≠ negligible: it is the whole origin of light's push.

Are radio waves, light, and X-rays really the same thing? They behave so differently.

Identically the same object at different $\omega$ — every one a transverse plane wave with the locked triad and $E = cB$ . The behavioral differences are about matter and scale, not about the wave: a 300 m AM wave diffracts around hills because the hills are smaller than λ, while light casts sharp shadows because everyday objects are $10^6$ wavelengths across; X-rays pass through flesh because their $\omega$ exceeds the resonances of the constituent atoms (the dispersion articles quantify this); and detectors differ because photon energy $\hbar\omega$ spans twelve orders of magnitude across the shelf. Slide any experiment coherently in scale with λ and the physics repeats — antenna engineers routinely prototype at scaled frequencies for exactly this reason.

Why is the plane wave the "right" basis rather than, say, spherical waves?

"Right" means "diagonalizes what you care about." Plane waves diagonalize derivatives — $\nabla \to i\mathbf{k}$ — so they turn Maxwell's PDEs into algebra, and they diagonalize translations, so they are the natural basis for infinite homogeneous space and for interfaces that preserve transverse momentum (the reflection articles live on this). Spherical waves diagonalize rotations and are the right basis around compact sources — the radiation chapter (E9) uses them constantly. Both are complete; conversion between them (partial-wave expansions) is standard machinery. Choose the basis whose symmetry matches the problem's, and for this chapter — homogeneous vacuum, flat interfaces — that is the plane wave.