Everything in classical electromagnetism, from Coulomb's law to the self-sustaining and wavelike propagation of electromagnetic field, is a mathematical consequence of four partial differential equations: Maxwell's equations.

The four equations

In SI units, with no exotic media in sight, Maxwell's equations are:

\[ \begin{aligned} \nabla \cdot \mathbf{E} &= \tfrac{\rho}{\varepsilon_0} && \text{(Gauss)} \\ \nabla \cdot \mathbf{B} &= 0 && \text{(no magnetic monopoles)} \\ \nabla \times \mathbf{E} &= -\tfrac{\partial \mathbf{B}}{\partial t} && \text{(Faraday)} \\ \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0\, \tfrac{\partial \mathbf{E}}{\partial t} && \text{(Ampère–Maxwell)} \end{aligned} \]

\(\rho\) is electric charge per unit volume, \(\mathbf{J}\) is electric current per unit area, and \(\varepsilon_0\), \(\mu_0\) are two constants of nature (the permittivity and permeability of empty space). The symbol \(\nabla\) is shorthand for the vector \((\partial_x, \partial_y, \partial_z)\) of spatial partial derivatives. These equations fully and precisely describe the relationship between electric field and magnetic field across space and time.

Divergence and curl, in five minutes

Divergence of a vector field \(\mathbf{F} = (F_x, F_y, F_z)\) is the scalar

\[ \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}. \]

Read it as the net outflow per unit volume at a point. A positive divergence means arrows on average point outward — there's a source there; negative means they point inward — a sink. For the electric field, \(\nabla \cdot \mathbf{E}\) measures the local density of sources, which is precisely where electric charges sit. That's Gauss's law in one breath.

Curl of \(\mathbf{F}\) is itself a vector:

\[ \nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z},\; \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x},\; \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right). \]

Geometrically, \(\nabla \times \mathbf{F}\) measures how rapidly \(\mathbf{F}\) varies as you step from a point to its immediate neighbours. Where neighbouring arrows are nearly identical, the curl is small or zero. Where they differ sharply in length or direction, the curl is large. The cross-product structure of the operator packages that rate of variation as a vector — perpendicular to both the field arrows and the direction along which they vary.

To anchor the abstract definition, take a concrete case we will return to in Section 2: a plane wave propagating along \(\hat{\mathbf{z}}\) whose electric field is everywhere directed along \(\hat{\mathbf{x}}\). As a plane wave, the field depends only on \(z\) and \(t\) — it takes the same value at every transverse \((x, y)\) position. So \(\mathbf{E}(z, t) = E_0\cos(kz - \omega t)\,\hat{\mathbf{x}}\) — a sinusoidal pattern stretched along the z-axis. Plugging into the curl formula, every term but one vanishes, and:

\[ \nabla \times \mathbf{E}(z, t) = -E_0\, k\, \sin(kz - \omega t)\, \hat{\mathbf{y}}. \]
The electric field (purple, along \(\hat{\mathbf{x}}\)) and its curl (amber, along \(\hat{\mathbf{y}}\)) sampled along a single line — the z-axis — at a frozen instant \(t = 0\). The plane wave fills all of space; because it is uniform across every \((x, y)\) plane, every line parallel to the z-axis carries an identical sequence of arrows, and the figure simply picks the z-axis itself to keep the picture readable. The two arrows are perpendicular at every \(z\) and \(\pi/2\) out of phase along \(z\): the curl peaks where \(\mathbf{E}\) crosses zero, and vanishes where \(\mathbf{E}\) is at a peak. Faint envelopes trace each spatial profile. (Curl arrows are scaled for visibility — their intrinsic magnitude is \(k\,E_0\).)

The diagram makes the "neighbours" intuition concrete. Where the purple arrows are at a peak or a trough — neighbouring arrows essentially the same length — the amber curl vanishes. Where the purple arrows pass through zero — adjacent arrows of nearly opposite sign — the curl is largest. The curl vector is perpendicular to \(\mathbf{E}\) and offset by a quarter wavelength along \(z\), exactly as differentiating a cosine produces a sine shifted by \(\pi/2\). By Faraday's law, this same \(\nabla \times \mathbf{E}\) is equal to \(-\partial \mathbf{B}/\partial t\), so the spatial undulation of \(\mathbf{E}\) is literally what drives \(\mathbf{B}\) to change in time. That handoff between the two fields, repeated at every point in space, is the engine that propels every electromagnetic wave.

Mathematical subtlety: coordinate independence

The formulas above are written in Cartesian coordinates \((x, y, z)\), where the scale along each axis is uniform. Divergence and curl are, however, intrinsic geometric operations — they measure physical properties (net outflow per unit volume; local rotation) that exist independently of any choice of coordinates. Their expressions change appearance in other systems. In cylindrical coordinates \((r, \theta, z)\) the divergence acquires a \(1/r\) factor; in spherical coordinates \((r, \theta, \varphi)\) both operators take noticeably more complex forms. (A note on terminology: "polar coordinates" are strictly two-dimensional — the \((r,\theta)\) system of the plane. In three dimensions the analogues are cylindrical and spherical, sometimes called "spherical polar".) More generally, divergence and curl can be expressed in any smooth, invertible coordinate system: orthogonal curvilinear coordinates introduce a trio of scale factors \(h_1, h_2, h_3\) to absorb the geometric distortion; fully general, non-orthogonal coordinates require the metric tensor. The physics — and the operators themselves — are invariant; only the notation varies.

One equation at a time

Gauss's law. \(\nabla \cdot \mathbf{E} = \rho/\varepsilon_0\). Charges are the sources and sinks of the electric field. Integrate this over a sphere around a single point charge and out drops Coulomb's inverse-square law — the entire electrostatics you learned in school is one geometric consequence of this equation.

No magnetic monopoles. \(\nabla \cdot \mathbf{B} = 0\). There is no analogue of electric charge for the magnetic field — no isolated north or south poles. Magnetic field lines never start or end; they always close on themselves.

Faraday's law. \(\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t\). A changing magnetic field induces a swirling electric field. This is why generators turn motion into electricity, why transformers work, why your phone charges wirelessly.

Ampère–Maxwell. \(\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0\, \partial \mathbf{E}/\partial t\). Currents (the \(\mathbf{J}\) term) and changing electric fields (the \(\partial \mathbf{E}/\partial t\) term) both produce a swirling magnetic field. That second term is Maxwell's own contribution — the displacement current — and it is the piece that makes light possible. Without it, a sloshing electric field couldn't seed a magnetic field, the two couldn't take turns generating each other, and electromagnetic waves would not exist.

Everything else is a consequence

Coulomb's law, the Biot–Savart law, the existence of light, its speed, the way energy flows through a beam, the very transverse character of light — none of these are independent postulates. They are all theorems whose proof starts with the four lines above. Before cashing that claim in, it is worth stating the two pre-Maxwell laws precisely so that their derivation is a demonstration, not just an assertion.

Coulomb's law

Coulomb's law. The electrostatic field of a point charge \(q\) at the origin is

\[ \mathbf{E}(\mathbf{r}) = \frac{q}{4\pi\varepsilon_0}\,\frac{\hat{\mathbf{r}}}{r^2}, \]

Here \(\hat{\mathbf{r}} = \mathbf{r}/r\) is the unit vector from source to field point, \(r\) the separation, \(q\) the source charge (signed), and \(\varepsilon_0\) the permittivity of free space. The \(4\pi\) spreads the charge's influence uniformly over spheres of area \(4\pi r^2\), in exact accord with Gauss's law. The \(r^2\) in the denominator is the inverse-square law: double the distance, quarter the field.

Electric field \(\mathbf{E}\) of a positive point charge \(+q\) at the origin. Each arrow's tail marks the observation point; the arrow points radially outward and its length is proportional to \(1/r^2\). The dashed circles are equipotential surfaces — they are perpendicular to every field arrow, as they must be.

Proof — from Gauss's law. Set \(\rho = q\,\delta^3(\mathbf{r})\) (a point charge at the origin). Spherical symmetry forces \(\mathbf{E}(\mathbf{r}) = E(r)\,\hat{\mathbf{r}}\). Apply the divergence theorem to Gauss's law over a sphere \(S_r\) of radius \(r\):

\[ \oint_{S_r} \mathbf{E} \cdot d\mathbf{A} = \int_{V_r} \frac{\rho}{\varepsilon_0}\,dV = \frac{q}{\varepsilon_0}. \]

The left side equals \(4\pi r^2 E(r)\), so \(E(r) = q/(4\pi\varepsilon_0 r^2)\) — Coulomb's law exactly.

Biot–Savart law

Biot–Savart law. The magnetic field contributed by an infinitesimal steady current element \(I\,d\mathbf{l}'\) at position \(\mathbf{r}'\) is

\[ d\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\, \frac{I\,d\mathbf{l}' \times (\mathbf{r} - \mathbf{r}')} {|\mathbf{r} - \mathbf{r}'|^3}. \]

\(\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T\,m\,A^{-1}}\) is the permeability of free space — the magnetic counterpart to \(\varepsilon_0\). The vector \((\mathbf{r} - \mathbf{r}')\) runs from source element to observation point; the cross product \(d\mathbf{l}' \times (\mathbf{r} - \mathbf{r}')\) encodes the field geometry: a current element in \(\hat{\mathbf{x}}\) generates a field wrapping around the \(x\)-axis, vanishing directly ahead or behind and peaking broadside to it. The \(|\mathbf{r} - \mathbf{r}'|^3\) denominator is not a steeper falloff than Coulomb's \(1/r^2\) — the cross product carries one implicit factor of distance, leaving net inverse-square decay in the broadside direction. Integrate \(d\mathbf{B}\) over the full circuit for the total field of any steady current loop or wire.

Cross-section of a long straight wire carrying current \(I\) toward the viewer (\(\odot\)). The magnetic field \(\mathbf{B}\) circulates counterclockwise around the wire (right-hand rule: thumb in current direction, fingers curl in the direction of \(\mathbf{B}\)). Arrow length is proportional to \(1/r\) — the falloff once Biot–Savart is integrated along an infinite wire. The dashed circles are contours of equal \(|\mathbf{B}|\).

Does the field blow up on the wire's surface? For a long straight wire of radius \(R\) with uniform current density \(\mathbf{J} = (I/\pi R^2)\,\hat{\mathbf{z}}\), source elements inside the wire can lie arbitrarily close to a surface point \(|\mathbf{r}| = R\), making the Biot–Savart integrand appear to diverge. The answer is nevertheless the finite

\[ B(R) = \frac{\mu_0 I}{2\pi R}. \]

The resolution is a counting argument. Set \(s = |\mathbf{r} - \mathbf{r}'|\) and write the volume element as \(d^3r' = s^2\,d\Omega\,ds\). The cross product contributes one factor of \(s\) — its magnitude is \(J\,s\sin\alpha\) — so the integrand near \(s = 0\) scales as

\[ \frac{J}{s^2}\,d^3r' \;\sim\; \frac{J}{s^2} \cdot s^2\,d\Omega\,ds = J\,d\Omega\,ds. \]

The powers of \(s\) cancel exactly: the \(1/s^2\) blow-up is offset by the volume measure, and the integral is locally bounded. This is a general property of \(\mathbb{R}^3\) — any \(1/r^2\) singularity is integrable against \(r^2\,dr\) — and the same argument makes the electric field at the surface of a uniformly charged sphere finite.

Proof — from Ampère–Maxwell and \(\nabla \cdot \mathbf{B} = 0\). For steady currents \(\partial\mathbf{E}/\partial t = 0\), so Ampère–Maxwell reduces to \(\nabla \times \mathbf{B} = \mu_0\mathbf{J}\). Because \(\nabla \cdot \mathbf{B} = 0\) one may write \(\mathbf{B} = \nabla \times \mathbf{A}\). Substituting and choosing the Coulomb gauge \(\nabla \cdot \mathbf{A} = 0\) yields Poisson's equation \(\nabla^2\mathbf{A} = -\mu_0\mathbf{J}\), whose solution is

\[ \mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\,d^3r'. \]

Taking the curl and identifying \(\mathbf{J}\,d^3r' = I\,d\mathbf{l}'\) for a thin wire recovers the Biot–Savart formula exactly.

\(\mathbf{E}\) and \(\mathbf{B}\) are perpendicular in a plane wave

Claim. For any monochromatic plane wave in vacuum, \(\mathbf{E} \perp \mathbf{B}\) at every point and every instant.

Proof. Take the propagation direction to be \(\hat{\mathbf{z}}\) and write the fields in complex notation, \(\mathbf{E} = \mathbf{E}_0\,e^{i(kz-\omega t)}\) (and likewise for \(\mathbf{B}\)), with physical fields given by the real parts. Because the operation \(\hat{\mathbf{z}}\times(\,\cdot\,)\) is real and linear, the argument transfers to the real fields directly.

Step 1 — E is transverse. In vacuum \(\rho = 0\), so Gauss's law gives \(\nabla \cdot \mathbf{E} = 0\). For a plane wave varying only in \(z\), this reduces to \(\partial E_z/\partial z = ik\,E_z = 0\). Since \(k \neq 0\) we get \(E_z = 0\), so \(\mathbf{E} = E_x\,\hat{\mathbf{x}} + E_y\,\hat{\mathbf{y}}\).

Step 2 — Faraday pins \(\mathbf{B}\) relative to \(\mathbf{E}\). For a plane wave, all \(x\)- and \(y\)-derivatives vanish, so the curl simplifies to \(\nabla \times \mathbf{E} = ik\,(\hat{\mathbf{z}} \times \mathbf{E})\). Faraday's law \(\nabla \times \mathbf{E} = -\partial\mathbf{B}/\partial t = i\omega\mathbf{B}\) then gives

\[ \mathbf{B} = \frac{k}{\omega}\,\hat{\mathbf{z}} \times \mathbf{E} = \frac{1}{c}\,\hat{\mathbf{z}} \times \mathbf{E}. \]

Step 3 — perpendicularity. Compute \(\mathbf{E} \cdot \mathbf{B}\):

\[ \mathbf{E} \cdot \mathbf{B} = \frac{1}{c}\,\mathbf{E} \cdot (\hat{\mathbf{z}} \times \mathbf{E}). \]

The scalar triple product identity \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \mathbf{v} \cdot (\mathbf{w} \times \mathbf{u})\) with \(\mathbf{u} = \mathbf{w} = \mathbf{E}\) and \(\mathbf{v} = \hat{\mathbf{z}}\) gives

\[ \mathbf{E} \cdot (\hat{\mathbf{z}} \times \mathbf{E}) = \hat{\mathbf{z}} \cdot (\mathbf{E} \times \mathbf{E}) = \hat{\mathbf{z}} \cdot \mathbf{0} = 0. \]

Therefore \(\mathbf{E} \cdot \mathbf{B} = 0\) for any plane-wave field, regardless of the orientation of \(\mathbf{E}\) in the transverse plane. Combined with step 1, the three vectors \(\hat{\mathbf{k}}\), \(\mathbf{E}\), and \(\mathbf{B}\) are mutually perpendicular — a fact that follows from Maxwell's equations alone, with no additional assumption. \(\blacksquare\)

The speed of light as a consequence of Maxwell's equations

The proof runs in three steps. Maxwell's curl equations couple \(\mathbf{E}\) and \(\mathbf{B}\) through each other's time derivatives. In vacuum that coupling forces each field to sustain the other — mutual induction. Combining the two coupled first-order equations into a single second-order equation gives a wave equation whose propagation speed is \(1/\sqrt{\mu_0\varepsilon_0}\) — the measured speed of light.

Concretely, a disturbance in \(\mathbf{E}\) at one point makes Ampère–Maxwell install a \(\mathbf{B}\) to match; the new \(\mathbf{B}\) is itself time-varying, so Faraday installs another \(\mathbf{E}\) a moment later — and the disturbance walks outward. The wave equation is what falls out when that bootstrap is written down algebraically.

Take the curl of Faraday's law \(\nabla \times \mathbf{E} = -\partial\mathbf{B}/\partial t\):

\[ \nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t}(\nabla \times \mathbf{B}). \]

Substitute Ampère–Maxwell on the right — in vacuum it reads \(\nabla \times \mathbf{B} = \mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t\):

\[ \nabla \times (\nabla \times \mathbf{E}) = -\mu_0\varepsilon_0\, \frac{\partial^2 \mathbf{E}}{\partial t^2}. \]

The left side reduces by the vector identity \(\nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla \cdot \mathbf{F}) - \nabla^2 \mathbf{F}\); the gradient term vanishes because Gauss's law gives \(\nabla \cdot \mathbf{E} = 0\) in vacuum. What remains is

\[ \nabla^2 \mathbf{E} - \mu_0\varepsilon_0\, \frac{\partial^2 \mathbf{E}}{\partial t^2} = \mathbf{0}. \]

The same manipulation applied to Ampère–Maxwell — take its curl, substitute Faraday on the right, use \(\nabla \cdot \mathbf{B} = 0\) — gives the identical equation for \(\mathbf{B}\). Both fields therefore satisfy

\[ \left(\nabla^2 - \frac{1}{c^2}\, \frac{\partial^2}{\partial t^2}\right)\mathbf{F} = \mathbf{0}, \qquad c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}. \]

This is the wave equation: spatial curvature \(\nabla^2 \mathbf{F}\) is locked to temporal acceleration \(\partial^2 \mathbf{F}/\partial t^2\) by the single constant \(c^2\), with no source on the right. Bend the field in space and it must accelerate in time — that is the formal content of the bootstrap sketched above, and the precise sense in which the wave is self-sustaining. The propagation speed \(c = 1/\sqrt{\mu_0\varepsilon_0} \approx 2.998 \times 10^{8}\ \mathrm{m\,s^{-1}}\) was not put in by hand — it fell out of two laboratory constants that have nothing a priori to do with optics. The wave Maxwell's equations admit is light. \(\blacksquare\)

A plane wave along the z-axis

Let's pick the simplest interesting solution: a wave propagating through empty space along the \(\hat{\mathbf{z}}\) direction. Empty space means \(\rho = 0\) and \(\mathbf{J} = \mathbf{0}\), and the four equations collapse into wave equations for both fields:

\[ \left( \nabla^2 - \frac{1}{c^2}\, \frac{\partial^2}{\partial t^2} \right) \mathbf{E} = \mathbf{0}, \qquad c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}. \]

The same equation holds for \(\mathbf{B}\). The constant \(c\) falls out as a bookkeeping combination of \(\mu_0\) and \(\varepsilon_0\) — two numbers that experimentalists can measure with capacitors and solenoids, completely independent of any optical apparatus. When Maxwell plugged them in, he got the already-measured speed of light. That was the moment light was unmasked as an electromagnetic wave.

The right mental picture. Think of an infinite flat wall of light — an unbounded sheet of synchronised oscillating field, extending in every transverse direction without limit at each \(z\), sliding forward along \(\hat{\mathbf{z}}\) at speed \(c\). Every point in the same \((x, y)\) slice has identical \(\mathbf{E}\) and \(\mathbf{B}\) vectors at the same instant; nothing in this solution distinguishes \((0, 0, z)\) from \((10^9, 10^9, z)\).

The simplest plane-wave solution of the wave equation for a wave travelling along \(\hat{\mathbf{z}}\) is

\[ \mathbf{E}(z, t) = a_x \cos(kz - \omega t + \alpha_x)\, \hat{\mathbf{x}} + a_y \cos(kz - \omega t + \alpha_y)\, \hat{\mathbf{y}}, \]

with \(\omega = c k\). Note the missing \(\hat{\mathbf{z}}\) component: Gauss's law in vacuum (\(\nabla \cdot \mathbf{E} = 0\)) forces the field to be transverse — perpendicular to the propagation direction. So the wave is described by four real numbers: two amplitudes \(a_x, a_y\) and two phase offsets \(\alpha_x, \alpha_y\).

The complex repackaging

Two cosines with independent phases are a slightly awkward thing to do algebra with. Physicists almost universally repackage this into a single complex-valued vector field:

\[ \tilde{\mathbf{E}}(z, t) = \mathbf{E}_0\, e^{i (kz - \omega t)}, \qquad \mathbf{E}_0 = E_{0x}\, \hat{\mathbf{x}} + E_{0y}\, \hat{\mathbf{y}} \in \mathbb{C}^3. \]

The physical field is recovered by taking the real part: \(\mathbf{E}(z, t) = \operatorname{Re}\{\tilde{\mathbf{E}}(z, t)\}\). Each complex amplitude is most naturally written in polar form,

\[ E_{0x} = a_x\, e^{i \alpha_x}, \qquad E_{0y} = a_y\, e^{i \alpha_y}, \]

and substituting back gives

\[ \begin{aligned} &\operatorname{Re}\{ a_x e^{i\alpha_x} e^{i(kz - \omega t)}\} \hat{\mathbf{x}} + \operatorname{Re}\{ a_y e^{i\alpha_y} e^{i(kz - \omega t)}\} \hat{\mathbf{y}} \\ &\quad = a_x \cos(kz - \omega t + \alpha_x)\, \hat{\mathbf{x}} + a_y \cos(kz - \omega t + \alpha_y)\, \hat{\mathbf{y}}, \end{aligned} \]

which is exactly the real ansatz we started with. The complex form isn't a new physical claim — it is the same field, repackaged. The modulus of each complex component is the amplitude; the argument is the phase offset. Both are physically meaningful.

Why bother? Four reasons, each genuinely load-bearing:

  1. Calculus becomes algebra. Time and space derivatives of \(\tilde{\mathbf{E}}\) simply pull down factors: \(\partial_t \to -i\omega\), \(\partial_z \to i k\). Substituting into Maxwell's equations turns differential constraints into algebraic ones.
  2. The imaginary axis has physical meaning. Time evolution multiplies \(\tilde{\mathbf{E}}\) by \(e^{-i\omega t}\), which is a literal rotation in the complex plane. The imaginary axis is the axis along which phase advances as time progresses — not a bookkeeping trick, an actual rotational coordinate.
  3. Interference is just addition. Add two complex numbers and their magnitudes and phases combine automatically into the correct interference of the underlying real waves. No trig identities required.
  4. Hidden structure becomes visible. The relative phase \(\alpha_y - \alpha_x\) between the two complex components is invisible inside any one real cosine, but it controls the qualitative behaviour of the field as a whole. The next two animations make this concrete.

A concrete case

Take a concrete example: unequal amplitudes \(a_x = 1\), \(a_y = 1.5\), and a relative phase of \(\pi/2\) (more generally this can be any number between 0 and \(2\pi\), do not confuse this with the perpendicularity of electric and magnetic field vectors). In complex notation,

\[ \mathbf{E}_0 = a_x\,\hat{\mathbf{x}} + i\,a_y\,\hat{\mathbf{y}} = \hat{\mathbf{x}} + 1.5\,i\,\hat{\mathbf{y}}. \]

Taking the real part of \(\tilde{\mathbf{E}}\) and multiplying out \((a_x\hat{\mathbf{x}} + i\,a_y\hat{\mathbf{y}})(\cos\theta + i\sin\theta)\) with \(\theta = kz - \omega t\), the physical field is

\[ \mathbf{E}(z, t) = a_x\cos(kz - \omega t)\,\hat{\mathbf{x}} - a_y\sin(kz - \omega t)\,\hat{\mathbf{y}}. \]

At any fixed point in space, the tip of the real-valued electric field arrow traces out an ellipse in the \((x, y)\) plane as time advances, with semi-axes \(a_x\) along \(\hat{\mathbf{x}}\) and \(a_y\) along \(\hat{\mathbf{y}}\). One factor of \(i\) in the complex amplitude bought us the entire phase relationship between the \(x\) and \(y\) components for free. That's the moment the complex form pays for itself.

The animation below shows this field at sample points along the propagation axis.

The electric-field arrow at sample points along the z-axis, drawn at the same instant. Each arrow is anchored on the z-axis and sweeps around inside its own dashed cross-sectional ellipse — the locus of the tip in the \((x, y)\) plane perpendicular to \(\hat{\mathbf{z}}\) at that \(z\), with semi-axes \(a_x = 1\) along \(\hat{\mathbf{x}}\) and \(a_y = 1.5\) along \(\hat{\mathbf{y}}\). The arrows rotate within these transverse planes; they do not orbit the z-axis. Snapped together, the tips form an elliptical helix, and as time advances the helix marches forward along \(\hat{\mathbf{z}}\). Each arrow is \(\operatorname{Re}\{\mathbf{E}_0\, e^{i(kz - \omega t)}\}\) with \(\mathbf{E}_0 = a_x\hat{\mathbf{x}} + i\,a_y\hat{\mathbf{y}}\).

Finally, once \(\mathbf{E}\) is known, \(\mathbf{B}\) is fixed by Faraday's law. Substituting the plane-wave ansatz reduces \(\nabla \times \tilde{\mathbf{E}} = -\partial \tilde{\mathbf{B}}/\partial t\) to

\[ \tilde{\mathbf{B}}(z, t) = \frac{1}{c}\, \hat{\mathbf{z}} \times \tilde{\mathbf{E}}(z, t), \]

so \(\mathbf{E}\), \(\mathbf{B}\), and \(\hat{\mathbf{z}}\) are mutually perpendicular at every point. The magnetic field carries no independent degrees of freedom for a plane wave — it is entirely determined by the electric field and the propagation direction.

Polarisation

The pattern that the tip of \(\mathbf{E}\) traces in the transverse \((x, y)\) plane at a fixed point in space is called the wave's polarisation state. It is fully encoded by the four real parameters \(a_x, a_y, \alpha_x, \alpha_y\) of the general ansatz: the amplitudes set the size of the trajectory, and the relative phase \(\delta = \alpha_y - \alpha_x\) sets its shape. Three regimes follow:

  • Linear (\(\delta = 0\) or \(\pi\)): the tip oscillates along a fixed line in the \((x, y)\) plane.
  • Circular (\(\delta = \pm\pi/2\) with \(a_x = a_y\)): the tip traces a circle.
  • Elliptical (any other case): the tip traces an ellipse — the regime our concrete example above illustrates, with semi-axes \(a_x\) and \(a_y\) since \(\delta = \pi/2\).

All three regimes are admissible plane-wave solutions of Maxwell's equations; nothing in classical EM prefers one polarisation state over another. Polarisation is therefore a genuine classical degree of freedom of the electromagnetic field — the orientation pattern of \(\mathbf{E}\) in the transverse plane is intrinsic to Maxwell's equations alone, not a quantum addition.

What Maxwell actually discovered

Maxwell published his equations in 1865 — forty years before quantum theory existed. He had no concept of photons. That word, and the idea behind it, arrived only after Planck fit the blackbody spectrum in 1900 and Einstein used it to explain the photoelectric effect in 1905. Maxwell's achievement was different: he showed that the electromagnetic field is a physical entity in its own right, filling all of space, obeying its own equations of motion, and capable of sustaining self-propagating waves without any material medium.

In that classical picture, "the speed of light" means the phase velocity — the rate at which surfaces of constant phase advance through space. For the monochromatic plane wave \(\mathbf{E}(z, t) = \mathbf{E}_0\cos(kz - \omega t)\), a surface of constant phase \(kz - \omega t = \mathrm{const}\) moves at \(dz/dt = \omega/k = c\). What propagates is not a particle but a spatial pattern of field values: the electric and magnetic vectors at position \(z\) at time \(t\) are identical to those at \(z + c\,\Delta t\) at time \(t + \Delta t\). No material medium moves; the abstract configuration of \(\mathbf{E}\) and \(\mathbf{B}\) across space translates rigidly at \(c\). The oscillating fields we have built are, in Maxwell's own terms, the whole story — particles are nowhere in it.

What classical EM does not address is the quantum granularity of these waves. Maxwell's equations permit the energy density of a monochromatic wave to take any positive value continuously, but in nature the energy carried in a mode of frequency \(f\) is always an integer multiple of a tiny indivisible quantum \(hf\) — Planck's relation, and the energy of a single photon. Each such photon also carries an intrinsic spin angular momentum of \(\pm\hbar\) along its propagation direction, whose two eigenstates correspond to the two circular polarisation states of the classical wave. Both facts — discrete energy quanta and quantised spin, are invisible to Maxwell's equations alone.

References

  1. Wikipedia contributors. Maxwell's equations. Wikipedia, The Free Encyclopedia. en.wikipedia.org/wiki/Maxwell's_equations
  2. Wikipedia contributors. Plane wave. Wikipedia, The Free Encyclopedia. en.wikipedia.org/wiki/Plane_wave