Maxwell’s equations specify a classical model of electric and magnetic fields. Their consequences include charge conservation, static fields under suitable source and boundary conditions, and electromagnetic waves. The mathematical prerequisite series develops the definitions and proofs from elementary real analysis. Its final chapter maps each result here to its proof.

The four equations

Work in fixed Cartesian coordinates in vacuum, allowing prescribed charges and currents. Write \(\mathbf E(x,t)\) and \(\mathbf B(x,t)\) for real three-component fields, \(\rho\) for charge density, and \(\mathbf J\) for current density. For the classical differential statements below, assume the fields are twice continuously differentiable. The positive constants \(\varepsilon_0,\mu_0\) are the vacuum permittivity and permeability. The adopted equations are

\[\begin{aligned} \nabla\cdot\mathbf E&=\rho/\varepsilon_0,&\nabla\cdot\mathbf B&=0,\\ \nabla\times\mathbf E&=-\partial_t\mathbf B,&\nabla\times\mathbf B&=\mu_0\mathbf J+\mu_0\varepsilon_0\partial_t\mathbf E. \end{aligned}\]

These equations are physical postulates, rather than conclusions of vector calculus. Material response, forces on matter, and experimental identification of the fields require additional model assumptions. See Alphabeta’s Maxwell postulate, sources, and separate Lorentz coupling postulate.

Divergence and curl

For a continuously differentiable field \(\mathbf F=(F_x,F_y,F_z)\), define

\[\nabla\cdot\mathbf F=\partial_xF_x+\partial_yF_y+\partial_zF_z,\] \[\nabla\times\mathbf F=(\partial_yF_z-\partial_zF_y,\ \partial_zF_x-\partial_xF_z,\ \partial_xF_y-\partial_yF_x).\]

Divergence is the limiting outward flux through a small box divided by its volume. The component of curl normal to a small oriented square is the limiting circulation around its boundary divided by its area. The flux proof and the circulation proof give explicit epsilon–delta error bounds. These limits explain the source and local-rotation intuitions. Alphabeta records the coordinate definitions.

Curl detects the antisymmetric part of spatial variation. A field can vary while having zero curl: \(\mathbf F=(x,-y,0)\) has zero curl and zero divergence. Curl is also not generally perpendicular to the field: \(\mathbf F=(-y,x,1)\) has curl \((0,0,2)\), and their dot product is two. For the particular plane-wave field \(\mathbf E=E_0\cos(kz-\omega t)\mathbf e_x\), direct differentiation does give

\[\nabla\times\mathbf E=-kE_0\sin(kz-\omega t)\mathbf e_y.\]

The snapshot displays this field in purple and its curl in amber; the curl arrows are rescaled for visibility.

At a cosine maximum its derivative is zero; at a zero crossing the derivative has maximal magnitude. That explains the quarter-wavelength offset in this example. The wave is uniform over each transverse plane, so sampling one line suffices to display its spatial profile.

Coordinates and field lines

The Cartesian formulas above use Euclidean lengths and a chosen orientation. Other coordinates require the corresponding geometric factors. Chapter 3 derives the cylindrical and spherical forms and explains general coordinate changes.

Zero magnetic divergence says that every sufficiently small closed surface has zero net magnetic flux. It does not force field lines to be closed loops. The constant field \(\mathbf B=(0,0,1)\) has zero divergence and straight field lines extending indefinitely. Singularities and excluded points also require a stated domain.

Static fields need source and boundary hypotheses

For a steady electrostatic field, Faraday’s law gives \(\nabla\times\mathbf E=0\). For an isolated point charge \(q\) at the origin, with spherical symmetry and the specified point-source flux, the field at \(r=|x|>0\) is

\[\mathbf E(x)=\frac{q}{4\pi\varepsilon_0}\frac{x}{r^3}.\]

The source at zero is interpreted distributionally as \(q\delta_0\), where \(\delta_0\) is the unit point source. The field is singular there. On a sphere of radius \(r\), symmetry makes the outward component constant, so the total flux is \(4\pi r^2 E(r)=q/\varepsilon_0\). Chapter 5 proves the point-source identity and the Coulomb result; Alphabeta gives the Coulomb solution. Gauss’s equation alone cannot exclude adding a constant field with zero divergence. Isolation, boundary conditions, or symmetry must be specified.

The diagram shows the radial field of a positive point charge; arrow lengths decrease as \(r^{-2}\).

For a smooth, compactly supported steady current \(\mathbf J\) with \(\nabla\cdot\mathbf J=0\), a decaying vector potential and its magnetic field are

\[\mathbf A(x)=\frac{\mu_0}{4\pi}\int_{\mathbb R^3}\frac{\mathbf J(y)}{|x-y|}\,dy,\qquad \mathbf B(x)=\frac{\mu_0}{4\pi}\int_{\mathbb R^3}\frac{\mathbf J(y)\times(x-y)}{|x-y|^3}\,dy.\]

The potential obeys \(\nabla\cdot\mathbf A=0\), \(-\Delta\mathbf A=\mu_0\mathbf J\), and \(\mathbf B=\nabla\times\mathbf A\). These statements and the resulting magnetostatic equations are proved in chapter 5. See Alphabeta’s vector potential and Biot–Savart theorem. Writing a divergence-free field as a curl globally also requires domain hypotheses; the construction above supplies a potential in its stated setting.

An ideal closed thin wire is a different source model, represented by a current measure along its curve. Its field away from the wire is

\[\mathbf B(x)=\frac{\mu_0 I}{4\pi}\oint\frac{d\mathbf l_y\times(x-y)}{|x-y|^3}.\]

The relation between the volume-current and thin-wire models is justified in the wire calculation and Alphabeta’s thin-current limit. A single current element does not by itself constitute a conserved steady circuit.

For an infinitely long straight wire, the azimuthal solution with no added external field has \(B(r)=\mu_0I/(2\pi r)\) outside the wire. If the wire has radius \(R\) and uniform current density, its interior field is \(B(r)=\mu_0 I r/(2\pi R^2)\). The two formulas agree at the surface. The diagram displays the exterior field, looking along the current direction.

Why a bounded volume current has a locally integrable kernel

If \(|\mathbf J|\le M\), the magnitude of the Biot–Savart kernel is at most \(M/|x-y|^2\). The contribution within distance \(\delta\) of an observation point is bounded by

\[\int_{|x-y|<\delta}\frac{M}{|x-y|^2}\,dy=4\pi M\delta.\]

This tends to zero with \(\delta\); chapter 5 justifies the integral and its continuity consequence. It explains why a bounded volume current can give a finite surface field even when its pointwise integrand is unbounded. An ideal zero-radius line current has a different singularity and its field diverges on the line.

For a uniformly charged solid sphere of radius \(R\) and total charge \(q\), spherical symmetry and enclosed charge give \(E(r)=qr/(4\pi\varepsilon_0R^3)\) inside and \(E(r)=q/(4\pi\varepsilon_0r^2)\) outside. They agree at \(R\), so the surface field is finite.

Wave equations follow from Maxwell’s equations

In a source-free region, assume \(\rho=0\) and \(\mathbf J=0\). The proved identity \(\nabla\times(\nabla\times\mathbf F)=\nabla(\nabla\cdot\mathbf F)-\Delta\mathbf F\) gives, by taking the curl of Faraday’s equation and substituting Ampère–Maxwell,

\[-\Delta\mathbf E=-\mu_0\varepsilon_0\partial_t^2\mathbf E.\]

The same calculation for \(\mathbf B\) gives

\[\partial_t^2\mathbf F=c^2\Delta\mathbf F,\qquad c=(\mu_0\varepsilon_0)^{-1/2},\qquad \mathbf F=\mathbf E\text{ or }\mathbf B.\]

Chapter 6 supplies every differentiation step and the hypotheses. Alphabeta’s corresponding result is the vacuum wave theorem. These wave equations are necessary consequences; they do not replace all four Maxwell constraints. For example, \(\mathbf E=\cos(kz-ck t)\mathbf e_x\), \(\mathbf B=0\), with \(k>0\), solves both wave equations but violates Faraday’s equation.

The identification of these waves with laboratory light is empirical. Mathematically, the domain-of-dependence proof shows finite propagation at speed \(c\). Removing the displacement-current term changes the model; in a source-free region the resulting curl equations no longer admit the nonzero-frequency plane waves derived below.

Plane waves and complex notation

A monochromatic plane wave is a field depending on one phase. For propagation along \(\mathbf e_z\), take \(k>0\), \(\omega>0\), and constant complex amplitudes:

\[\widetilde{\mathbf E}=\mathbf E_0e^{i(kz-\omega t)},\qquad \widetilde{\mathbf B}=\mathbf B_0e^{i(kz-\omega t)}.\]

The physical fields are their real parts. Euler’s formula and the derivative rules are proved in the complex-analysis prerequisite. This notation reduces spatial and time derivatives to multiplication by \(ik\) and \(-i\omega\). Maxwell’s equations then give

\[\omega=ck,\qquad \mathbf e_z\cdot\mathbf E_0=0,\qquad \mathbf B_0=\frac1c\mathbf e_z\times\mathbf E_0.\]

Thus \(\mathbf B=(\mathbf e_z\times\mathbf E)/c\) for the real fields, since the cross product with a real fixed vector commutes with taking real parts. The scalar triple-product identity gives \(\mathbf E\cdot\mathbf B=0\), and both fields are transverse to the propagation direction. Chapter 6 proves the dispersion relation and perpendicularity, including the complex-to-real step. See Alphabeta’s plane-wave Maxwell theorem.

These conclusions concern the specified nonzero-frequency mode. A constant longitudinal field or a static magnetic background can be added and still satisfy source-free Maxwell equations; such a background is excluded by this ansatz. Faraday’s equation alone fixes the time-dependent magnetic contribution, not an arbitrary time-independent addition.

Writing \(E_{0x}=a_xe^{i\alpha_x}\) and \(E_{0y}=a_ye^{i\alpha_y}\), with \(a_x,a_y\ge0\), gives

\[\mathbf E=a_x\cos(kz-\omega t+\alpha_x)\mathbf e_x+a_y\cos(kz-\omega t+\alpha_y)\mathbf e_y.\]

For \(\mathbf E_0=\mathbf e_x+1.5i\mathbf e_y\), this becomes \(\mathbf E=\cos(kz-\omega t)\mathbf e_x-1.5\sin(kz-\omega t)\mathbf e_y\). At a fixed observation point the tip traces an ellipse with semi-axes 1 and 1.5. The animation displays this classical field at several positions; its plotting units set an illustrative speed.

Polarisation

Polarisation describes the trajectory of the real electric-field tip at a fixed position. Put \(\delta=\alpha_y-\alpha_x\). When both amplitudes are positive and \(\sin\delta\ne0\), eliminating the phase gives the ellipse

\[\frac{E_x^2}{a_x^2}+\frac{E_y^2}{a_y^2}-\frac{2\cos\delta\,E_xE_y}{a_xa_y}=\sin^2\delta.\]
  • If \(\sin\delta=0\), or exactly one amplitude is zero, the tip oscillates on a line.
  • If \(a_x=a_y>0\) and \(\delta=\pm\pi/2\) modulo \(2\pi\), it traces a circle.
  • The remaining nondegenerate cases are ellipses. Their semi-axes equal \(a_x,a_y\) in the quadrature case \(\cos\delta=0\); in general the axes rotate and their lengths change.
  • If both amplitudes vanish, the trajectory is a single point.

The classification proof checks every degeneracy. All these fields satisfy the same plane-wave Maxwell constraints. Polarisation already occurs in classical electromagnetism; interpreting a preparation as a quantum state requires additional postulates. Alphabeta gives the polarisation definition.

Propagation and energy

A fixed phase \(kz-\omega t\) moves at \(dz/dt=\omega/k=c\). More generally, a smooth transverse profile \(\mathbf E(z,t)=\mathbf f(z-ct)\), paired with \(\mathbf B=\mathbf e_z\times\mathbf E/c\), solves source-free Maxwell equations by direct differentiation. The identity \(\mathbf E(z+c\Delta t,t+\Delta t)=\mathbf E(z,t)\) proves rigid translation. The translation proof distinguishes this travelling-profile family from arbitrary superpositions of waves travelling in different directions. An ideal monochromatic plane wave extends over all transverse space and has infinite total energy.

With the model’s assigned energy density and flux

\[u=\frac{\varepsilon_0|\mathbf E|^2}{2}+\frac{|\mathbf B|^2}{2\mu_0},\qquad \mathbf S=\frac{\mathbf E\times\mathbf B}{\mu_0},\]

Maxwell’s equations imply \(\partial_tu+\nabla\cdot\mathbf S=-\mathbf J\cdot\mathbf E\). Chapter 6 proves this identity; Alphabeta records the energy and flux definitions and Poynting balance. The interpretation as energy transfer to matter uses the adopted coupling model.

Classical equations allow arbitrary amplitude scaling: if a source-free pair is a solution, multiplying both fields by any real \(\lambda\) gives another, with energy density multiplied by \(\lambda^2\). They therefore do not select a discrete quantum of energy. Photons and quantisation require additional quantum postulates.

Model, measurement, and historical context

Maxwell’s 1865 paper connects electromagnetic disturbances with light. That historical identification and the measured propagation speed are evidence about nature, rather than mathematical consequences of calculus alone. In the current SI, \(c\) has the exact defined value \(299\,792\,458\,\mathrm{m\,s^{-1}}\), while \(\mu_0\) is experimentally determined. The familiar \(4\pi\times10^{-7}\,\mathrm{H\,m^{-1}}\) is no longer an exact definition. See Alphabeta’s SI conventions and the BIPM SI Brochure.

For the complete prerequisites and the status of each claim, read Mathematics for classical electromagnetism. Continue the physics series with Energy carried by electromagnetic wave.