Derive the wave equation from Maxwell
From the four source-free vacuum Maxwell equations to propagating transverse E and B fields: derive both vector wave equations, identify c from SI constants, check the missing constraints and state exactly where the argument fails in matter.
- State the source-free, homogeneous vacuum assumptions and all four Maxwell equations.
- Derive the vector E and B wave equations without dropping either divergence law.
- Explain why the vector wave equations alone are not sufficient to reconstruct a physical Maxwell field.
- Check transverse plane-wave E/B directions, amplitude ratio, propagation speed and units.
- Solve a numerical wave example and diagnose an unphysical longitudinal trial solution.
Why can a changing electric field travel through empty space when no conductor or external current is present? Here the answer is derived from ALL four Maxwell equations. The vector wave equation is a necessary consequence, but only the original Maxwell constraints tell us whether a proposed wave is physically admissible.
0101.1 — Choose a precise physical domain before taking curls
Consider an open region of homogeneous vacuum containing no electric charge or physical charge-current density: ρ=0 and J=0. Fields are sufficiently differentiable for spatial and time derivatives to commute. Vacuum constants ε₀ and μ₀ are spatially and temporally constant. No claim is made that the entire universe is source-free: remote antennas may set the boundary/initial fields.
Write the FOUR local Maxwell laws, rather than starting from a memorized wave equation. Gauss's electric law and Gauss's magnetic law are constraints. Faraday's and Ampère–Maxwell's laws couple time variation to spatial curl. There are no material polarization, magnetization, free/conduction-current or dispersion terms inside the chosen region.
0201.2 — Derive the electric-field vector wave equation, line by line
Take the curl of Faraday's law, not the divergence. Since time and spatial derivatives commute, move the time derivative outside the curl. Replace the curl of B using the source-free Ampère–Maxwell law.
Use the vector identity for the curl of a curl. The term ∇(∇·E) vanishes ONLY because Gauss's law in this region gives ∇·E=0. Rearrange without changing either minus sign. The vector Laplacian acts separately on each Cartesian component in fixed Cartesian coordinates.
The result is a second-order hyperbolic equation. It follows from Maxwell; its converse is not automatic because the curl and divergence constraints were used in deriving it.
0301.3 — Derive the magnetic-field equation independently
Curl Ampère–Maxwell's law and commute derivatives. Substitute Faraday's law for ∇×E. Here the gradient-divergence term vanishes by Gauss's magnetic law, ∇·B=0, not by assuming B has zero amplitude.
Both field equations propagate at the same characteristic speed. For a vacuum region the material-independent expression is c=1/√(μ₀ε₀). Its SI units follow from μ₀ε₀ having dimensions s²/m². The modern SI fixes c exactly at 299 792 458 m/s; μ₀ and ε₀ individually are not both assigned independent exact numerical values.
0401.4 — Put a candidate plane wave back into Maxwell: transverse fields and signs
A wave-equation trial E=E₀ cos(k·r−ωt) admits ω²=c²|k|² but does NOT by itself determine E₀·k, B₀ or E/B handedness. Insert the trial E and B into the original divergence and curl laws. For nonzero ω, their amplitudes obey k·E₀=k·B₀=0 and k×E₀=ωB₀. Hence the physically admissible vacuum solution is transverse and satisfies |E₀|=c|B₀|.
The following existing AoN field diagram uses a right-handed x-right, y-up, z-out-of-page frame: propagation +x, E along +y and B along +z (out of page, dot). For these signed directions k×E points +z and E×B points +x. The arrows are physical phase/amplitude references for the shown wave, not independent arbitrary voltage/current arrows.
If E is longitudinal (E₀ parallel to k) and nonzero, its scalar components can satisfy the vector wave equation, but Gauss's law gives k·E₀≠0. Thus it is NOT an electromagnetic vacuum plane wave. A second independent check is the magnitude relation B₀=E₀/c, not B₀=cE₀.
For the convention cos(kx−ωt), choose k∥+x, E∥+y and B∥+z (out of page). Maxwell–Faraday gives B=(1/ω)k×E, so E×B∥+x and S=E×B/μ0 travels with the wave. The arrows depict one instant when E>0, not a longitudinal electric field or a spatial wave profile. In vacuum |B0|=|E0|/c.
0501.5 — Calculate a wave and delimit the model
Worked example: a single-frequency vacuum wave travels along +x with electric peak amplitude 3.00 V/m, frequency 1.00 GHz and electric polarization +y. The wave travels at c, its wavelength is c/f≈0.300 m and its magnetic peak amplitude is 3.00/c≈1.00×10⁻⁸ T. The magnetic direction is +z from the right-hand rule; replacing it by −z would reverse the propagation sign of E×B.
Check SI dimensions: [c/f]=m, [(V/m)/(m/s)]=V·s/m²=T. The 3 V/m amplitude is a PEAK, not root-mean-square value. The field-wave statements do not assume any antenna geometry or detector hardware.
In a material, ∇·E need not vanish even without free charge, and Ampère's law generally involves D, H and conduction/polarization current. Only after specifying linearity, homogeneity, isotropy, loss, dispersion and sources may one derive a corresponding material wave equation. Do not replace c with c/√εᵣ by default in arbitrary lossy, anisotropic or dispersive matter. The next lesson will study plane-wave solutions and their polarization systematically.
Practice — solve first, then open the worked solutions
- Derive the electric wave equation but deliberately omit Gauss's law. Which residual term prevents the advertised source-free equation?
- Take a plane wave moving in +z with E along +x. Determine the SIGNED direction of B and check E×B.
- At f=500 MHz and electric peak amplitude 12 V/m in vacuum, calculate wavelength and magnetic peak amplitude with units.
- Test a nonzero longitudinal trial E=ẑ E₀ cos(kz−ωt), B=0 with ω=ck. Which Maxwell equation rejects it?
- A student claims the vector wave equations alone imply Maxwell's equations. Explain which constraints/initial data are missing and why curls cannot supply them.
- A dielectric has frequency-dependent complex permittivity. State which vacuum steps cannot be transported unchanged, and which material information must be supplied.
Worked solutions
1. Without ∇·E=0 the identity gives ∇(∇·E)−∇²E=−μ₀ε₀∂t²E, hence ∇²E−μ₀ε₀∂t²E=∇(∇·E), not zero in general. In a charged region one must also retain ρ and J.
2. k̂=+ẑ and E=+x̂ give B=(1/c)(ẑ×x̂)E₀=+ŷ E₀/c. E×B=+ẑ E₀²/c: energy flows with the chosen propagation vector.
3. λ=299 792 458/(500×10⁶)≈0.600 m; B₀=12/299 792 458≈4.00×10⁻⁸ T. Peak E must be paired with peak B.
4. Gauss's law gives ∇·E=−kE₀ sin(kz−ωt), nonzero for generic points. Moreover k×E₀=0 requires B₀=0 but Ampère–Maxwell with nonzero ∂tE then fails.
5. Second-order E and B equations admit independently chosen components. Maxwell additionally requires ∇·E=∇·B=0 and first-order curl coupling, with initial/boundary data compatible with them. Deriving a necessary equation never proves the converse.
6. Use ∇·D=ρ_free and ∇×H=J_free+∂tD, supply constitutive D(E,ω), B(H,ω) and conductivity/boundaries; dispersion and absorption can produce complex k(ω). The vacuum step ∇·E=0 and constant scalar μ₀ε₀ coefficient need not hold.
Bridge: the next chapter will derive amplitude, phase, arbitrary polarization and the time-averaged wave-energy flux from a common Maxwell-compatible plane-wave ansatz.