Electromagnetic Wave Equation

Mathematical consequence of Maxwell: wave equation for E and B in vacuum and derivation of c.

Conceptual context

Maxwell's equations are four laws governing the fields $\mathbf{E}$ and $\mathbf{B}$ at every point in space and time. Taken individually, each equation describes a local relationship: how much divergence or curl the field has at each point. But when combined — applying the curl operator to Faraday's law and substituting Ampère–Maxwell's law — something unexpected emerges: a wave equation.

This derivation was the greatest theoretical achievement of the nineteenth century. Maxwell measured nothing new: he simply manipulated the equations already known and found that the electric and magnetic fields could propagate independently, without material support, at a speed that turned out to match exactly the measured speed of light. Electromagnetism and optics, two apparently separate branches of physics, were unified within a single framework.

The concrete result is the electromagnetic wave equation, which states that $\mathbf{E}$ and $\mathbf{B}$ propagate as waves at speed $c$, and that $c$ is a direct consequence of the vacuum constants $\varepsilon_0$ and $\mu_0$.

🟢 Essential level

A light wave is an electric field $\mathbf{E}$ and a magnetic field $\mathbf{B}$ that oscillate simultaneously and sustain each other. A varying $\mathbf{E}$ generates $\mathbf{B}$, and a varying $\mathbf{B}$ generates $\mathbf{E}$: the two fields feed each other and propagate through vacuum without any material medium.

The key is speed: the wave propagates at $c$, approximately three hundred million meters per second. This speed is not measured directly; it is derived from the values of $\varepsilon_0$ and $\mu_0$, the constants that describe how vacuum responds to electric and magnetic fields. The fact that this number agreed with the speed of light measured by Fizeau in 1849 was the signal that light is an electromagnetic wave.

An electromagnetic wave in vacuum is always transverse: $\mathbf{E}$ and $\mathbf{B}$ oscillate perpendicularly to the direction of propagation and perpendicularly to each other. There is no oscillation along the direction of travel; all energy moves forward. This transverse character is what makes polarizing filters possible: a filter aligned across the oscillation direction blocks the wave, while one aligned along it lets the light through.

🔵 Formal level

Basic formulas

The derivation starts from Maxwell's differential equations in vacuum (no charges or currents). The curl operator is applied to Faraday's law:

Wave equation for the electric field

The right-hand side contains the second temporal derivative of $\mathbf{E}$. The coefficient multiplying this derivative is the product $\mu_0$·$\varepsilon_0$. Comparing with the standard scalar wave equation ∇²f = (1/v²)∂²f/∂t², the propagation speed is immediately identified:

Speed of light from μ₀ and ε₀

The numerical value $c$ = 1/√(μ₀ε₀) ≈ 2.998×10⁸ m/s matches the experimentally measured speed of light.

Real student questions

Is the speed of light a postulate or is it derived? It is derived. Maxwell did not postulate $c$; he simply combined Gauss's, Faraday's, and Ampère's equations and obtained the wave equation. The speed $c$, equal to 1/√(μ₀ε₀), is a mathematical consequence of those laws. What surprised Maxwell was that this value matched the experimentally measured $c$.

Why are E and B perpendicular to each other? In a plane wave in vacuum, Gauss's law with no charges requires $\mathbf{E}$ to be perpendicular to the propagation direction. Faraday's law then forces $\mathbf{B}$ to be perpendicular to both $\mathbf{E}$ and that direction. The transversality of both fields is a direct consequence of Maxwell's equations, not an additional assumption.

Can there be a wave with only E or only B? No, not in vacuum. The two fields are coupled: varying $\mathbf{E}$ generates $\mathbf{B}$ (Faraday) and varying $\mathbf{B}$ generates $\mathbf{E}$ (Ampère–Maxwell). A wave with only $\mathbf{E}$ would violate Faraday's law. The two fields always coexist in the wave and are inseparable.

What happens to the wave equation in vacuum if there are charges? If charges or currents are present, source terms appear in Maxwell's equations and the homogeneous wave equation no longer holds directly. An inhomogeneous equation with sources is obtained, whose solution includes the waves emitted by accelerated charges (radiation).

Final synthesis

The electromagnetic wave equation is the deepest consequence of Maxwell's equations: it demonstrates that the fields $\mathbf{E}$ and $\mathbf{B}$ can exist and propagate alone in vacuum at the universal speed $c$ = 1/√(μ₀ε₀), without material support. This prediction unified optics and electromagnetism, and was the first indication that Newtonian mechanics needed to be reformulated.