On Physical Lines of Force/Modern derivations/The electromagnetic wave equation

On Physical Lines of Force → Modern derivations → The wave equation

What Maxwell had

Prop. XVI of Part III, which carries no printed equation numbers. His argument is three steps:

$V = \sqrt{\frac{m}{ρ}}, ρ = μ, m = E^{2} ⟹ V = \frac{E}{\sqrt{μ}}$

with $V = E$ in air or vacuum, where $μ = 1$ . Here $m$ is the coefficient of transverse elasticity of the cell substance, $ρ$ its density, $μ$ the capacity for magnetic induction, and $E$ the dielectric coefficient of Prop. XIII.

This is the wave speed of an elastic solid, applied to a mechanical medium. It is not the derivation given today, and Maxwell himself replaced it three years later.

The modern derivation

Start from the four equations in a region with no charges and no conduction currents — $ρ = 0$ , $𝐉 = 0$ — and a linear, homogeneous, isotropic medium:

$\begin{matrix} \nabla \cdot 𝐄 & = 0 \\ \nabla \cdot 𝐁 & = 0 \\ \nabla \times 𝐄 & = - \frac{\partial 𝐁}{\partial t} \\ \nabla \times 𝐁 & = μ ε \frac{\partial 𝐄}{\partial t} \end{matrix}$

Take the curl of the third:

$\nabla \times (\nabla \times 𝐄) = - \frac{\partial}{\partial t} (\nabla \times 𝐁) = - μ ε \frac{\partial^{2} 𝐄}{\partial t^{2}}$

Now use the vector identity

$\nabla \times (\nabla \times 𝐄) = \nabla (\nabla \cdot 𝐄) - \nabla^{2} 𝐄$

The first term vanishes, because $\nabla \cdot 𝐄 = 0$ in the source-free region. What is left is

$\nabla^{2} 𝐄 = μ ε \frac{\partial^{2} 𝐄}{\partial t^{2}}$

and the identical operation on the fourth equation gives

$\nabla^{2} 𝐁 = μ ε \frac{\partial^{2} 𝐁}{\partial t^{2}}$

Reading the result

Both are the wave equation

$\nabla^{2} ψ = \frac{1}{v^{2}} \frac{\partial^{2} ψ}{\partial t^{2}}$

whose solutions are disturbances travelling at speed $v$ . Comparing,

$v = \frac{1}{\sqrt{μ ε}}, and in vacuum c = \frac{1}{\sqrt{μ_{0} ε_{0}}}$

Two constants, both measured in laboratories with no optical apparatus, combine to give a velocity. See the unit-ratio page for why that had to happen dimensionally, and why the number is what it is.

Where the displacement current enters

This is the point of the whole derivation, and it is easy to miss because the term is doing its work quietly.

Delete $\partial 𝐃 / \partial t$ from the fourth equation. Then in a source-free region $\nabla \times 𝐁 = 0$ , and taking the curl of the third equation gives

$\nabla^{2} 𝐄 = 0$

— Laplace's equation. No time derivative. No waves. Fields would adjust instantaneously and nothing would propagate.

The second time derivative in the wave equation comes from applying $\partial / \partial t$ to a term that is already a time derivative. One of those comes from Faraday's law, the other from the displacement current. Remove either and the wave equation collapses. Faraday supplied the first in 1831; Maxwell supplied the second in Prop. XIV of January 1862, to fix a mechanical difficulty about spinning cells.

The transverse condition

Transverse undulations of the electromagnetic medium

Take a plane wave travelling along $z$ , so every field depends on $z$ and $t$ only. Then $\nabla \cdot 𝐄 = 0$ reduces to $\partial E_{z} / \partial z = 0$ , and $\nabla \cdot 𝐁 = 0$ likewise. A component that does not vary along the direction of travel is not part of the wave. So

$E_{z} = 0, B_{z} = 0$

The wave is transverse. Nothing wobbles along the direction of travel.

Feeding a plane wave $𝐄 = 𝐄_{0} f (z - v t)$ into Faraday's law gives further

$𝐁 = \frac{1}{v} \hat{𝐳} \times 𝐄$

so $𝐄$ , $𝐁$ and the direction of travel are mutually perpendicular, and the two field amplitudes are in a fixed ratio.

That the wave must be transverse matters historically. The wave theory of light required transverse vibrations in order to explain polarization, and this had been a persistent embarrassment: transverse waves need a medium with rigidity, and a rigid ether filling all space is hard to accept. Maxwell's equations deliver transversality with no assumption about rigidity at all. It falls out of the two divergence equations.

Why Maxwell replaced his own derivation

The Prop. XVI argument depends on:

the medium being an elastic solid in the ordinary mechanical sense;
the 6-to-5 ratio of elastic moduli adopted in Prop. XII — a theory of elasticity later abandoned;
the identification of the density of the vortex substance with $μ$ .

Remove the cells and the argument does not survive.

The derivation on this page needs none of that. It uses only the four field equations and two constitutive relations. This is why A Dynamical Theory of the Electromagnetic Field (1865) exists: Maxwell re-derived the same result from the field equations alone, and it is the 1865 derivation that stands in textbooks. The 1862 calculation is where the answer was first found; the 1865 one is where it was secured.

What it says today

	form
wave equation, vacuum	$\nabla^{2} 𝐄 = \frac{1}{c^{2}} \frac{\partial^{2} 𝐄}{\partial t^{2}}$
speed	$c = \frac{1}{\sqrt{μ_{0} ε_{0}}}$
speed in a medium	$v = \frac{c}{n}$ , with $n = \sqrt{μ_{r} ε_{r}}$
transversality	$𝐄 ⊥ 𝐁 ⊥$ direction of travel
field ratio	$\| 𝐄 \| = c \| 𝐁 \|$ in vacuum

The relation $n = \sqrt{μ_{r} ε_{r}}$ is Maxwell's Prop. XVII prediction that the refractive index is the square root of the dielectric constant — which holds when both are measured at the same frequency, and fails spectacularly for polar liquids when they are not.

Where the conventions differ

	Maxwell 1861	modern
the wave speed	$V = \sqrt{m / ρ}$ , an elastic-solid formula	$v = 1 / \sqrt{μ ε}$ , from the field equations
the elasticity	$m = E^{2}$ , the stiffness of a cell wall	$1 / ε$
the density	$ρ = μ$ , the density of the vortex substance	$μ$ , a permeability
transversality	assumed, from the mechanical analogy with light	derived, from the divergence equations
the numerical value	310,740,000,000 mm/s	299,792,458 m/s, exactly, by definition since 1983

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Plain-language version: How Maxwell found the speed of light