The Wave Equation in Vacuum

The Idea in Plain Language

Imagine a ripple on the surface of a pond. The water moves up and down, but the disturbance travels outward. In electromagnetism, there is no water, but there is a field. If you wiggle an electric field, it creates a magnetic field. That changing magnetic field, in turn, creates a new electric field further away. This chain reaction continues, carrying energy through the vacuum. The key insight is that the "stiffness" of the electric field and the "inertia" of the magnetic field determine how fast this ripple travels. Just as the tension and density of a guitar string set its wave speed, the constants \(\epsilon_0\) and \(\mu_0\) set the speed of light.

Deriving the Wave Equation

To see the wave nature explicitly, we must eliminate one field from Maxwell's equations to obtain a single equation for the other. We start with the source-free vacuum conditions: \(\rho = 0\) and \(\mathbf{J} = 0\). The relevant equations are Faraday’s law and the Ampère-Maxwell law:

We take the curl of Faraday’s law. Using the vector identity \(\nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}\), and noting that \(\nabla \cdot \mathbf{E} = 0\) in vacuum, we get:

Now we substitute the Ampère-Maxwell law for \(\nabla \times \mathbf{B}\). Since the fields are smooth, we can interchange the spatial and temporal derivatives:

Rearranging gives the standard wave equation for the electric field. A parallel derivation starting with the curl of the Ampère-Maxwell law yields the identical form for the magnetic field.

The Speed of Propagation

The general form of a second-order wave equation in three dimensions is \(\nabla^2 \psi = \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2}\), where \(v\) is the phase velocity. Comparing this with our derived equation:

We identify the propagation speed \(c\) as:

Using the SI values \(\epsilon_0 \approx 8.854 \times 10^{-12} \ \mathrm{F/m}\) and \(\mu_0 = 4\pi \times 10^{-7} \ \mathrm{H/m}\), we calculate:

This value matches the experimentally measured speed of light. This was not a coincidence; it was the theoretical prediction that confirmed light is an electromagnetic wave. The speed is fixed by the vacuum permittivity and permeability, meaning it is a fundamental property of spacetime itself, independent of the source or observer (in the context of special relativity).

Structure of the Solution

The wave equation is linear, meaning any linear combination of solutions is also a solution. The most fundamental solutions are plane waves. For a wave traveling in the \(z\)-direction, the spatial dependence is \(e^{i(kz - \omega t)}\). Substituting this into the wave equation reveals the dispersion relation:

Here, \(\omega\) is the angular frequency and \(k\) is the wavenumber. This linear relationship between \(\omega\) and \(k\) indicates that vacuum is a non-dispersive medium; all frequencies travel at the same speed \(c\). The electric and magnetic fields are coupled such that \(\mathbf{B} = \frac{1}{c} \hat{\mathbf{k}} \times \mathbf{E}\). This ensures that the fields are perpendicular to each other and to the direction of propagation, a property we will explore in detail in the next lecture.