Electromagnetic Waves in Vacuum

Maxwell's greatest discovery was that the curl equations couple a time-varying electric field to a circulating magnetic field, and a time-varying magnetic field to a circulating electric field. In empty space, these fields sustain each other in a self-propagating loop: an Electromagnetic Wave.

1. Waves as a Consequence

In regions far from charges ($\rho=0$) and currents ($\vec{J}=0$), Maxwell's equations take a symmetric, source-free form:

\nabla \cdot \vec{E} = 0, \quad \nabla \cdot \vec{B} = 0 $$ $$ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \vec{B} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}

The coupling between the third and fourth equations is the key. A spatial circulation of $\vec{E}$ is tied by Faraday's law to a temporal change in $\vec{B}$, while a spatial circulation of $\vec{B}$ is tied by the Ampère-Maxwell law to a temporal change in $\vec{E}$. As those coupled changes repeat through space, the fields sustain the cycle. By applying the curl operator again, we decouple them into independent wave equations:

\nabla^2 \vec{E} = \frac{1}{c^2} \frac{\partial^2 \vec{E}}{\partial t^2}, \quad \nabla^2 \vec{B} = \frac{1}{c^2} \frac{\partial^2 \vec{B}}{\partial t^2}

The predicted speed $c$ is exactly the speed of light, revealing that light itself is an electromagnetic phenomenon.

Deep Dive: The Wave Equation

2. Properties of Plane Waves

The simplest solution to the wave equation is the monochromatic plane wave. These waves have several rigid properties that are not merely empirical, but are strictly required by Maxwell's equations:

$ \perp \hat{k} $ Transverse Nature. Both $\vec{E}$ and $\vec{B}$ are perpendicular to the direction of propagation. This is a direct consequence of $\nabla \cdot \vec{E} = 0$ and $\nabla \cdot \vec{B} = 0$.

$ \vec{E} \perp \vec{B} $ Mutual Perpendicularity. The fields are orthogonal to each other: $\vec{B} = \frac{1}{c} (\hat{k} \times \vec{E})$. They form a right-handed triplet with the propagation vector.

$ \Delta \phi = 0 $ In Phase. $\vec{E}$ and $\vec{B}$ reach their peaks and zeros at the exact same time and location. There is no lag between them in vacuum.

$ E_0/B_0 $ Fixed Ratio. The magnitudes are locked by the speed of light: $E_0 = c B_0$. Note that $E$ is much larger in standard units, but they carry equal energy.

Interactive: EM Wave Configuration

3. Polarization

Since EM waves are transverse, the electric field vector $\vec{E}$ must live in the plane perpendicular to the direction of travel. Polarization refers to the path the tip of the $\vec{E}$ vector traces over time.

By the Principle of Superposition, any state of polarization can be described as the sum of two orthogonal linear waves (e.g., one in $x$ and one in $y$). Depending on the relative phase between these two components, we get:

Linear: Components are in-phase ($0$ or $180^\circ$).
Circular: Components have equal magnitude and a $90^\circ$ phase shift.
Elliptical: The general case with unequal magnitudes or arbitrary phase.

Interactive: Polarization Visualization

Deep Dive: Polarization