Electromagnetic Waves

Light is an electromagnetic wave — a coupled oscillation of electric and magnetic fields that propagates through space without needing a material medium. Understanding its wave nature is the essential classical starting point before we ask why the classical picture breaks down.

Maxwell's prediction

In the 1860s James Clerk Maxwell unified electricity and magnetism into four equations. A remarkable consequence of those equations is that a changing electric field produces a changing magnetic field, which in turn regenerates the electric field, creating a self-sustaining propagating disturbance. Applying the curl equations to a region of free space yields the same wave equation for both the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ :

\nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}.

The wave speed that falls out is $c = 1/\sqrt{\mu_0 \epsilon_0}$ . Inserting the measured values of the electric permittivity $\epsilon_0$ and magnetic permeability $\mu_0$ of free space gives $c \approx 3.0 \times 10^8\,\text{m/s}$ — exactly the measured speed of light. Maxwell concluded that light is an electromagnetic wave.

Plane waves and their properties

The simplest solution is a monochromatic plane wave travelling in the $x$ -direction:

\mathbf{E}(x,t) = E_0 \cos(kx - \omega t)\,\hat{y}, \qquad \mathbf{B}(x,t) = \frac{E_0}{c} \cos(kx - \omega t)\,\hat{z}.

Several features deserve notice:

Transversality. Both $\mathbf{E}$ and $\mathbf{B}$ oscillate perpendicular to the direction of propagation. Electromagnetic waves in vacuum are always transverse.
In-phase oscillation. $\mathbf{E}$ and $\mathbf{B}$ reach their peaks and zeros simultaneously. Their amplitudes satisfy $|\mathbf{B}| = |\mathbf{E}|/c$ at every instant.
Angular frequency and wave number. $\omega = 2\pi\nu$ (where $\nu$ is the ordinary frequency) and $k = 2\pi/\lambda$ (where $\lambda$ is the wavelength). These are linked by the dispersion relation $\omega = ck$ , which is just another way to write $c = \nu\lambda$ .

The electromagnetic spectrum

The same equations describe every electromagnetic wave regardless of frequency. Visible light is just a narrow slice of the full electromagnetic spectrum, spanning roughly $400\,\text{nm}$ (violet) to $700\,\text{nm}$ (red). Moving to shorter wavelengths (higher frequencies) gives ultraviolet, X-rays, and gamma rays; moving to longer wavelengths gives infrared, microwaves, and radio waves. All travel at $c$ in vacuum and differ only in $\nu$ (and therefore $\lambda$ ).

Polarisation

Because $\mathbf{E}$ is a vector, it has a direction of oscillation: the polarisation of the wave. In the plane wave above, $\mathbf{E}$ always points along $\hat{y}$ — this is linear polarisation. Superposing two linearly polarised waves with a relative phase difference produces other polarisation states (circular, elliptical). Polarisation will reappear later as a concrete physical realisation of a qubit: two orthogonal polarisation directions map directly onto the two basis states $|0\rangle$ and $|1\rangle$ .

Energy transport: the Poynting vector

A propagating wave carries energy. The rate of energy flow per unit area is described by the Poynting vector $\mathbf{S} = (\mathbf{E} \times \mathbf{B})/\mu_0$ . For the plane wave above, the time-averaged intensity (power per unit area) is

I = \langle S \rangle = \frac{E_0^2}{2\mu_0 c}.

This shows that intensity is proportional to the square of the electric-field amplitude. The same amplitude-squared rule reappears in quantum mechanics as the Born rule — a deep structural echo worth keeping in mind.

Where classical physics breaks down

The wave picture is extraordinarily successful, yet it fails at the quantum level. It predicts that intensity (and therefore energy delivery) should depend only on amplitude, not on frequency. The photoelectric effect — covered in Module 5 — contradicts this directly: low-frequency light ejects no electrons no matter how intense it is, while even faint high-frequency light can eject electrons immediately. That failure motivates the photon picture, where light also has a particle nature with energy $E = h\nu$ . The classical electromagnetic wave is not wrong; it describes interference, diffraction, and polarisation perfectly. But it is incomplete.