Electromagnetic Waves

Maxwell's equations contain one of the most stunning predictions in the history of science. When Maxwell completed his equations by adding the displacement current, he noticed that the coupled equations for $\mathbf{E}$ and $\mathbf{B}$ admitted wave solutions, disturbances that propagate through empty space at a speed determined entirely by two electrical constants. When he calculated that speed, he found it was the speed of light. In a single stroke, Maxwell unified electricity, magnetism, and optics.

"We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena," Maxwell wrote. This was not a guess. It was a mathematical inevitability.

Deriving the Wave Equation

Let's derive this from scratch. In free space (no charges or currents: $\rho = 0$, $\mathbf{J} = 0$), Maxwell's equations simplify to:

$$\nabla \cdot \mathbf{E} = 0 \qquad \nabla \cdot \mathbf{B} = 0$$ $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \qquad \nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

Take the curl of Faraday's law:

$$\nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t}(\nabla \times \mathbf{B})$$

Use the vector identity $\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}$. Since $\nabla \cdot \mathbf{E} = 0$ in free space:

$$-\nabla^2 \mathbf{E} = -\frac{\partial}{\partial t}\left(\mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right)$$

This is a standard wave equation $\nabla^2 f = (1/v^2) \partial^2 f / \partial t^2$ with wave speed $v = 1/\sqrt{\mu_0\varepsilon_0}$. The numerical values $\mu_0 = 4\pi \times 10^{-7}$ T·m/A and $\varepsilon_0 = 8.854 \times 10^{-12}$ C²/(N·m²) give precisely $c = 3 \times 10^8$ m/s, the measured speed of light.

Notice the profound implication: the speed of light is determined by $\varepsilon_0$ (the strength of electric forces) and $\mu_0$ (the strength of magnetic forces). These are measured in tabletop experiments with capacitors and solenoids. Yet they predict the speed of something as exotic as light traveling from distant stars. The unity of physics could not be clearer.

Plane Wave Solutions

The simplest solutions to the wave equation are plane waves, waves with flat wavefronts extending to infinity. For a wave propagating in the $z$-direction:

$$\mathbf{E}(z, t) = E_0 \cos(kz - \omega t) \, \hat{\mathbf{x}}$$ $$\mathbf{B}(z, t) = B_0 \cos(kz - \omega t) \, \hat{\mathbf{y}}$$

where $k = 2\pi/\lambda$ is the wave number, $\omega = 2\pi f$ is the angular frequency, and the dispersion relation is:

$$\omega = ck$$

Maxwell's equations impose strict constraints on these waves:

Polarization

Since $\mathbf{E}$ must be perpendicular to the propagation direction, a wave traveling in the $z$-direction can have $\mathbf{E}$ pointing in any direction in the $xy$-plane. This freedom is called polarization.

The most general plane wave propagating along $z$ is:

$$\mathbf{E} = E_{0x}\cos(kz - \omega t)\,\hat{\mathbf{x}} + E_{0y}\cos(kz - \omega t + \phi)\,\hat{\mathbf{y}}$$

The relative amplitude and phase $\phi$ between the $x$ and $y$ components determine the polarization state:

• Linear polarization ($\phi = 0$ or $\pi$): $\mathbf{E}$ oscillates along a fixed direction
• Circular polarization ($E_{0x} = E_{0y}$, $\phi = \pm\pi/2$): the tip of $\mathbf{E}$ traces a circle; right-handed or left-handed
• Elliptical polarization (general case): $\mathbf{E}$ traces an ellipse

Polarization is not merely a mathematical curiosity. It carries information (LCD screens work by controlling polarization), it reveals the geometry of astrophysical sources, and in quantum mechanics, the two polarization states of the photon are a natural qubit, the simplest quantum system.

Energy in Electromagnetic Fields

Electromagnetic fields carry energy. The energy density stored in the fields is:

$$u = \frac{1}{2}\varepsilon_0 E^2 + \frac{1}{2\mu_0} B^2$$

For an electromagnetic wave, the relation $E = cB$ means the electric and magnetic contributions are equal: the energy is shared equally between $\mathbf{E}$ and $\mathbf{B}$.

The Poynting vector satisfies an energy conservation equation (Poynting's theorem):

$$\frac{\partial u}{\partial t} + \nabla \cdot \mathbf{S} = -\mathbf{J} \cdot \mathbf{E}$$

This is the continuity equation for electromagnetic energy. The term $-\mathbf{J} \cdot \mathbf{E}$ represents the rate at which the field does work on charges (transferring energy from the field to matter). In the absence of currents, electromagnetic energy is perfectly conserved.

The time-averaged intensity (power per unit area) of a plane wave is:

$$\langle S \rangle = \frac{1}{2} c \varepsilon_0 E_0^2 = \frac{E_0^2}{2\mu_0 c}$$

Momentum and Radiation Pressure

Electromagnetic waves carry not only energy but also momentum. The momentum density is:

$$\mathbf{g} = \frac{\mathbf{S}}{c^2} = \mu_0 \varepsilon_0 \mathbf{S}$$

When light hits a surface, it exerts a pressure: radiation pressure. For light fully absorbed by a surface, the pressure is $P = \langle S \rangle / c$. For perfect reflection, it doubles to $P = 2\langle S \rangle / c$. This pressure is tiny for everyday light, but it is what pushes comet tails away from the Sun and is the principle behind solar sails.

Radiation from Accelerating Charges

What creates electromagnetic waves? The answer is beautifully simple: accelerating charges. A charge at rest creates a static electric field. A charge moving at constant velocity creates static-looking fields (in the charge's rest frame). But an accelerating charge creates a disturbance in the electromagnetic field that propagates outward at the speed of light: radiation.

The power radiated by a non-relativistic accelerating charge is given by the Larmor formula:

$$P = \frac{q^2 a^2}{6\pi \varepsilon_0 c^3}$$

where $q$ is the charge and $a$ is the acceleration. The power scales as $a^2$. Double the acceleration and you quadruple the radiation.

The radiation pattern has a characteristic $\sin^2\theta$ dependence, where $\theta$ is the angle from the acceleration direction. Maximum radiation is emitted perpendicular to the acceleration; none is emitted along the acceleration axis. This is the dipole radiation pattern.

The Electromagnetic Spectrum

Maxwell's equations place no restriction on the frequency of electromagnetic waves. The entire electromagnetic spectrum, from radio waves to gamma rays, consists of the same phenomenon: oscillating electric and magnetic fields propagating at speed $c$. What differs is only the frequency and wavelength, related by $c = f\lambda$.

Type	Wavelength	Frequency	Source
Radio	$> 1$ mm	$< 300$ GHz	Electronic circuits, antennas
Infrared	$700$ nm $ - 1$ mm	$300$ GHz $- 430$ THz	Thermal radiation
Visible	$400 - 700$ nm	$430 - 750$ THz	Atomic transitions
Ultraviolet	$10 - 400$ nm	$750$ THz $- 30$ PHz	Hot stars, atomic transitions
X-rays	$0.01 - 10$ nm	$30$ PHz $- 30$ EHz	Inner electron transitions
Gamma rays	$< 0.01$ nm	$> 30$ EHz	Nuclear transitions, particle physics

The boundaries between these categories are conventional, not fundamental. A photon does not "know" whether it is infrared or visible. The physics is identical across the entire spectrum. Only the wavelength changes.

Electromagnetic Waves in Media

In a linear dielectric medium with permittivity $\varepsilon$ and permeability $\mu$, the wave speed becomes:

$$v = \frac{1}{\sqrt{\mu\varepsilon}} = \frac{c}{n}$$

where $n = \sqrt{\mu\varepsilon/(\mu_0\varepsilon_0)}$ is the index of refraction. In most materials $\mu \approx \mu_0$, so $n \approx \sqrt{\varepsilon/\varepsilon_0} = \sqrt{\kappa_e}$, where $\kappa_e$ is the dielectric constant. Light slows down in matter because the electric field in the wave drives oscillations of bound charges, which re-radiate, and the superposition of incident and re-radiated waves travels at the reduced speed $v = c/n$.

Key Insights

• Maxwell's equations predict electromagnetic waves propagating at $c = 1/\sqrt{\mu_0\varepsilon_0}$, the speed of light
• The wave equation emerges from the mutual coupling of $\mathbf{E}$ and $\mathbf{B}$: each generates the other when it changes
• EM waves are transverse: $\mathbf{E} \perp \mathbf{B} \perp \hat{\mathbf{k}}$, with $E = cB$
• The Poynting vector $\mathbf{S} = \mathbf{E} \times \mathbf{B}/\mu_0$ gives the energy flux direction and magnitude
• Accelerating charges radiate electromagnetic waves (Larmor formula)
• The entire electromagnetic spectrum (radio to gamma rays) is the same physics at different frequencies
• EM waves carry both energy and momentum, exerting radiation pressure on surfaces

Looking Ahead

We've derived the electromagnetic field and its waves from Maxwell's equations. But these equations have a hidden freedom: the fields $\mathbf{E}$ and $\mathbf{B}$ can be expressed in terms of potentials, and these potentials are not unique. This gauge freedom may seem like a mathematical curiosity, but it turns out to be one of the deepest principles in all of physics, the foundation of every modern field theory. That's our next topic.