Electromagnetic radiation §

Step 1 — a self-sustaining feedback loop. In free space there are no charges or currents. Two of Maxwell’s equations then say: a changing magnetic field produces an electric field (Faraday’s law), and a changing electric field produces a magnetic field (the Maxwell–Ampère term). Each field regenerates the other, and the pair detaches from its source and travels. This is the whole mechanism of radiation.

Step 2 — the fields are transverse. Take a wave moving in the $+\hat{\mathbf{x}}$ direction. Faraday’s law forces $\mathbf{E}$ to point, say, along $\hat{\mathbf{y}}$; the Maxwell–Ampère law then forces $\mathbf{B}$ along $\hat{\mathbf{z}}$. So $\mathbf{E}\perp\mathbf{B}\perp\mathbf{k}$: neither field has a component along the direction of travel. This is why light is a transverse wave, unlike a sound wave (a longitudinal pressure wave).

Step 3 — in phase, with a fixed ratio. The two fields reach their maxima and zero-crossings at the same place and time, and their amplitudes obey $|\mathbf{E}_0|=c\,|\mathbf{B}_0|$. The ratio $E/B=c$ is a fingerprint of a free-space wave: measure one field and you know the other.

Example. A plane wave in vacuum has $E_0=300\ \text{V/m}$. Then $B_0=E_0/c=300/(3.0\times10^8)=1.0\times10^{-6}\ \text{T}=1\ \mu\text{T}$ — comparable to the Earth’s magnetic field.

Interactive: drag the frequency slider to see the wavelength and the in-phase $E$, $B$ fields, and the energy-flow (Poynting) direction.

Authoritative sources

• Maxwell, A Dynamical Theory of the Electromagnetic Field (1865)

Step 1 — decouple the equations. Start from Faraday’s law $\nabla\times\mathbf{E}=-\partial\mathbf{B}/\partial t$. Take the curl of both sides: $\nabla\times(\nabla\times\mathbf{E})=-\partial(\nabla\times\mathbf{B})/\partial t$.

Step 2 — substitute the source-free Maxwell–Ampère law. In vacuum $\nabla\times\mathbf{B}=\mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t$, so the right side becomes $-\mu_0\varepsilon_0\,\partial^2\mathbf{E}/\partial t^2$.

Step 3 — use the vector identity. With $\nabla\cdot\mathbf{E}=0$ in free space, $\nabla\times(\nabla\times\mathbf{E})=-\nabla^2\mathbf{E}$. The two negatives cancel and you get the wave equation. The identical steps work for $\mathbf{B}$.

Step 4 — read off the speed. The standard wave equation $\nabla^2\psi=(1/v^2)\partial^2\psi/\partial t^2$ has speed $v=1/\sqrt{\mu_0\varepsilon_0}$. Plugging in $\mu_0=4\pi\times10^{-7}\ \text{H/m}$ and $\varepsilon_0=8.85\times10^{-12}\ \text{F/m}$ gives $v\approx3.0\times10^8\ \text{m/s}$ — the measured speed of light. This is the moment Maxwell recognized light as an electromagnetic wave.

The general solution of this equation (superposition, standing waves, beats, dispersion) is developed in Wave Propagation; the full field machinery and the near-field limit are in Electrodynamics. Here we only need the traveling-wave solution of Section 1.

Authoritative sources

• Maxwell, A Dynamical Theory of the Electromagnetic Field (1865)

Step 1 — where the energy goes. The Poynting vector $\mathbf{S}=\tfrac{1}{\mu_0}\mathbf{E}\times\mathbf{B}$ is the electromagnetic analogue of a fluid’s velocity times density: it is power per unit area, and it points where the energy is going. For the plane wave of Section 1, $\mathbf{E}\times\mathbf{B}$ points exactly along $\mathbf{k}$, so the energy travels with the wave.

Step 2 — the intensity. Because $E$ and $B$ oscillate, $S$ oscillates too. The physically relevant quantity is the cycle average. Using $B=E/c$ and $\langle\cos^2\rangle=\tfrac12$:

Example. Sunlight at Earth has $I\approx1360\ \text{W/m}^2$. The corresponding field amplitude is $E_0=\sqrt{2I/(c\varepsilon_0)}\approx1.0\times10^3\ \text{V/m}$ — a surprisingly large field for something that feels like a gentle warm glow.

Step 3 — momentum and pressure. A wave carrying energy $U$ at speed $c$ carries momentum $p=U/c$. When it is absorbed, it hands that momentum to the surface, producing a pressure $p_{\text{rad}}=I/c$; a mirror reverses the momentum, doubling the impulse to $2I/c$. This is the principle behind solar sails.

Authoritative sources

• Maxwell, A Dynamical Theory of the Electromagnetic Field (1865)

Step 1 — one phenomenon, many names. There is no physical boundary between “radio” and “microwave” or between “visible” and “ultraviolet”; the bands are human conventions chosen because different technologies and different interactions with matter dominate in each range. A 10 GHz microwave and a 10 PHz infrared wave are both just solutions of the same wave equation.

Step 2 — the frequency–wavelength trade-off. In vacuum $\lambda=c/\nu$, so higher frequency always means shorter wavelength. Visible light is a narrow slice: roughly $400$–$700\ \text{nm}$, i.e. $\nu\approx4.3\times10^{14}$ to $7.5\times10^{14}\ \text{Hz}$.

Step 3 — why the bands matter. The interaction with matter changes across the spectrum: radio waves pass through walls and drive antennas; microwaves rotate water molecules (heating); infrared matches molecular vibrational modes (thermal emission); visible matches electronic transitions in the eye and in pigments; X-rays and gamma rays are energetic enough to ionize atoms. Those material interactions are the subject of Electromagnetic Media.

Example. A 915 MHz microwave (a common oven/RF band) has $\lambda=c/\nu=3.0\times10^8/9.15\times10^8\approx0.33\ \text{m}$ — about a third of a meter, which is why microwave cavities and antennas are that size.

Authoritative sources

• Maxwell, A Dynamical Theory of the Electromagnetic Field (1865)

Step 1 — what is being polarized. Only the electric field is usually meant, because $\mathbf{E}$ is what pushes charges in a detector. As the wave travels, watch the tip of the $\mathbf{E}$ vector at one fixed point in space.

Step 2 — linear polarization. If $\mathbf{E}$ always points along one fixed line (say $\hat{\mathbf{y}}$), the tip traces a line. This is the simplest state and what a straight wire antenna naturally produces.

Step 3 — circular and elliptical. A general wave has two perpendicular components, $E_x$ and $E_y$, that can differ in amplitude and phase. If they are equal in amplitude and $90^\circ$ out of phase, the tip traces a circle (circular polarization, left- or right-handed). If the amplitudes differ, the tip traces an ellipse. Linear and circular are the two special cases of the general elliptical state.

Step 4 — why it matters. Polarization is a degree of freedom you can use: polarizing sunglasses block one linear direction to cut glare; satellite and radio links use circular polarization to reject reflections; and the polarization of light from a star or galaxy carries information about magnetic fields and scattering.

Example. A wave with $E_x=E_0\cos(kz-\omega t)$ and $E_y=E_0\sin(kz-\omega t)$ has equal amplitudes and a $90^\circ$ phase difference, so it is circularly polarized.

Authoritative sources

• Maxwell, A Dynamical Theory of the Electromagnetic Field (1865)

Step 1 — from a continuous wave to discrete quanta. Classically the wave’s energy is continuous and set by its amplitude. Quantum mechanically, the energy of a mode of frequency $\nu$ comes in integer multiples of a single quantum, $E=h\nu$, where $h=6.626\times10^{-34}\ \text{J·s}$ is Planck’s constant. Each quantum is a photon.

Step 2 — energy and momentum of one photon. A photon of frequency $\nu$ has

and, being massless, always moves at $c$. Higher frequency means a more energetic photon — this is why X-rays and gamma rays can ionize atoms while radio photons cannot.

Step 3 — where the classical wave comes from. A bright beam is simply an enormous number of photons; the classical intensity $I$ of Section 3 is the photon flux times the energy per photon. The full theory — photons as the quanta of the electromagnetic field, gauge symmetry, and the spin-1 nature — is developed in Quantum Field Theory. Here we only need the single-photon energy–frequency relation.

Example. A green photon of $\lambda=550\ \text{nm}$ has $E=hc/\lambda=(6.626\times10^{-34})(3.0\times10^8)/(5.5\times10^{-7})\approx3.6\times10^{-19}\ \text{J}\approx2.3\ \text{eV}$ — just enough to trigger the photoreceptor molecules in your eye.

Authoritative sources

• Planck, quantum hypothesis (1900)
• Quantum Field Theory

Step 1 — why accelerating charges radiate. A charge at rest makes a static field; a charge moving at constant velocity drags its field along. But an accelerated charge (an oscillating current in an antenna) shakes its field, and that disturbance propagates outward at $c$ as a wave. This is the source term behind the free-space wave of Section 1. The full near-field and radiation-zone analysis lives in Electrodynamics.

Step 2 — the antenna as a transducer. A half-wave dipole driven by an AC current of frequency $\nu$ radiates a wave of that same frequency, with $\lambda=c/\nu$ setting the antenna’s size. Reversing the roles, an incoming wave pushes the electrons in the antenna, producing a current you can amplify — the same device transmits and receives.

Step 3 — detection. At low frequencies the wave drives currents (antenna, radio receiver). At optical and higher frequencies the photons are energetic enough to eject electrons from a material — the photoelectric effect — which is how solar cells, photodiodes, and photomultipliers turn light into electricity.

Example. A 100 MHz FM radio wave has $\lambda=3\ \text{m}$, so a half-wave dipole is about $1.5\ \text{m}$ long — exactly the scale of a car radio antenna.

Authoritative sources

• Hertz, experimental confirmation of electromagnetic waves (1887)
• Electrodynamics

A monochromatic plane wave in vacuum has a frequency of $\nu=6.0\times10^{14}\ \text{Hz}$ and an electric-field amplitude $E_0=100\ \text{V/m}$.

1. What is its wavelength, and which band of the spectrum is it in?
2. What is the magnetic-field amplitude $B_0$?
3. What is its intensity $I$?
4. What is the energy of a single photon of this wave, in eV?