Electromagnetic waves and light

Lessons 1 to 9 treated electric and magnetic fields separately or linked through induction. Their full connection has an astonishing consequence: a changing electric field creates a magnetic field and a changing magnetic field creates an electric field, so together they can sustain each other and travel through empty space as an electromagnetic wave. Radio, Wi-Fi, mobile phones, radar, microwave ovens, infrared thermometers, visible light, X-rays and fibre-optic communication are all the same phenomenon at different frequencies. This lesson describes how electromagnetic waves arise, the electromagnetic spectrum and its engineering uses, the energy and intensity the waves carry, polarization, reflection and refraction, the total internal reflection that guides light along optical fibres, and the photon picture of light that explains solar cells and LEDs (Lesson 11).

From fields to waves

Faraday's law (Lesson 7) says a changing magnetic field induces an electric field. Maxwell added the complementary statement: a changing electric field produces a magnetic field, just as a current does. Combining the two laws gives a wave equation for both fields (MATH-209), with a speed that depends only on the electric and magnetic constants:

This is the measured speed of light, and in 1865 the agreement told Maxwell that light is an electromagnetic wave. In such a wave the electric and magnetic fields are perpendicular to each other and to the direction of travel (a transverse wave), oscillate in phase, and have magnitudes related by E = cB. Any accelerating charge radiates: the oscillating currents in an antenna launch radio waves; the vibrating charges in hot objects emit infrared and visible light; electrons decelerating in a target produce X-rays. Like all waves (PHYS-130), electromagnetic waves obey \(c = f\lambda\) in vacuum.

The electromagnetic spectrum

Different parts of the spectrum interact with matter in very different ways, which decides how each is used. Long radio waves bend around hills and reflect from the ionosphere; microwaves pass through clouds but are absorbed by water, which is how a microwave oven heats food; infrared is emitted by everything warm, so thermal cameras can find overheating electrical connections and heat leaks in buildings; and X-rays pass through soft tissue and thin metal but are stopped by denser material, revealing bones and hidden weld defects.

Energy and intensity

Electromagnetic waves carry energy, shared equally between the electric and magnetic fields. The power per unit area carried by a wave, its intensity, is the average of the Poynting vector (MATH-209):

where \(E_0\) is the amplitude of the electric field, and 377 Ω is the impedance of free space, \(\sqrt{\mu_0/\varepsilon_0}\). From a small source radiating power P equally in all directions, the intensity falls with the inverse square of distance, \(I = P/(4\pi r^2)\), as for sound (PHYS-130).

Electromagnetic waves also carry momentum and exert radiation pressure, I/c on an absorbing surface: only about 3 µPa for full sunlight, but enough to steer solar sails and to require correction in the orbits of satellites with large solar panels.

Polarization

The direction of a wave's electric field defines its polarization. Antennas emit polarized waves, and a receiving antenna must be aligned with them: a vertical antenna picks up little from a horizontally polarized signal. Light from the Sun or a lamp is unpolarized, a random mixture of directions, but a polarizing filter transmits only one component. Ideal polarizers transmit half of unpolarized light, and polarized light passing through a polarizer at angle θ to its polarization direction follows Malus's law:

Light reflected from water, roads and glass is partly polarized horizontally, which is why polarized sunglasses, with vertical transmission axes, cut glare. Liquid crystal displays work by using an electric field to rotate the polarization of light between crossed polarizers, switching each pixel between bright and dark. Engineers also use polarized light to reveal stress patterns in transparent models and in glass (photoelasticity).

Reflection and refraction

Light slows down in transparent materials, travelling at v = c/n, where n is the refractive index: about 1.33 for water, 1.5 for glass and 1.46 to 1.48 for the silica in optical fibres. At a boundary between materials, some light reflects, with angle of reflection equal to the angle of incidence, and the rest refracts, changing direction according to Snell's law:

with angles measured from the normal. Light entering a denser medium bends toward the normal. Because n varies slightly with wavelength (dispersion), a prism separates white light into colours, and pulses of light in a fibre spread out over long distances, limiting data rates.

Total internal reflection and optical fibres

When light travels from a denser medium toward a less dense one (\(n_1 > n_2\)), it bends away from the normal. At the critical angle \(\theta_c\), with \(\sin\theta_c = n_2/n_1\), the refracted ray skims along the boundary; beyond it, no light escapes and all of it is reflected. This total internal reflection is lossless, which is what makes optical fibres possible. A fibre has a core of slightly higher refractive index than the surrounding cladding, so light entering within a certain cone is trapped and guided along the core for kilometres, around gentle bends.

Antennas and links

An antenna converts guided currents into radiated waves and back again. The simplest efficient design, the half-wave dipole, is two conductors each a quarter wavelength long, fed at the centre; a monopole of a quarter wavelength over a ground plane, such as a car's whip antenna, works the same way. Antennas radiate more strongly in some directions than others, described by their gain: a dish antenna concentrates its power into a narrow beam, which is why satellite dishes must be aimed accurately. Antennas also work identically in transmission and reception (reciprocity), so the same gain helps in both directions.

For a radio link in free space, the received power falls with the square of distance and, for antennas of fixed gain, with the square of frequency. The Friis transmission equation summarizes this: \(P_r = P_tG_tG_r\left(\lambda/4\pi d\right)^2\). A 0.1 W Wi-Fi transmitter at 2.4 GHz with simple antennas (gains near 1) delivers about \(0.1 \times (0.125/(4\pi \times 30))^2 = 1.1 \times 10^{-8}\) W, about −50 dBm, to a receiver 30 m away in open space, still comfortably above the sensitivity of a typical receiver. Walls, floors and people absorb and reflect the signal, which is why indoor coverage falls off much faster, and why engineers plan wireless networks with surveys and models rather than the free-space formula alone.

Light as photons

Some properties of light cannot be explained by waves. In the photoelectric effect, light ejects electrons from a metal only if its frequency exceeds a threshold, however intense the light. Einstein explained this in 1905: light energy comes in packets, photons, each with energy

A convenient form is \(E\ (\text{eV}) = 1240/\lambda\ (\text{nm})\). A red photon at 650 nm carries 1.9 eV, a blue one at 450 nm 2.8 eV, and an ultraviolet photon at 254 nm, used for disinfection, 4.9 eV, enough to break chemical bonds in DNA. Whether a material absorbs or emits light, and at what colour, depends on how photon energies compare with the energy levels of its electrons, the principle behind LEDs, lasers, solar cells and digital camera sensors, developed in Lesson 11.