Light, radio waves, X-rays, and microwaves are all electromagnetic waves - transverse waves consisting of oscillating electric and magnetic fields that travel through space at the speed of light. In 1865, James Clerk Maxwell unified electricity, magnetism, and optics by showing that a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field, allowing self-sustaining waves to propagate.
Maxwell predicted that these waves travel at a speed determined by the constants of electricity and magnetism, c = 1/sqrt(mu_0 epsilon_0) = 3 x 10^8 m/s, exactly the speed of light. This established light as an electromagnetic wave. In 1888, Heinrich Hertz generated and detected electromagnetic waves experimentally, confirming Maxwell's theory.
This chapter studies the properties of electromagnetic waves, their energy and momentum, and the full electromagnetic spectrum from radio waves to gamma rays. Each region of the spectrum has distinct sources and applications, from broadcasting and radar to medical imaging and cancer therapy.
Amphere's circuital law, B dot dl = mu_0 I, is consistent only when there is no changing electric field between capacitor plates. During charging, current stops at the plates, yet a magnetic field exists between them. Maxwell resolved this by adding a new term to Ampere's law:
B dot dl = mu_0 (I + I_d)
where the displacement current I_d is proportional to the rate of change of electric flux:
I_d = epsilon_0 d Phi_E / d t
Maxwell called the changing electric flux a displacement current because it produces a magnetic field just like a conduction current. With this correction, the set of four Maxwell's equations became consistent and complete, and they predicted the existence of electromagnetic waves.
Maxwell's equations describe how electric and magnetic fields are produced and how they interact: 1. Gauss's law for electricity: the flux of E equals the enclosed charge divided by epsilon_0. 2. Gauss's law for magnetism: the flux of B through any closed surface is zero (no magnetic monopoles). 3. Faraday's law: a changing magnetic field produces an electric field. 4. Ampere-Maxwell law: a changing electric field (or a current) produces a magnetic field.
In an electromagnetic wave, the electric and magnetic fields oscillate sinusoidally and are perpendicular to each other and to the direction of propagation. If the wave travels along the x-axis, then:
E = E0 sin(kx - omega t), B = B0 sin(kx - omega t)
with E along the y-axis and B along the z-axis. The magnitudes are related by:
E / B = c
The wave is transverse: the oscillations of E and B are perpendicular to the direction of propagation. The speed of the wave is:
c = 1 / sqrt(mu_0 epsilon_0) = 3 x 10^8 m/s
The wave length lambda and frequency f are related by c = f lambda. Electromagnetic waves do not need a medium; they travel through vacuum, which is why light reaches us from the Sun across empty space.
Electromagnetic waves carry energy. The energy densities of the electric and magnetic fields are equal in a wave:
u_E = (1/2) epsilon_0 E^2 = (1/2) B^2 / mu_0 = u_B
The total energy density is u = epsilon_0 E^2 = B^2/mu_0. The energy flux, the energy crossing unit area per unit time, is given by the Poynting vector:
S = E x B / mu_0
The average intensity of the wave is:
I = u_avg c = (1/2) epsilon_0 E0^2 c
Electromagnetic waves also carry momentum. The momentum per unit volume is S/c^2, and when a wave is absorbed by a surface, it exerts radiation pressure equal to u, while for a perfectly reflecting surface the pressure is 2u. Radiation pressure is used in the concept of solar sails and explains phenomena like the tails of comets pointing away from the Sun.
The electromagnetic spectrum spans a vast range of frequencies and wavelengths:
Radio waves have the longest wavelengths, from about 1 m to several kilometres, and are produced by oscillating circuits. They are used in radio and television broadcasting and in wireless communication.
Microwaves have wavelengths of about 1 mm to 1 m. They are used in radar, microwave ovens (which heat water molecules in food), and long-distance communication.
Infrared radiation has wavelengths from about 700 nm to 1 mm. It is emitted by hot objects and is used in night-vision devices, remote controls, and thermal imaging.
Visible light occupies the narrow band from about 400 nm (violet) to 700 nm (red) and is the only part of the spectrum detected by our eyes.
Ultraviolet radiation, with wavelengths from about 10 nm to 400 nm, is emitted by the Sun and hot bodies. It causes tanning and can harm skin; it is used in sterilization and for detecting forged currency.
X-rays have wavelengths from about 0.01 nm to 10 nm. They penetrate soft tissue and are absorbed by bone, making them invaluable for medical imaging and for studying crystal structure.
Gamma rays have the shortest wavelengths, below 0.01 nm, and the highest energies. They are emitted by radioactive nuclei and are used in radiotherapy for cancer and in sterilization of medical equipment.
Different regions of the spectrum are produced by different sources: - Radio waves: accelerating charges in oscillating circuits and antennas. - Microwaves: klystrons and magnetrons. - Infrared: hot bodies, molecules with rotational and vibrational transitions. - Visible light: excited atoms, such as in the Sun and incandescent lamps. - Ultraviolet: very hot bodies and specific atomic transitions. - X-rays: electrons accelerated and stopped at high-energy targets (X-ray tubes), and inner-shell atomic transitions. - Gamma rays: radioactive decay and nuclear reactions.
An accelerated charge is the fundamental source of electromagnetic radiation. When charges oscillate, they radiate electromagnetic waves of the corresponding frequency. This is why AC circuits and alternating currents produce radio waves, and why accelerated electrons in an X-ray tube produce X-rays.
Electromagnetic waves have common properties across the spectrum: 1. They are transverse waves. 2. They travel at c in vacuum and slow down in a medium. 3. They can travel through vacuum and do not require a medium. 4. They transport energy, momentum, and information. 5. They obey the wave equation and can be reflected, refracted, diffracted, and polarized.
Applications include radio and television broadcasting, mobile communication, radar, satellite communication, microwave ovens, infrared thermometers, optical communication through optical fibres, ultraviolet sterilization, X-ray imaging, and gamma-ray radiotherapy. The study of the electromagnetic spectrum has made modern communication and medicine possible.
| Region | Wavelength range | Production | Typical use |
|---|---|---|---|
| Radio waves | > 1 m | Oscillating circuits | Broadcasting, communication |
| Microwaves | 1 mm to 1 m | Klystron, magnetron | Radar, microwave oven |
| Infrared | 700 nm to 1 mm | Hot bodies | Thermal imaging, remotes |
| Visible | 400 to 700 nm | Excited atoms | Vision, optical fibre |
| Ultraviolet | 10 to 400 nm | Very hot bodies | Sterilization, forensics |
| X-rays | 0.01 to 10 nm | X-ray tubes | Medical imaging |
| Gamma rays | < 0.01 nm | Radioactive decay | Radiotherapy |
| Property | Value/Expression |
|---|---|
| Speed in vacuum | c = 1/sqrt(mu_0 epsilon_0) = 3 x 10^8 m/s |
| Relation of E and B | E/B = c |
| Energy density | u = epsilon_0 E^2 = B^2/mu_0 |
| Intensity | I = (1/2) epsilon_0 E0^2 c |
| Radiation pressure (absorbed) | P = u |
| Radiation pressure (reflected) | P = 2u |
Maxwell's equations, completed by the displacement current, unified electricity and magnetism and predicted electromagnetic waves travelling at c = 1/sqrt(mu_0 epsilon_0) = 3 x 10^8 m/s. These transverse waves have perpendicular electric and magnetic fields satisfying E/B = c, and they carry energy with density u = epsilon_0 E^2 and momentum that exerts radiation pressure. The electromagnetic spectrum spans radio waves to gamma rays, each region produced by accelerating charges, atomic transitions, or nuclear decays, with applications from broadcasting and radar to X-ray imaging and radiotherapy. Electromagnetic waves are the foundation of modern communication, medicine, and our understanding of light itself.