Maxwell's equations and electromagnetic waves
James Clerk Maxwell unified electricity and magnetism and predicted electromagnetic waves. His equations explain light, radio, and the limits of circuit theory.
Maxwell's synthesis
In the 1860s, James Clerk Maxwell combined the laws of Gauss, Faraday, and Ampere into a single framework, adding a crucial term: a changing electric field, the displacement current, also creates a magnetic field. His equations, published in 1865 and later written in compact vector form by Oliver Heaviside, describe all classical electromagnetic phenomena.
The four equations
In words: Gauss's law, electric field lines begin and end on charges; Gauss's law for magnetism, there are no magnetic monopoles, so magnetic field lines form closed loops; Faraday's law, a changing magnetic field creates a circulating electric field; and the Ampere-Maxwell law, currents and changing electric fields create circulating magnetic fields.
Electromagnetic waves
Maxwell's equations predict self-sustaining waves: a changing electric field creates a changing magnetic field, which creates a changing electric field, and so on, travelling through space. In vacuum their speed is \(c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \approx 3 \times 10^{8}\) m/s, matching the measured speed of light, which showed that light is an electromagnetic wave. Heinrich Hertz generated and detected radio waves in 1887 to 1888, confirming the prediction.
Wavelength and frequency
For electromagnetic waves, \(c = f\lambda\). Mains at 50 Hz has a wavelength of 6,000 km; FM radio around 100 MHz, 3 m; Wi-Fi at 2.4 GHz, 12.5 cm; visible light, about 400 to 700 nanometres. The electromagnetic spectrum spans radio, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays, all the same phenomenon at different frequencies.
Antennas
An antenna converts guided electrical signals into radiated waves and back. Accelerating charges radiate, so alternating currents in a conductor launch waves. A half-wave dipole, two rods totalling about half a wavelength, is a basic efficient antenna, so higher frequencies allow smaller antennas, one reason phones use gigahertz bands. Antenna gain and directivity describe how strongly an antenna focuses energy in particular directions.
Transmission lines
When a circuit's dimensions approach a significant fraction of the signal wavelength, voltages vary along wires and Kirchhoff's lumped approximations fail. Coaxial cables, twisted pairs, and circuit-board traces then behave as transmission lines with a characteristic impedance, commonly 50 or 75 ohms for coaxial cable. Terminating a line in its characteristic impedance prevents reflections that would distort signals.
Signal integrity and EMC
In fast digital systems, signals with sub-nanosecond edges contain gigahertz frequencies, so board traces act as transmission lines and antennas. Engineers control trace impedances, keep return paths short, and use ground planes to preserve signal integrity. Electromagnetic compatibility (EMC) rules limit how much interference products emit and how well they tolerate it; testing for compliance is part of bringing electronics to market.
Worked example
A quarter-wave antenna for 433 MHz, a common licence-free band in many regions: the wavelength is \(\lambda = \frac{3 \times 10^{8}}{433 \times 10^{6}} \approx 0.69\) m, so a quarter wavelength is about 17 cm. For Wi-Fi at 5.8 GHz, \(\lambda \approx 5.2\) cm, and a quarter wave is about 1.3 cm, small enough to fit inside a phone or as a trace on a circuit board.
Key takeaways
Maxwell (1865) unified electricity and magnetism, adding displacement current; Heaviside wrote the four equations compactly. They state that charges source electric fields, magnetic monopoles do not exist, changing magnetic fields create electric fields, and currents plus changing electric fields create magnetic fields. They predict electromagnetic waves travelling at c = 1/root(mu0 epsilon0), about 3 x 10^8 m/s, so light is electromagnetic; Hertz confirmed radio waves in 1887 to 1888. c = f lambda links frequency and wavelength across the spectrum. Antennas radiate waves; half-wave dipoles are basic designs. When circuits approach wavelength scale, transmission lines with characteristic impedances such as 50 or 75 ohms apply, and signal integrity and EMC matter.
Flashcard
Four laws of classical electromagnetism
Check your understanding
Why do mobile phones use gigahertz frequencies partly for practical antenna reasons?