In an introductory circuit, a wire is a perfect line: whatever voltage you put in at one end appears at the other end at the same instant. That is a lie, and a useful one. Electrical signals travel at a large fraction of the speed of light, which sounds fast until you notice that light covers 30 cm in a billionth of a second. A modern processor's signals change in less than that. On the circuit board, a wire a few centimetres long is already a long journey.
Then things go odd. A signal can bounce back from the far end of a cable as an echo. A TV aerial needs a cable of a particular "impedance". A radio aerial is a rod of a particular length. One subject explains all of it: a wire is not a wire, it is a guide for a wave.
Who takes it: electrical and computer engineers, usually in year 3; physicists take a theory-heavy cousin of it.
What it's for: predicting how electric and magnetic fields move through space and along wires: waves, reflections, antennas and shielding.
Comes after: Physics II: Electricity & Magnetism (34/07) and Multivariable Calculus (34/02), because fields are functions of position and the equations use divergence and curl.
Leads to: Communication Systems (34/40), since every radio link begins with an antenna and a wave. It also feeds Power Systems & Electric Machines (34/39), where long transmission lines are the same maths at 50 or 60 Hz, and Embedded Systems (34/41), where board layout decides whether the chip works.
Honest line: you won't be able to design an antenna or a high-speed board after this. You will see why a cable has an impedance, why an antenna has a length, and why fast edges, not fast clocks, are what make a circuit board hard. For a gentler first read, see Physics II (34/07).
In the 1860s James Clerk Maxwell gathered what was known about electricity and magnetism into a small set of equations and added one missing piece. Oliver Heaviside later rewrote them into the four compact vector equations in textbooks today. In words, each one is a statement about how fields behave around a point in space.
The last term is Maxwell's own, and it is the one that matters for this subject. Faraday says a changing B makes an E. Maxwell's term says a changing E makes a B. Each makes the other, so once you shake the fields they can keep each other going without any wire or charge, and leave the source behind as a travelling wave. Hertz showed in experiments in 1887 and 1888 that such waves exist, can be reflected and refracted, and travel at the speed of light.
Put numbers into Maxwell's equations and a wave solution drops out with a speed you can calculate from two constants that you can measure with capacitors and coils on a bench. The speed is c = 1 / √(μ₀ε₀) = 299,792,458 m/s. Maxwell noticed that this was the speed of light, and concluded that light is an electromagnetic wave. Since 1983 the number is exact, because the metre is now defined from it: a metre is the distance light travels in 1/299,792,458 of a second.
One relation, and the entire spectrum follows. Slow waves are long and fast ones are short: at 50 Hz the wavelength is 6,000 km, at 100 MHz (FM radio) it is 3 m, at 2.45 GHz (Wi-Fi and microwave ovens) 12 cm, and for green light about 555 nanometres. Radio, microwaves, infrared, light, ultraviolet and X-rays are the same thing at different frequencies. What changes is how they interact with matter, and that decides what each band is used for. Inside a material the wave is slower: a coax cable with a plastic insulator carries it at about two-thirds of c, a glass fibre about 0.68 c, and a signal in a buried layer of a typical circuit board about half c (these depend on the insulator's permittivity, as 1/√εr).
λ = (speed in the medium) ÷ f. Band edges follow the usual ITU naming; the uses are examples, and allocations differ between countries.
Now take a two-wire cable and connect a battery with a switch to one end. When you close the switch, the far end does not know yet. The wave has to travel down the cable, and at each instant the part of the line the wave has reached carries a voltage and a current in a fixed ratio: its characteristic impedance Z₀, set only by the geometry and the insulator, not by the length. Common ones: 50 Ω (lab, radio), 75 Ω (TV, video), about 100 Ω (Ethernet twisted pair).
The wave then reaches the far end and meets whatever is connected there. If the load resistance is exactly Z₀, the wave sees what looks like more line, and all of it is absorbed. If it is anything else, the load cannot take that voltage and current ratio, and the difference goes back as a reflected wave:
The reflection runs back to the source, and if the source impedance does not match the line either, it reflects again. The result is a staircase of voltage with ringing that settles only after several round trips. Worked example, in the interactive below: a driver of 20 Ω pushing a 1 V step down a 50 Ω line into an open end (a chip input draws almost no current). The first wave is 1 × 50/(20+50) = 0.714 V. At the open end the voltage doubles to 1.43 V, an overshoot of 43% above the 1 V it will finally settle at. It then rings down: 0.82, 1.08, 0.97, 1.01 V.
When does this matter? A common rule of thumb is that a connection needs treating as a transmission line once it is longer than about a tenth of the wavelength of the highest frequency present. A signal edge that rises in 1 ns has significant content up to about 0.35/(1 ns) = 350 MHz. On a circuit board, where waves travel at about 0.48 c, that wavelength is about 41 cm, and a tenth of it is about 4 cm. So a signal trace longer than that, carrying a 1 ns edge, can ring. The clock speed does not matter. The edge speed does, which is why a slow clock on a fast chip is still a signal-integrity problem.
A 1 V step on a lossless 50 Ω line, computed with the standard bounce (lattice) method. T is the one-way travel time: 5 ns for a 1 m coax cable, about 7 ps per millimetre in a typical circuit board. "Matched source" is 50 Ω at the source and an open far end: one reflection, then silence.
Every reflection is power that was not delivered. Engineers measure a mismatch in a few equivalent ways: the reflection coefficient Γ, the fraction of power reflected |Γ|², the standing-wave ratio (VSWR, which is what the little meter on an amateur radio shows), or the return loss in decibels. A 75 Ω TV cable plugged into a 50 Ω instrument has Γ = 25/125 = 0.2: that is 4% of the power reflected, a VSWR of 1.5, and 14 dB of return loss. For an antenna that is a small price; for a pulse train in a computer it can be fatal, because the echo arrives at the receiver just as the next bit does.
The cures are all ways to give the wave something that looks like more line: a termination resistor equal to Z₀ at the end, a matching network, a transformer, or lines cut to exact fractions of a wavelength.
A real line with a real (resistive) load. The dashed green line marks the matched value, where nothing is reflected.
At steady DC, current spreads through the whole cross-section of a wire. As the frequency rises, the changing magnetic field inside the wire induces eddy currents that push the current out toward the surface. The current crowds into a thin layer, the skin depth:
For copper at 60 Hz this is 8.4 mm, so the middle of a thick cable carries little: this is why high-voltage lines use steel-cored aluminium with the current in the outside strands, and why large conductors for 50 or 60 Hz are made of insulated strands or hollow tubes. At 1 MHz the skin is 65 µm, as thick as a hair. At 1 GHz it is 2 µm, so the inside of the conductor is irrelevant and only its surface finish matters, which is why microwave hardware is silver- or gold-plated. A 2 mm radius copper wire at 1 MHz has an AC resistance about 16 times its DC value in the thin-skin approximation.
Resistivities at about 20 °C: copper 1.68×10⁻⁸, aluminium 2.65×10⁻⁸, silver 1.59×10⁻⁸ Ω·m. Skin depths ignore temperature and magnetism. The resistance ratio is the thin-skin approximation; it is accurate when the skin is much thinner than the wire.
A wire carrying a steady current does not radiate. One carrying a current that changes fast enough does: the changing magnetic field makes an electric field, which makes a magnetic field, and the disturbance leaves for good. The trouble is efficiency. A rod much shorter than a wavelength has the charge slosh back and forth across so small a distance that it radiates little and mostly heats up the wire.
The special lengths come from the fact that current has to go to zero at the open end of a rod: there is nowhere for it to go. A rod that is half a wavelength long (a half-wave dipole) carries a standing wave with its current largest in the middle and zero at both ends, like a guitar string vibrating at its lowest note. At that length the rod resonates, its input looks like a pure resistance of about 73 Ω (so you can drive it without a matching coil) and both halves radiate in step. A quarter-wave rod above a metal plane does the same job, because the plane makes a mirror image of the missing half: a car's radio whip for 100 MHz is about 75 cm, a quarter of 3 m. Practical rods are cut a few per cent shorter than the free-space number because of end effects.
Worked example: for Wi-Fi at 2.45 GHz, a wavelength is 12.2 cm, so a half-wave rod is 6.1 cm and a quarter-wave stub is 3.1 cm. That is why a router antenna is a stubby rod and an FM aerial is a long one.
A dipole radiates nothing along its own axis and most at right angles to it, which is why you tilt a TV aerial.
Ideal thin rod, centre-fed, sine-shaped current (the textbook model). The pattern is from the standard formula [cos(kL/2·cosθ) − cos(kL/2)] / sinθ; directivity and feed resistance are integrated numerically (half-wave: 1.64 and 73.1 Ω).
Every circuit is also an antenna, and every antenna is also a receiver. Electromagnetic compatibility (EMC) is the discipline of making sure a device neither radiates enough to disturb others nor is easily upset by them. The law follows the physics: the EU's EMC Directive (2014/30/EU) requires equipment to neither generate nor be disturbed by electromagnetic disturbances beyond set levels, and in the US the FCC's Part 15 limits what unintentional radiators may emit.
The main tool is a conducting shield. Charges in a closed metal box move until the fields inside cancel, so a closed box shields its contents, and a mesh does nearly as well as long as the holes are much smaller than the wavelength. The microwave oven is the classic case: its door window has a metal mesh with holes a millimetre or two wide, far below the 12 cm wavelength at 2.45 GHz, so the microwaves reflect off it while visible light, whose wavelength is a few hundred thousand times smaller, passes through.
Loops are the enemy: by Faraday's law a loop picks up noise in proportion to its area, and a twisted pair (Ethernet) has loops that alternate in direction and cancel.
Maxwell's equations say that changing electric and magnetic fields make each other, so they can travel as a wave at c = fλ. A wire is a wave guide with an impedance: any mismatch reflects part of the wave, and fast edges, not fast clocks, make a board behave like a transmission line. At high frequency current stays near the surface, antennas are sized by the wavelength (half or a quarter), and shields work when their holes are small against it.
Try this: look at the back of your Wi-Fi router, then at the car aerial or the FM radio's wire. The router's rod is a few centimetres long and the FM wire is about a metre: one is a half or quarter of 12 cm, the other of 3 m. Then check your microwave: shine a torch at the door and see the mesh.
Sources. Maxwell's equations, transmission-line theory, skin effect and dipole antenna theory are standard textbook material (Ulaby; Griffiths; Pozar; Balanis, Antenna Theory). Maxwell's field theory dates from 1861 to 1865 (see Wikipedia, "Maxwell's equations", which credits Heaviside with the modern vector form); Hertz's experiments 1886 to 1889 (Wikipedia, "Heinrich Hertz"). The speed of light exactly 299,792,458 m/s since the 1983 redefinition of the metre (17th CGPM, Resolution 1; BIPM and NIST). Copper skin depth of 8.4 mm at 60 Hz, 65 µm at 1 MHz, 2 µm at 1 GHz: Wikipedia, "Skin effect"; resistivities from standard tables (CRC Handbook). FM 87.5 to 108 MHz (88 to 108 in the Americas) and AM medium wave 531 to 1602 kHz (530 to 1700 in the Americas): ITU and Wikipedia, "FM broadcast band" and "Medium wave". EU EMC Directive 2014/30/EU and FCC 47 CFR Part 15: the regulations themselves (eCFR; EUR-Lex). The "tenth of a wavelength" and 0.35/rise-time rules are rules of thumb. Computed in node: every number in the interactives, including the bounce voltages, the 73.1 Ω and 1.64 dipole values, skin depths and the 16-fold AC resistance.