Electromagnetic Waves

Ampere's circuital law connects the magnetic field circulating around a closed loop to the conduction current I_c passing through any surface bounded by that loop: ∮B·dl = μ₀I_c. This works perfectly for steady, unbroken currents such as the current in a long straight wire. But consider a wire carrying a charging current into one plate of a capacitor. If we choose a flat surface that cuts straight through the wire, the enclosed current is I. If instead we choose a bulged surface that passes through the gap between the plates (where no charge actually crosses), the enclosed conduction current is zero. Ampere's law, applied to the same loop, now gives two different answers for the same magnetic field — a contradiction.

Maxwell resolved this by noticing that although no charge flows across the gap, the electric field E between the plates is changing with time as charge accumulates, so the electric flux Φ_E through the bulged surface is also changing. He proposed that a time-varying electric flux is exactly as effective at producing a magnetic field as a conduction current. He defined a new quantity, the displacement current, I_d = ε₀(dΦ_E/dt), and generalised Ampere's law to the Ampere–Maxwell law: ∮B·dl = μ₀I_c + μ₀ε₀(dΦ_E/dt). With this correction, both surfaces give the same, consistent magnetic field, because on the bulged surface I_c = 0 but I_d = I, while on the flat surface I_c = I and I_d = 0.

Displacement current is not a flow of charge; it is a bookkeeping device that makes the total current (conduction + displacement) continuous around any closed circuit, even where an actual gap exists. It has the same units as current (ampere) and produces a genuine magnetic field, but it involves no moving charge and no heat dissipation (no I²R loss). This single correction was the missing piece that let Maxwell show that electric and magnetic fields could sustain each other in empty space, travelling as a wave — the electromagnetic wave.

Worked example. A parallel-plate capacitor with circular plates of radius R = 6.0 cm is being charged by a constant current I = 0.15 A. Find the rate of change of the electric field between the plates and confirm the displacement current equals the charging current.

1. Plate area: A = πR² = π(0.06)² = 1.131 × 10⁻² m².
2. The field between the plates is E = Q/(ε₀A), so dE/dt = (dQ/dt)/(ε₀A) = I/(ε₀A) = 0.15/(8.85 × 10⁻¹² × 1.131 × 10⁻²) ≈ 1.5 × 10¹² V/(m·s).
3. Displacement current: I_d = ε₀A(dE/dt) = ε₀A × I/(ε₀A) = I = 0.15 A.

So I_d exactly equals the conduction current I flowing in the wires — this is true in general, not just for this geometry, and is what keeps the current continuous through the capacitor gap.

Maxwell combined Gauss's law for electricity, Gauss's law for magnetism, Faraday's law of electromagnetic induction, and his own Ampere–Maxwell law into a single consistent set of equations. Faraday's law says a changing B produces an E; the Ampere–Maxwell law says a changing E produces a B. Together, these mean that in a region with no charges or conduction currents, an oscillating electric field creates an oscillating magnetic field, which in turn recreates an oscillating electric field a little further away, and so on — the disturbance propagates outward on its own, without needing a medium. This self-sustaining disturbance is an electromagnetic wave, and Maxwell calculated its speed in vacuum from purely electric and magnetic constants: c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s — a value that matched the already-measured speed of light so closely that Maxwell concluded light itself is an electromagnetic wave. Electromagnetic waves are produced whenever a charge is accelerated, most commonly by charges oscillating in a circuit or antenna; the German physicist Heinrich Hertz first generated and detected such waves (radio waves) in the laboratory in 1887, experimentally confirming Maxwell's theory.

A plane electromagnetic wave has three key features. First, it is transverse: the electric field E and magnetic field B oscillate perpendicular to the direction in which the wave travels (unlike sound, which is longitudinal). Second, E and B are also perpendicular to each other, and the direction of propagation is along E × B. Third, E and B oscillate in phase — they reach their maximum and zero values at the same instant and the same point in space — with their amplitudes always related by E₀ = cB₀. Electromagnetic waves carry energy, stored partly in the electric field and partly in the magnetic field; at every instant the two contributions are equal (u_E = u_B), and the total energy density is u = ½ε₀E² + B²/(2μ₀). The time-averaged rate at which this energy crosses unit area (the intensity) is I = ½ε₀cE₀². Electromagnetic waves also carry momentum, so they exert a (very small) radiation pressure on any surface they strike.

Worked example 1. The electric field amplitude of a plane electromagnetic wave in vacuum is E₀ = 60 V/m. Find the amplitude of the magnetic field.

B₀ = E₀/c = 60/(3 × 10⁸) = 2.0 × 10⁻⁷ T = 200 nT. Note how small B₀ is compared with E₀ numerically — this is simply because c is a large number; the energies stored in the E and B fields are nevertheless equal.

Worked example 2. An FM radio wave has frequency 91 MHz. Find its wavelength using c = νλ.

λ = c/ν = (3 × 10⁸)/(91 × 10⁶) ≈ 3.3 m. This wavelength (a few metres) is why FM radio antennas are of that order in length.

Electromagnetic waves exist over an enormous continuous range of wavelengths and frequencies, collectively called the electromagnetic spectrum. Moving from long wavelength/low frequency to short wavelength/high frequency, the spectrum is broadly divided into radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays. These bands overlap and have no rigid boundaries; a wave is classified mainly by how it is produced and detected. All of them travel at the same speed c in vacuum and differ only in wavelength/frequency (and hence photon energy).

Radio waves have wavelengths roughly from about 0.1 m to several hundred metres (frequencies from a few hertz up to about 10⁹ Hz). They are generated by accelerating charges in a conducting wire or antenna — an oscillating current in a circuit — and are used for AM/FM radio and television transmission. Longer-wavelength radio waves (AM band) travel further by reflecting off the ionosphere, while shorter waves (FM/TV band, VHF/UHF) travel more by line of sight.

Microwaves have shorter wavelengths, roughly from about 1 mm to 0.1 m (frequencies of order 10⁹–10¹¹ Hz). They are produced by special vacuum-tube devices such as klystrons, magnetrons and Gunn diodes rather than ordinary circuits, because such short wavelengths need very rapidly oscillating currents. Microwaves are used in radar systems (for detecting the speed and location of objects such as aircraft and vehicles), microwave ovens (their frequency is tuned to be absorbed efficiently by water molecules in food, heating it), and in analysing the fine structure of atoms and molecules.

Worked example. A microwave oven operates at a frequency of 2.45 GHz. Find its wavelength, and state which spectral band this falls in.

λ = c/ν = (3 × 10⁸)/(2.45 × 10⁹) ≈ 0.122 m ≈ 12.2 cm. Since this lies between about 1 mm and 0.1 m (up to a few tens of cm, by common convention), it is squarely in the microwave region — consistent with its use in microwave ovens.

Infrared (IR) waves have wavelengths roughly from 700 nm up to about 1 mm. They are produced by hot bodies and by vibrating/rotating molecules, and are often called heat waves because they are readily absorbed by most materials, increasing their thermal motion (heating them). The Sun is a strong source of infrared. Applications include remote controls, thermal/night-vision imaging, physiotherapy for muscular treatment, and infrared astronomy for studying cool objects hidden behind interstellar dust. Infrared radiation is also central to the greenhouse effect: incoming sunlight (mostly visible) warms the Earth's surface, which re-radiates the absorbed energy as infrared; greenhouse gases such as CO₂ and water vapour absorb part of this outgoing infrared and re-emit it, trapping heat in the atmosphere.

Visible light occupies a very narrow band, roughly 400 nm (violet) to 700 nm (red), and is the only part of the spectrum the human eye can detect. It is produced mainly by electrons making transitions between energy levels in atoms and by hot, glowing bodies. Different wavelengths within this band are perceived as different colours, and a mixture of all of them appears white. Because it is such a small slice of the full spectrum, most of the universe's electromagnetic activity is invisible to us and can only be studied using instruments sensitive to other bands.

Ultraviolet (UV) radiation has wavelengths from about 1 nm to 400 nm, produced by very hot objects (like the Sun), specially designed lamps, and electric discharges through gases. UV carries more energy per photon than visible light, is capable of ionising atoms, and can damage living cells — prolonged exposure causes sunburn, premature skin ageing and can trigger skin cancer, and it can harm the eyes. Useful applications include sterilising water and surgical instruments (its germicidal effect kills bacteria), checking for forged banknotes/documents (fluorescence), LASIK eye surgery, and detecting invisible ink/fingerprints. Most of the harmful UV from the Sun is absorbed by the ozone (O₃) layer in the upper atmosphere before it reaches the ground, which is why depletion of the ozone layer (caused by chlorofluorocarbons, CFCs) is an environmental concern.

Worked example. A hot filament radiates strongly at a wavelength of 550 nm. Find the frequency of this radiation and identify which part of the spectrum it belongs to.

ν = c/λ = (3 × 10⁸)/(550 × 10⁻⁹) ≈ 5.45 × 10¹⁴ Hz. Since 550 nm lies between 400 nm and 700 nm, this radiation is visible light — it would appear green to the eye.

X-rays have wavelengths roughly from 10⁻⁸ m down to about 10⁻¹³ m (0.0001 nm to 10 nm), making them far more energetic than UV. They are produced when high-speed electrons, accelerated through a large potential difference in an X-ray tube, are suddenly decelerated on striking a heavy metal target. Because of their short wavelength, X-rays can penetrate soft tissue but are absorbed more by denser matter like bone, which is why they are used for medical diagnostic imaging (radiography) of bones and teeth, for airport baggage/security scanning, and in industry and crystallography to study the atomic arrangement inside crystals. X-rays are ionising radiation, so excessive or repeated exposure can damage or kill living cells; their medical use is always kept to the minimum necessary dose with proper shielding for staff.

Gamma rays lie at the extreme high-frequency, short-wavelength end of the spectrum, with wavelengths below about 10⁻¹⁰–10⁻¹⁴ m, overlapping the X-ray region but generally shorter. Unlike X-rays, which come from electron processes, gamma rays originate in the nucleus — emitted during radioactive decay of unstable nuclei and in nuclear reactions. They are the most penetrating and energetic of all electromagnetic radiations, requiring thick shielding (lead, concrete) to stop. Controlled use of gamma radiation includes cancer radiotherapy (destroying malignant cells), sterilising medical equipment and food, and gamma-ray astronomy, but uncontrolled exposure is highly hazardous to living organisms because of its strong ionising power.

Taken together, the full electromagnetic spectrum — radio, microwave, infrared, visible, ultraviolet, X-ray and gamma ray — is a single family of transverse waves differing only in wavelength/frequency, all travelling at speed c in vacuum, all obeying c = νλ, and all originating from accelerated charges (in circuits, atoms or nuclei, depending on the band). As frequency increases across this list, wavelength decreases and the radiation generally becomes more penetrating and more capable of ionising matter, which is why the higher bands (UV, X-ray, gamma) need careful handling while the lower bands (radio, microwave, infrared) are comparatively benign in normal use.

Worked example. A gamma ray emitted during a radioactive decay has wavelength 2 × 10⁻¹³ m. Compare its frequency with that of a typical X-ray of wavelength 2 × 10⁻¹⁰ m.

ν(gamma) = c/λ = (3 × 10⁸)/(2 × 10⁻¹³) = 1.5 × 10²¹ Hz. ν(X-ray) = (3 × 10⁸)/(2 × 10⁻¹⁰) = 1.5 × 10¹⁸ Hz. The gamma ray's frequency is 1000 times higher, consistent with gamma rays generally being far more energetic and penetrating than X-rays.