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By the mid-19th century the wave theory of light was experimentally established (lesson 2). The next question was: what kind of wave is light, and in what medium does it propagate? The answer — that light is a self-supporting oscillation of electric and magnetic fields, propagating through vacuum at a speed determined by two laboratory-measurable electromagnetic constants — was given by James Clerk Maxwell in the 1860s and confirmed experimentally by Heinrich Hertz in 1887. The Maxwell-Hertz synthesis is one of the cleanest examples in physics of a theoretical prediction (an entire family of waves at all frequencies, of which visible light is a narrow band) followed by laboratory confirmation a generation later. It transformed physics from a collection of separate phenomena — electricity, magnetism, optics — into a single unified theory of the electromagnetic field, and it set up almost every 20th-century technology that uses electromagnetic radiation: radio, microwaves, television, mobile telephony, radar, fibre-optic communications.
Spec mapping: This lesson covers part of AQA 7408 section 3.12.2 — Maxwell's prediction of electromagnetic waves travelling at c = 1/√(ε₀μ₀), the unification of light with electromagnetism, Hertz's 1887 experimental demonstration of radio waves, and the historical relationship between theoretical prediction and experimental confirmation. (Refer to the official AQA specification document for exact wording.)
Synoptic links:
- Section 3.7 (electric and magnetic fields): ε₀ (permittivity of free space) appears in Coulomb's law and capacitor formulae; μ₀ (permeability of free space) appears in the magnetic field of a current-carrying wire. Both are measured in essentially static laboratory experiments — yet their combination yields the speed of light. The bridge from static fields to wave propagation is the deep result.
- Section 3.3 (waves): EM waves are transverse, with E and B perpendicular to the direction of propagation and to each other. The speed v = fλ relation and the wave equation v² = 1/(εμ) generalise the wave-on-a-string c² = T/μ that students meet in Year 12.
- Section 3.12.2 (photoelectric effect): Maxwell's wave theory was so spectacularly successful between 1865 and 1900 that the failure of the wave picture to explain the photoelectric effect (lesson 4) was an enormous shock. Without Maxwell's triumph there is no photoelectric "crisis."
By 1860, four major experimental laws of electromagnetism had been established by Ampère, Faraday, Gauss and others:
| Law | Statement (informal) |
|---|---|
| Gauss's law (electric) | Electric flux through a closed surface = enclosed charge / ε₀ |
| Gauss's law (magnetic) | Magnetic flux through a closed surface = 0 (no magnetic monopoles) |
| Faraday's law | A changing magnetic flux induces an EMF (electric field circulation) |
| Ampère's law | Magnetic field circulation = μ₀ × enclosed current |
These four were known in pieces and used in practical engineering — telegraphy, electromagnets, induction motors — but they were not yet a theory in the sense of a small set of equations from which all phenomena could be derived. They contained subtle internal inconsistencies, the most important of which Maxwell would fix.
The key inconsistency concerned Ampère's law in the case of a charging capacitor. Consider a parallel-plate capacitor being charged through a wire. Ampère's law says the magnetic field around the wire is related to the current passing through any surface bounded by the loop around the wire. But you can choose a surface that passes between the capacitor plates — through which no actual current flows, yet the magnetic field is the same as if you had chosen a surface intersecting the wire. The law is inconsistent: it gives different B for different surfaces bounded by the same loop.
Maxwell's 1861-65 contribution was to add a new term to Ampère's law: the displacement current, proportional to the rate of change of electric flux. The amended Ampère-Maxwell law reads (informally):
Magnetic field circulation = μ₀ × (enclosed real current + ε₀ × rate of change of electric flux)
The new ε₀ × dΦ_E/dt term — which has the units of a current but is not a transport of charge — fixes the surface-choice inconsistency: between the capacitor plates, the electric field is changing as charge builds up, and the displacement-current term contributes exactly the same magnetic field as the real current does on the wire side.
This looks like a minor mathematical tidying. It is not. The displacement current produces the symmetry between electric and magnetic fields that Maxwell needed for his deepest result.
Faraday: changing B → circulating E (induction).
Maxwell: changing E → circulating B (displacement current).
A changing electric field generates a magnetic field; a changing magnetic field generates an electric field. So a disturbance in either field can sustain itself by continually generating the other. The mathematical consequence — derived in A Dynamical Theory of the Electromagnetic Field (1865) and developed in A Treatise on Electricity and Magnetism (1873) — is that the equations support travelling-wave solutions in which E and B oscillate perpendicular to each other and to the direction of propagation, at a speed:
c = 1/√(ε₀μ₀)
Substituting the laboratory values measured by Weber and Kohlrausch in the 1850s (ε₀ ≈ 8.85 × 10⁻¹² F m⁻¹; μ₀ = 4π × 10⁻⁷ H m⁻¹, exact by old definition):
c = 1/√[(8.85 × 10⁻¹²) × (4π × 10⁻⁷)] = 1/√(1.112 × 10⁻¹⁷) = 1/(3.34 × 10⁻⁹) ≈ 3.00 × 10⁸ m s⁻¹
This was, at the time, exactly the speed of light as measured by Fizeau (1849) and Foucault (1850-62) — within experimental error. Maxwell drew the conclusion that is now textbook: light is an electromagnetic wave. Two phenomena that had been studied separately for 150 years — electricity-magnetism and optics — were the same thing.
Maxwell wrote in 1865 that the agreement between the measured speed of light and the value of 1/√(ε₀μ₀) computed from electromagnetic measurements was so close that it could scarcely be regarded as accidental. The unification of light and electromagnetism is, in his words, the central conclusion of the theory.
Maxwell died in 1879 without seeing his prediction directly tested. The prediction was that all frequencies of EM wave should propagate at c — visible light was just a narrow band, and waves of much lower frequency should be producible by purely electrical means. Heinrich Hertz, working at Karlsruhe in 1886-88, set out to make and detect them.
Hertz's transmitter was an induction coil driving a spark gap between two metal spheres connected to two short wires (an antenna), with two larger spheres attached to the wire ends to store charge. When the coil charged the system to breakdown voltage, a brief, highly damped oscillating current flowed across the spark gap at the antenna's natural electrical resonance frequency — roughly 50 MHz to 500 MHz, depending on the antenna geometry, corresponding to wavelengths from about 0.6 m to 6 m.
His detector was a resonant loop of wire bent into an open circle with a small spark gap of its own. When the loop was placed several metres from the transmitter, faint sparks appeared in the detector gap each time the transmitter fired — only when the loop's geometry was tuned to the transmitter's frequency, only when the orientation matched, and only when no metal shielding intervened.
graph LR
A["Induction coil<br/>(~kV pulses)"] --> B["Spark gap S₁<br/>+ antenna spheres"]
B -.->|"EM radiation<br/>at antenna<br/>resonance"| C["Resonant loop<br/>antenna"]
C --> D["Detector spark gap S₂"]
style B fill:#ef4444,color:#fff
style C fill:#1d4ed8,color:#fff
style D fill:#27ae60,color:#fff
Hertz's experiments demonstrated:
Each of these properties is shared with visible light. The result, by 1888, was overwhelming experimental confirmation of Maxwell's prediction: light and Hertz's "radio waves" are the same kind of wave, differing only in frequency.
Within a decade of Hertz's confirmation, Guglielmo Marconi had turned the discovery into a working wireless telegraph (first patent 1896; first transatlantic transmission 1901). The path from Maxwell's 1865 equations to Marconi's 1901 signal across the Atlantic — about 36 years — is one of physics's fastest journeys from pure theoretical prediction to commercial technology. By 1920, broadcast radio existed in the US; by 1936 the BBC was broadcasting television; by 1946 radar had been the decisive technology of World War II. Every one of these is a direct descendant of Hertz's spark and Maxwell's equation.
Calculate the speed of EM waves in vacuum using ε₀ = 8.854 × 10⁻¹² F m⁻¹ and μ₀ = 4π × 10⁻⁷ H m⁻¹.
Solution.
ε₀μ₀ = (8.854 × 10⁻¹²) × (4π × 10⁻⁷) = (8.854 × 10⁻¹²) × (1.2566 × 10⁻⁶) = 1.1127 × 10⁻¹⁷
c = 1/√(1.1127 × 10⁻¹⁷) = 1/(3.3358 × 10⁻⁹) = 2.998 × 10⁸ m s⁻¹
This matches the modern defined value c = 2.998 × 10⁸ m s⁻¹ (since 1983 the metre is defined in terms of c, so this equality is now exact by definition).
A Hertz-style antenna consists of two metal rods, each 0.75 m long, end-to-end with a spark gap in the middle. The dipole behaves as a half-wave antenna with the full antenna length L = 1.50 m corresponding to one half wavelength, λ/2 = 1.50 m.
(a) What is the wavelength of the EM radiation produced? (b) What is the frequency, assuming propagation at c?
Solution.
(a) λ = 2L = 2 × 1.50 = 3.00 m.
(b) f = c/λ = (3.00 × 10⁸)/3.00 = 1.00 × 10⁸ Hz = 100 MHz.
This falls within the VHF range (today's commercial FM broadcasting), much lower frequency than visible light (~5 × 10¹⁴ Hz). Hertz's actual frequencies were in this rough region — his antennas were geometrically very similar to a modern FM transmitter.
Maxwell's theory predicts EM waves at any frequency, since c is independent of f and the wave equation has no preferred scale. The EM spectrum, ordered by increasing frequency, is:
| Band | Frequency (Hz) | Wavelength | Typical source / use |
|---|---|---|---|
| Radio | 10⁴ – 10⁹ | 10⁴ – 0.3 m | Broadcast, mobile telephony, Wi-Fi |
| Microwave | 10⁹ – 10¹² | 0.3 m – 0.3 mm | Radar, microwave ovens, satellite links |
| Infrared | 10¹² – 4 × 10¹⁴ | 0.3 mm – 700 nm | Thermal radiation, IR cameras |
| Visible | 4 – 7.5 × 10¹⁴ | 700 – 400 nm | Atomic transitions in outer-shell electrons |
| Ultraviolet | 10¹⁵ – 10¹⁶ | 400 – 10 nm | UV lamps, sunburn, astronomy |
| X-ray | 10¹⁶ – 10¹⁹ | 10 nm – 10 pm | Inner-shell transitions, medical imaging |
| Gamma | > 10¹⁹ | < 10 pm | Nuclear transitions, cosmic-ray events |
The whole spectrum spans more than 24 orders of magnitude in frequency. Every part is described by the same Maxwell equations, propagates at the same speed in vacuum, and shows the same general phenomena (reflection, refraction, diffraction, interference, polarisation). Hertz's spark transmitter in 1887 and a modern 5G mobile-phone tower at 28 GHz produce qualitatively the same kind of wave as a candle flame.
Maxwell's 1865 derivation showed that electromagnetic disturbances must propagate at a speed determined entirely by two electromagnetic constants — the permittivity of free space ε₀ and the permeability of free space μ₀:
c = 1 / √(ε₀ μ₀)
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