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James Clerk Maxwell Sticker

Field Goals Achieved | Physics

\[\nabla\cdot\vec E = \frac{\rho}{\varepsilon_0}, \quad \nabla\cdot\vec B = 0\]
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Four equations, and light turned out to be electromagnetism. Maxwell unified electricity, magnetism and light: his equations predict waves that travel at the speed of light. A pixel-art portrait on a 3" × 3" sticker; the physics panel below has the equation, every symbol and the sources.

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3″ × 3″ kiss-cut sticker.

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The physics

Maxwell added one missing term, the displacement current, to the known laws of electricity and magnetism. With it the four equations are consistent and predict waves moving at the speed of light, which showed that light itself is an electromagnetic wave.

The sticker says Field Goals Achieved. The goal was a missing term.

By the 1860s the laws of electricity and magnetism were known piece by piece. Charges make electric fields. There are no magnetic charges. Changing magnetic fields make electric fields, which was Faraday's discovery. Currents make magnetic fields, which was Ampère's. James Clerk Maxwell noticed that Ampère's law, as written, broke the conservation of charge wherever charge piles up, as on the plates of a capacitor. He repaired it by adding a term: a changing electric field makes a magnetic field too, just as a current does. It is the last term of the second line on this page.

With that term the equations do something new. A changing electric field makes a changing magnetic field, which makes a changing electric field, and the disturbance carries itself through empty space. Maxwell worked out its speed from two constants measured on a bench with coils and capacitors, and the answer matched the measured speed of light. In his 1865 paper he concluded that light is an electromagnetic disturbance. The third formula is that calculation. Heinrich Hertz generated and detected such waves with spark gaps in the late 1880s, which is where radio begins.

Two footnotes. The tidy four-line vector form is not how Maxwell wrote it; Oliver Heaviside and others condensed his longer set afterwards. And since 2019 the logic runs the other way round: the speed of light is exact by definition, ε₀ and μ₀ are measured, and only their product is fixed.

Maxwell also studied colour vision; in 1861 he presented a colour photograph of a tartan ribbon, made from three exposures through red, green and blue filters.

He did not only describe fields. He found what they were missing, and it was light.

Equations

\[\nabla\cdot\vec E = \frac{\rho}{\varepsilon_0}, \quad \nabla\cdot\vec B = 0\]
\[\nabla\times\vec E = -\frac{\partial\vec B}{\partial t}, \quad \nabla\times\vec B = \mu_0\vec J + \mu_0\varepsilon_0\frac{\partial\vec E}{\partial t}\]
\[c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}\]
SymbolMeaningUnit
\(\vec E\) electric field \(\mathrm{V/m}\)
\(\vec B\) magnetic flux density \(\mathrm{T}\)
\(\rho\) charge density \(\mathrm{C/m³}\)
\(\vec J\) current density \(\mathrm{A/m²}\)
\(\varepsilon_0, \mu_0\) vacuum permittivity, permeability \(\mathrm{F/m, N/A²}\)
\(c\) speed of light \(\mathrm{m/s}\)

Sources

  1. Maxwell (1865) A dynamical theory of the electromagnetic field, Phil. Trans. R. Soc. 155, 459
  2. Encyclopaedia Britannica — James Clerk Maxwell
  3. Encyclopaedia Britannica — Maxwell’s equations
  4. Feynman Lectures on Physics Vol. II Ch. 18: The Maxwell Equations
  5. NIST CODATA 2022 — vacuum electric permittivity ε0
  6. NIST CODATA 2022 — vacuum magnetic permeability μ0
  7. NIST CODATA 2022 — speed of light in vacuum c

Source Maxwell, J. C. (1865). A dynamical theory of the electromagnetic field. Philosophical Transactions of the Royal Society of London, 459–512.doi.org/10.1098/rstl.1865.0008

Field notes

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