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In the electromagnetic waves chapter of Class 12 physics, you are told that light is an electromagnetic wave and that its speed is c = 1/√(μ₀ε₀). It can feel like a formula handed down from nowhere. It came from one paper, published in 1865, in which a Scottish physicist noticed that two numbers from very different experiments were almost the same. This post covers what James Clerk Maxwell actually wrote, how his equations became the four in your textbook, and how to get the speed of light yourself.
Who was James Clerk Maxwell?
Maxwell was born in Edinburgh on 13 June 1831 and died in Cambridge on 5 November 1879, according to his MacTutor biography at the University of St Andrews. He worked on colour vision, the rings of Saturn and the kinetic theory of gases as well as electromagnetism. In 1871 he became the first Cavendish Professor of Physics at Cambridge and designed the Cavendish Laboratory, which opened in 1874.
Before Maxwell, Michael Faraday had described electric and magnetic effects using lines of force. Maxwell's aim was to turn those pictures into mathematics that could make predictions.
What did Maxwell's 1865 paper claim?
The paper is A Dynamical Theory of the Electromagnetic Field, published in the Philosophical Transactions of the Royal Society in 1865. In it Maxwell set out the full system. In his own words, "Between these twenty quantities we have found twenty equations", covering magnetic force, currents, electromotive force and more. One of his quantities was the "total current (including variation of displacement)": the idea that a changing electric field acts like a current. This displacement current is what lets electric and magnetic fields keep regenerating each other in empty space.
From these equations he found that a disturbance would travel as a wave with a speed v, which could be worked out from purely electrical measurements. Weber and Kohlrausch had measured the ratio that gives this speed. Maxwell quoted their value as 310,740,000 metres per second, and set it beside measured speeds of light: 314,858,000 m/s from Fizeau and 298,000,000 m/s from Foucault.
He then wrote: "This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws."
How did twenty equations become four?
Maxwell wrote his equations in components, one equation per direction, with potentials as extra variables. The compact version came later. According to MacTutor's biography of Oliver Heaviside, Heaviside, a former telegraph operator, "reduced Maxwell's 20 equations in 20 variables to 4 equations in 2 variables" using vector notation. Those four are what you now call Maxwell's equations: Gauss's law for electricity, Gauss's law for magnetism, Faraday's law and the Ampère–Maxwell law.
The prediction still needed testing. Heinrich Hertz, working at Karlsruhe, produced and detected long-wavelength electromagnetic waves in 1888, as his MacTutor biography records. Radio, which you use every day, is the practical result.
Worked example: the speed of light from two constants
Find the speed of an electromagnetic wave in vacuum from μ₀ and ε₀.
Take μ₀ ≈ 4π × 10⁻⁷ N A⁻² and ε₀ ≈ 8.85 × 10⁻¹² F m⁻¹. Both are close to the current NIST values.
Multiply: μ₀ε₀ = 4π × 10⁻⁷ × 8.85 × 10⁻¹² ≈ 1.112 × 10⁻¹⁷ s²/m².
Take the square root: √(1.112 × 10⁻¹⁷) ≈ 3.335 × 10⁻⁹ s/m.
Invert: c ≈ 1 / (3.335 × 10⁻⁹) ≈ 3.00 × 10⁸ m/s.
The exact value of c is 299,792,458 m/s. Maxwell's figure from Weber and Kohlrausch, 310,740,000 m/s, was about 3.7% too high, which was close enough for him to see the link.
Common mistakes to avoid:
Forgetting the square root. 1/(μ₀ε₀) gives about 9 × 10¹⁶, which is c², not c.
Dropping the 4π. Using μ₀ = 10⁻⁷ gives a speed about 3.5 times too large.
Mixing up units. Check that μ₀ε₀ comes out in s²/m². Our guide to dimensional analysis as a fast check shows how.
A useful extension: in a material, c is replaced by v = 1/√(μ₀μᵣε₀εᵣ), so the refractive index is n = √(μᵣεᵣ). For most transparent materials μᵣ ≈ 1, so n ≈ √εᵣ, provided you use εᵣ at the frequency of light.
Seeing the wave, not just the formula
The hardest part of this chapter is picturing an electric field and a magnetic field at right angles, both oscillating and moving forward together. Drawing it once in your notebook helps. Watching it change helps more. The Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read, which helps when a static diagram is not enough. Our post on learning waves through simulation explains what to look for.
What can a student learn from how Maxwell worked?
Compare numbers across topics. Maxwell's insight came from putting an electrical result next to an optical one. When two chapters give you the same number, ask why.
Write the whole system down. He listed every quantity and every equation and checked that they were enough to determine the rest. When you solve a problem, list your unknowns and your equations before you start.
Expect your notation to improve. Heaviside made Maxwell's work far easier to use without changing the physics. Rewriting a messy derivation in cleaner form is real understanding, not decoration.
Tonight, compute c from μ₀ and ε₀ by hand, then write one sentence on why a changing electric field must produce a magnetic field.
