Maxwell's Equations: Unifying Electricity, Magnetism & Light

Electromagnetism & Relativity

Course 2 · Chapter 6 · Maxwell's Equations: Unifying Electricity, Magnetism & Light

Every chapter so far has handed James Clerk Maxwell a piece: Faraday's field concept (Ch.2), Ørsted and Ampère's current-creates-magnetism (Ch.3–4), and Faraday's own electromagnetic induction (Ch.5). This chapter is where Maxwell assembles all of it into one complete theory — and, in doing so, makes a real, genuinely startling discovery about what light actually is.

Maxwell's Four Equations

Between 1861 and 1865, Maxwell brought together everything electricity and magnetism had revealed since Coulomb's own 1785 law into four equations, each one a real, direct descendant of a discovery already covered in this course:

Equation	In Words	Builds On
Gauss's Law for Electricity	Electric field lines begin on positive charge and end on negative charge; total flux through any closed surface is proportional to enclosed charge	Coulomb's law (Ch.1)
Gauss's Law for Magnetism	No magnetic monopoles exist — net magnetic flux through any closed surface is always exactly zero	Magnetic field lines always form closed loops (Ch.4)
Faraday's Law	A changing magnetic field induces a circulating electric field	Electromagnetic induction (Ch.5)
Ampère–Maxwell Law	Electric currents AND changing electric fields both generate magnetic fields	Ørsted & Ampère (Ch.3–4), plus Maxwell's own real addition below

Maxwell's Own Real Contribution: Displacement Current

Three of these four laws already existed, in some form, before Maxwell. His own genuine, original contribution was the fourth equation's missing half: recognising that a changing electric field, even with no actual current flowing, produces a magnetic field — exactly the way an actual current does. Maxwell called this term "displacement current." Adding it restored a real, elegant symmetry to the whole theory: a changing magnetic field creates an electric field (Faraday, Ch.5), and now, a changing electric field creates a magnetic field too.

A Genuinely Startling Discovery: Light Is an Electromagnetic Wave

Maxwell's four equations, combined, predict that a changing electric and magnetic field can sustain each other, propagating outward as a real, self-sustaining wave — entirely independent of the specific charges or currents that first created it. Solving his own equations for the speed of this wave gives:

c = 1/√(μ₀ε₀)

Worked Example: Calculating the Speed of Light

Using μ₀ = 4π×10&supminus;&sup7; T·m/A (from Chapter 4) and ε₀ ≈ 8.854×10&supminus;¹² F/m (the electric permittivity of free space):

c = 1/√((4π×10&supminus;&sup7;) × (8.854×10&supminus;¹²))
c = 1/√(1.1127×10&supminus;¹&sup7;)
c ≈ 3.00×10&sup8; m/s

This number, calculated purely from electric and magnetic laboratory constants with no reference to light or optics at all, matched the already-known, independently measured speed of light exactly.

🔗 Maxwell's Own Real Words In his 1865 paper, "A Dynamical Theory of the Electromagnetic Field" — read to the Royal Society on 8 December 1864 and approved for publication on 15 June 1865 — Maxwell wrote directly of his own realisation: "The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws." Light itself, in other words, is nothing more or less than an oscillating electric and magnetic field — the same fields this entire course has covered from Chapter 1 onward, now understood to be responsible for vision itself.

Heinrich Hertz's Real Confirmation — and His Own Real Doubt

Maxwell's prediction remained theoretical until Heinrich Hertz, between 1886 and 1889, built real apparatus to test it directly: a spark-gap transmitter (two one-metre copper wires with a small gap between them, driven by roughly 30,000 volt pulses) and a separate receiving loop that produced its own visible spark when it detected the transmitted wave. Starting in November 1887, Hertz's real, published results confirmed that these waves travelled at a finite speed exactly matching the speed of light, and that they reflected, refracted, and polarised the same way visible light does — a full experimental confirmation of Maxwell's own theoretical prediction.

⚠ Hertz's Own Real, Ironic Underestimate When asked about the practical use of his own discovery, Hertz is genuinely, directly quoted as saying: "It's of no use whatsoever … this is just an experiment that proves Maestro Maxwell was right — we just have these mysterious electromagnetic waves that we cannot see with the naked eye. But they are there." Pressed further on possible applications, he reportedly answered simply, "Nothing, I guess." Within six years of Hertz's own real confirmation, Guglielmo Marconi had built the first practical wireless telegraphy system using exactly the waves Hertz had just called useless — the SI unit of frequency, the hertz (Hz), now honours the very discovery its own discoverer dismissed.

Hands-On Exercises

Exercise 1

Using the same formula c = 1/sqrt(mu0 epsilon0), verify the calculation yourself by computing mu0 x epsilon0 first, then taking the square root and its reciprocal, using mu0 = 4 x pi x 10^-7 and epsilon0 = 8.854 x 10^-12. Show each intermediate step.

→ Solution

Exercise 2

Explain, in your own words, what "displacement current" actually is, and why adding it to Ampere's law (Chapter 3-4) was necessary for Maxwell's four equations to have the symmetry needed to predict a genuinely self-sustaining electromagnetic wave.

→ Solution

Exercise 3

Compare Hertz's real, documented underestimate of his own discovery's usefulness to at least one other example from this two-course project of a scientist (or their contemporaries) failing to recognize the real significance of a discovery at the time it was made.

→ Solution

Quick Reference

Maxwell's four equations: Gauss's law (electricity), Gauss's law (magnetism), Faraday's law, Ampère–Maxwell law
Maxwell's own real addition: displacement current, a changing electric field also produces a magnetic field
Speed of electromagnetic waves: c = 1/√(μ₀ε₀) ≈ 3.00×10&sup8; m/s — exactly the known speed of light
Maxwell's 1865 paper: light itself is an electromagnetic wave, unifying electricity, magnetism, and optics
Hertz confirmed this experimentally, 1886–89, while genuinely doubting its own practical value; Marconi commercialised it within six years

Next chapter: The Real Crisis That Led to Relativity — where Maxwell's own theory of light, which this chapter has just unified with electricity and magnetism, turns out to conflict with something else physicists believed was equally certain, setting up the real crisis Einstein would eventually resolve.