James Clerk Maxwell · Science
How four compact equations captured everything electric and magnetic in the universe - and how Maxwell’s addition of a single missing term, the displacement current, completed them, made them consistent, and forced the existence of electromagnetic waves.
By the 1850s, electricity and magnetism were a sprawling jungle of experimental laws, each won by a different investigator. Coulomb had measured the force between charges. Gauss had related electric fields to the charges that produce them. Ampère had found how currents create magnetism. Faraday had discovered that a changing magnetic field induces a current. There were rules for capacitors, for coils, for magnets, for batteries - dozens of separate results, no single framework. It looked like a collection of distinct phenomena that happened to involve sparks and lodestones.
Maxwell’s supreme achievement was to gather this entire jungle into four equations. Not four equations about charges and wires, but four equations about fields - the electric field and the magnetic field at every point of space. Each equation states one fundamental fact about how these fields behave, and together they contain, as special cases, every law that the experimenters had found. This is unification of the highest order: a reduction of bewildering variety to a handful of exact, universal statements. After Maxwell, you did not need to remember Coulomb’s law and Ampère’s law and Faraday’s law as separate facts. You needed to remember four equations, and everything else followed.
Strip away the symbols and each of Maxwell’s four equations makes one plain physical claim. First: electric charges create electric fields that spread out from them - pile up charge, and field lines stream outward (this contains Coulomb’s and Gauss’s laws). Second: there are no magnetic charges; magnetic field lines form closed loops with no source (the no-monopoles law). Third: a changing magnetic field creates a swirling electric field around it - move a magnet through a coil and you drive a current (this is Faraday’s law of induction, the principle of every generator). Fourth: an electric current creates a magnetic field that loops around it - and, Maxwell added, so does a changing electric field (Ampère’s law, completed).
Look at the symmetry hidden in the third and fourth laws. A changing magnetic field makes an electric field; a changing electric field makes a magnetic field. Each kind of field, when it changes, gives birth to the other. This reciprocal creation is the engine of everything that follows. It means the two fields are not independent - they are partners in a single electromagnetic field, each able to call the other into being. And it hints at something extraordinary: if a changing electric field makes a magnetic field, and that changing magnetic field makes an electric field, and so on, the two could in principle keep regenerating each other endlessly, sustaining a self-propagating wave that needs no wire and no magnet to keep it going.
Once the displacement current was in place, Maxwell combined his four equations and asked what they permitted. The answer astonished him. The equations allowed a disturbance in which a changing electric field creates a magnetic field, whose change in turn recreates the electric field, the two fields leapfrogging through empty space, each sustaining the other - a self-propagating electromagnetic wave, needing no wire, no charge, no medium of moving matter to keep it alive. The equations even dictated how fast such a wave must travel: a speed fixed entirely by two constants that experimenters had already measured in the laboratory, one from electricity and one from magnetism.
When Maxwell put in those measured numbers and computed the speed, he got roughly 310,000 kilometres per second. The known speed of light was about 315,000 kilometres per second - the same number, within the errors of measurement. This was no coincidence; the agreement was far too close. Maxwell drew the only possible conclusion: light is an electromagnetic wave. Optics, the ancient science of light, was suddenly a branch of electromagnetism. A man working with chalk and equations had deduced the nature of light from the behaviour of magnets and currents - one of the supreme moments of theoretical prediction in the history of science. The next lesson follows this discovery in full.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that Maxwell’s equations bundle all of electromagnetism into four laws: charges make electric fields, there are no magnetic charges, changing magnetic fields make electric fields, and currents (plus changing electric fields) make magnetic fields. Explain in your own words what the displacement current is,…
Leads to Isaac Newton.
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