Maxwell's equations

By the 1860s, the mathematical unification of electricity, magnetism, and light was waiting to be written. Michael Faraday's experimental work had established that changing magnetic fields induce electric currents, and changing electric fields produce magnetic effects. Hans Christian Ørsted had shown the connection between electricity and magnetism in 1820. The vector calculus developed by Hamilton and others provided the mathematical vocabulary. James Clerk Maxwell assembled these components into the four equations that would reshape physics.

Maxwell's genius lay not in discovering new phenomena but in recognizing that existing experimental results, properly expressed mathematically, revealed a deeper unity. Working at King's College London and later at his family estate Glenlair in Scotland, Maxwell translated Faraday's intuitive 'lines of force' concept into rigorous mathematical form. Where Faraday visualized fields, Maxwell quantified them.

The key insight came in 1861: Maxwell added a 'displacement current' term to Ampère's law—a mathematical necessity to maintain consistency, not an experimental discovery. This addition had a stunning consequence. When Maxwell worked through the mathematics, he found that his equations predicted waves of oscillating electric and magnetic fields propagating through space at a speed determined by two experimentally measured constants. That speed matched the measured speed of light almost exactly.

'We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena,' Maxwell wrote. Light itself was electromagnetic radiation. This was not a hypothesis—it emerged mathematically from the equations.

The full treatise, 'A Treatise on Electricity and Magnetism,' appeared in 1873, during Maxwell's tenure as the first Cavendish Professor at Cambridge. The work synthesized decades of experimental findings—Coulomb's law, Gauss's law, Faraday's law of induction, Ampère's circuital law—into a unified framework. The equations could be written in different forms: the original twenty equations, later condensed by Oliver Heaviside into the four vector equations used today.

The adjacent possible for this synthesis had been accumulating for decades. Ørsted's 1820 demonstration of electromagnetism, Faraday's 1831 discovery of electromagnetic induction, the development of vector calculus, and the precision measurement of electrical constants—each was necessary. Maxwell himself noted how dependent his work was on Faraday's experimental foundation.

Convergent development was occurring. Hermann von Helmholtz in Germany, working from different premises, was approaching similar conclusions about the relationship between light and electromagnetism. The mathematical tools existed; the experimental data existed; the conceptual framework was crystallizing across European physics.

Maxwell's equations predicted the existence of electromagnetic waves at frequencies other than visible light—a prediction confirmed by Heinrich Hertz in 1887, eight years after Maxwell's death. This confirmation opened the path to radio, radar, wireless communication, and the entire electromagnetic technology stack of the twentieth century. The equations also contained the seeds of special relativity: the speed of light appearing as a fundamental constant hinted at the spacetime transformations Einstein would formalize in 1905.

Richard Feynman later called Maxwell's discovery 'the most significant event of the 19th century.' The equations demonstrated that mathematical synthesis could reveal physical truths invisible to direct observation—a template for theoretical physics that persists today.