"From a long view of the history of mankind — seen from, say, ten thousand years from now — there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade." - Richard Feynman
I have so often talked about the 23-year old English gentleman who asked the question, "If an apple fall, does the moon also falls?" But Isaac Newton could not answer one question, why a piece of amber fricated on silk attracts a needle. Today, we all know that it is the effect of electrostatics. Another thing not understood in Newton's time was magnetism. In fact, magnetism have been known since the Ancient Greeks and Thales of Miletus is considered as one of the first person to have studied it. Yet, for more than 2000 years nobody could explain the phenomenon. However, Newton studied a subject very close to electricity and magnetism, that is, optics. Optics, the physics of light. How are magnets, electricity and light related to each other? This question is answered by the Maxwell's equations.
Albert Einstein kept in his study wall portraits of isaac newton, Michael Faraday and James Clerk Maxwell. James Clerk Maxwell, largely unknown to the general public, is considered as the greatest theoretical physicist to have lived in the era between Newton and Einstein. But first of all, we will need to look at the work of an experimentalist, Michael Faraday. Faraday was a self-educated English physicist. Working on the works of French physicists Charles-Augustin de Coulomb and André-Marie Ampère, American polymat Benjamin Franklin, amongst others, he did a lot of physical and chemical experiments. But we can all agree that his most important work was to find a link between electricity and magnetism. This link is what runs our economy. I was asked a few weeks ago by an economics student why I say that "Faraday's law of electromagnetic induction" runs our economy. Because, one of the most important resources in the modern world is electricity. In fact, without electricity there would not have been any industrial revolution. There is a legend that once, the British minister of finance asked Faraday about the practical value of electricity, he answered, "Why, sir, there is every probability that you will soon be able to tax it." I can't guarantee how true this story is but I can guarantee that Michael Faraday was one of the few legendary scientists(includes Charles Darwin) not good at maths. Faraday could not come up with a mathematical description of his discovery. This was done a few decades later by Scottish physicist James Clerk Maxwell.
In order to do so, first Maxwell had to introduce the hypothesised displacement current and then he found the symmetry between electricity and magnetism. From that, he derived a set of about 20 differential equations which explain the two phenomenon together. In the years 1861-1862 he published a set of papers called On Physical Lines of Force. He also concluded that light was an electromagnetic wave from the solutions of his equations. But it wasn't until his next paper in 1864, A Dynamical Theory of the Electromagnetic Field that Maxwell used this displacement current term to derive the electromagnetic wave equation.
"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"
- James Clerk Maxwell (On Physical Lines of Force)
Later, English physicist Oliver Heaviside reformulated Maxwell's equations using vector analysis into a set of four equations. Maxwell's equations have been consistent with experiments and are the basis of the theory of classical electrodynamics. In the beginning of the twentieth century, a few problems arose. Maxwell's equations were not invariant under Galilean transformation. And for a theory to be true, it needs to work anywhere else in the universe and at any time. Hendrik Lorentz came up with transformations in which the Maxwell's equations were invariant. Of course, it was assumed that something must be wrong with Maxwell's equations rather than with Galileo's equivalence principle. However, in 1905, Albert Einstein published a paper, On the electrodynamics of moving bodies, in which he declared that Maxwell's equations are right, the Lorentz transformations are right but for that we need to redefine our equivalence principle and laws of motion. If time and space are not absolute and the speed of light is constant, all the asymmetries are solved. Special relativity was born. Experiments proved Einstein right.
Today, Maxwell's equations have been reduced into a set of two equations using exterior algebra and differential topology. And they are the basis of classical electromagnetism. On a side note, I would like to add that James Clerk Maxwell and photographer Thomas Sutton took the first colour photography in 1861. And he also studied the kinetic theory of gases.