Maxwell Equations – the First Unification

(Last Updated On: January 27, 2018)

Anyone who has any association with physics has surely heard about, read, studied and solved the 4 Maxwell Equations of Electromagnetism. Maxwell built upon the works of his predecessors, such as Ampere and Faraday, and gave a set of relations that unified the realms of optics, electricity and magnetism into one – electrodynamics.


Maxwell Equations


Maxwell’s Equations incorporate Gauss Law, Ampere’s Law and Faraday’s Law.


Maxwell equations are essentially a set of four equations that co-relate electric and magnetic fields. Long before, electricity and magnetism were thought to be of different realms of science – one created lightning in the sky, the other moved iron nails. It was so because there was no experimental correlation at that time that the fields are actually unified, and more glaringly, while you could obtain free electric charges, positive and negative, free magnetic charges or monopoles, had not been observed (and still have not been)


It all started to change with a fateful experiment conducted by Hans Oersted. He found out that a current carrying wire causes the needle of a magnetic compass to deflect, thus giving the first signs that electricity and magnetism may not be that different. This was the basis of Ampere Circuital law, that could relate mathematically the field generated by a current carrying wire and the current. The next piece of evidence came from  Michael Faraday, who observed how dragging a loop of wire through a magnetic field could generate electricity.

So the basics were there – magnetic as well as electric fields could be generated via one another, and by then Gauss’ Law in electricity and magnetism, which could calculate the field due to an electric charge, was joined by Ampere Circuital Law and Faradays’ Law of Electromagnetic Induction. However, there was much to be done – Ampere’s law was not completely accurate, and gave incorrect results based on what loop you chose to calculate the net magnetic flux. It was James Clerk Maxwell, who then set on to unify these equations into one, and also rectify Ampere’s Law.

Maxwell included the term J – a displacement current – into Ampere’s Law, which removed all calculative discrepancies. Then, Maxwell formulated the four equations into one model which could unify electric and magnetic fields, but it was a messy affair. In fact, the Maxwell version of his equations had 20 equations and 20 variables of electromagnetic potentials. It was then Heaviside, who sorted up the mess, and put the electric and magnetic fields at the center of it all. He then arranged the equations using vector calculus into the beautiful form we are familiar with.


The equations could beautifully predict the results of every experiment you could try them on, and it was being established on a solid footing, but then came the big problem – Maxwell’s equations were not consistent with the Galilean principle of relativity. Since both theories were astoundingly accurate, the two not agreeing with each other created a big headache – either one of them was incorrect, or at least incomplete. Also, Maxwell’s equations predicted that light was an electromagnetic wave, and traveled through space without needing any medium. Physicists, most notably Lord Kelvin, did not like this preposition, and hence proposed that there existed a medium called the Ether, which permeated all space and helped light travel.

However, the more you tried to modify Maxwell’s equations, the more it backfired – the ether model had so many loopholes and conditional corrections that it was more like an adjustment than a theory. Also, the equations still remained inconsistent with Galileo and his principles. And this deadlock was broken by probably the greatest physicist ever born – Albert Einstein.

Galilean Relativity proposed that light traveled at infinite speeds. Albert Einstein challenged this very principle, and proposed that light actually had a fixed speed, as proposed by Maxwell’s equations. Light was the fastest thing in the Universe, and nothing could travel as fast as or faster than light. Einstein also rejected the flawed ether hypothesis, claiming that light did not need any medium to travel. These two assumptions, and some calculations aided by Lorentz mechanics, helped Einstein to formulate a theory which challenged the very nature of reality – the Special Theory of Relativity.


Maxwell’s equations, while inconsistent with Galilean Relativity, were fully compatible with STR. As experimental proofs kept in coming, the laws were established on a more solid footing than ever, and soon, Maxwell’s equations made Electrodynamics the first fully complete and predictive theory.


The predictive power of Maxwell’s equations is profound. The set of equations not only predict that light is an electromagnetic wave, but also establish the speed of all electromagnetic waves. This was proved in his experiments by Heinrich Hertz, and soon, entire unseen electromagnetic spectra were uncovered. Maxwell’s equations were the inspiration behind Einstein’s Special and General Theories of Relativity as well. Production of electricity by a dynamo, working of an electromagnet, attraction between atoms of a substance – all can be explained by a set of four equations.


Maxwell Equations


Covariant Notation of Maxwell’s equations using F – the electromagnetic field tensor.


Maxwell’s equations have since been written in more convenient covariant notations using tensorial notations. They have also been joined with Quantum Mechanics to form Quantum Electrodynamics – an astoundingly accurate theory that explains all interactions between electricity and magnetic fields. In addition, electrodynamics is one of the theories that can be applied without modifications in the Riemann manifolds of GTR. Today, it remains the only common link between two unrelated fields of research – quantum field theory and general theory of relativity. And it is likely that as it has happened before, Maxwell will come to aid us in order to unify the two into a single unified theory.

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