History of Maxwell’s Equation

In 1865, James Clerk Maxwell wrote his famous (or infamously difficult) Maxwells equations. These equations were to transform our world.  For example, when Einstein was asked if he stood on the shoulders of Newton, he replied, “No, on the shoulders of Maxwell”.

But who was James Maxwell, why are his equations important and why did he write them? It all has to do with a great scientist named Michael Faraday who had no math skills and a young Lord Kelvin (of temperature fame) who inspired them both.

Table of Contents

James Maxwell’s Early Life

James Maxwell’s Life at Cambridge

Cambridge: On Faraday’s Lines of Force

Maxwell’s Theory on Electromagnetic Waves


James Maxwell’s Early Life

History of Maxwells Equation
Portrait of James Clerk Maxwell and his wife Katherine Dewar Maxwell, circa 1860s.

Let’s start with a little bit of backstory.  In 1845, a 21-year-old Scottish student named William Thompson (later knighted Lord Kelvin) inspired the already famous 54 yr old Michael Faraday to conduct an experiment that found a link between electricity, magnetism, and light. 

The next year Faraday postulated that maybe light was really vibrations of electric and/or magnetic fields.  This was a radical idea and most legitimate scientists thought it was balderdash. 

To make matters worse, Faraday was rapidly losing his memory and suffering from severe depression (probably from mercury poisoning) and struggled to conduct research in the lab.  Also, many scientists at the time looked down on Faraday for his low-class background and his complete lack of mathematical ability. 

James Maxwell’s Life at Cambridge

Portrait of James Clerk Maxwell at Cambridge

Fast-forward 9 years to 1854.  James Clerk Maxwell was a 22-year-old Scottish math and physics student at Cambridge famous for his intelligence, wit and late hours (supposedly when he heard there was a required 6 am chapel meeting he replied, “Aye, I suppose I can stay up that late”). 

1853 was an important year for Maxwell as he moved into his own apartment or as he put it, “entered the unholy estate of bachelorhood[1]” and had more time to do independent research.  He therefore decided, “to attack electricity” and asked his friend William Thompson for advice on who to read, looking for, “any notions I could screw into my head”. 

Maxwell specifically wondered if he should start with mathematical scientists or start with Faraday (who never used math).  Thompson, the same person who inspired Faraday earlier, advised Maxwell to start with Faraday. Surprisingly Maxwell felt that Faraday’swork and concepts were secretly mathematical.  

“As I proceeded with the study of Faraday, I perceived that his method of conceiving the phenomena was also a mathematical one, though not exhibited in the conventional form of mathematical symbols.[2]”  Moreover, Maxwell felt that he had the ability to convert Faraday’s “math” into “the ordinary mathematical forms,” which he promptly did.

Cambridge: On Faraday’s Lines of Force

History of Maxwells Equation

Two years later, Maxwell published “On Faraday’s Lines of Force” and in a bold move, Maxwell sent his paper to Faraday himself.  Faraday, was impressed and wrote Maxwell a plea to, “translate them out of their hieroglyphics [i.e. math] that we almost might work upon them by experiment.” 

Faraday’s work is “generally supposed to be of an indefinite and unmathematical character” “I am not attempting to establish any physical theory of a science which I have hardly made a single experiment”

Meanwhile, Maxwell had become, at age 25, a full time professor at Marishal College in Aberdeen.  To fit in with the older staff he grew a big bushy beard.  He didn’t feel comfortable at Aberdeen saying, “No jokes of any kind are understood here.  I have not made a joke for 2 months, and if I feel one coming on I shall bite my tongue.”  

However, he did end up meeting Katherine Dewar, the daughter of a history professor.  It was an instant match and they were married in June of 1859.  They never had kids but were a little obsessed with their dog Toby. 

Katherine was a scientist herself.  We don’t know how much Katherine did as a fire burned much of their original work and biographers were hesitant to mention Katherine’s contributions in that sexist society. 

However, we do know, for example, that when a friend asked James Maxwell for some data he replied, “my better 1/2, who did all the real work of the kinetic theory, is at present engaged in other researches.  When she is done I will let you know her answer to your enquiry”. 

The Maxwells moved to London and James Maxwell became a professor at King’s College.   At this time Maxwell and his wife did pioneering work on color (including a method for creating the first color photograph), molecular theory, dimensional analysis and how gasses flow.

Maxwell continued his studies into Faraday’s theories and in his attempts to add mathematics to it.  He was particularly interested in Faraday’s theory from 1846 that light was really just a vibrating wave of electric and magnetic fields.

At this time William Thompson told Maxwell to read two German physicists, Wilhelm Weber and Rudolf Kohlrausch who had experimentally connected the units of electricity to the units of magnetism. 

Weber et all.experimentallyderived a speed, which they labeled “c”, probably because they already had an “a” and a “b” in their paper.  This speed “c” turned out to be the speed of light!  This, as a side note, is why the speed of light is labeled “c” in Einstein’s famous E=mc2 equation.

Maxwell’s Theory on Electromagnetic Waves

Anyway, in 1865, Maxwell wrote a paper on electromagnetic forces where he coined the term “fields” as compared to Faraday’s “lines of force”.  Maxwell excitedly wrote a friend that, “till I am convinced to the contrary, I hold [my new paper] to be great guns.”

And he was right!  In this paper Maxwell said that Weber’s results are “so nearly that of light, that it seems we have strong reason to conclude that light itself is an electromagnetic wave.” To Maxwell, his equations were “the same in substance” as the one created by Faraday, “except that in 1846 there were no data to calculate the velocity of propagation.”

Maxwell’s equations were more then “just” an equation for light, however, they were equations that explained all of electrical and magnetic theory in one “beautiful” package all based on interconnected magnetic and electric fields.

For example, in 1820, Orsted found that current (or moving charges) will move a magnet.  In Maxwell’s equations, the current or moving charges make a moving electric field, which creates a magnetic field.  Or, in another example, Faraday found in 1831, that moving a magnet in a coil would create a current in the coil. 

Maxwell’s equations describe how a changing magnetic field will create an Electric field, which creates a current.  It is hard to overstate how important Maxwell’s equations are to Physicists.  A hundred years after Maxwell the great Physicist Richard Feynman poetically wrote that, “10,000 years from now the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electromagnetism.  The American Civil War will pale into provincial insignificance in comparison.”

Although Maxwell’s theories were heralded as genius by the very late 1800s at the time they were mostly ignored.  The theory was bizarre and the mathematics truly arduous. In fact, the original form of Maxwell’s had a whopping 20 interconnected equations! However, soon a group of young people calling themselves “Maxwellians” started promoting Maxwell’s theories and simplifying it.

For example, the modern form of Maxwell’s equations with“just” 4 equations was created by a “Maxwellian” named Oliver Heaviside in 1884 by using something he created called vector calculus.

Back in the 1860s, Faraday was pleased with Maxwell’s attention but was getting older and more and more feeble and confused.  He gave his last free Christmas lecture for kids in 1860 and his last experiment (on the magnetic influence on the light produced by a glowing gas)on March 12, 1862. 

By the way, Faraday was unsuccessful, but a man named Pieter Zeeman redid the experiment with more sensitive equipment and won a Nobel Prize for it in 1902. Faraday died on August 25, 1867 and, as was his wish, had “a plain simple funeral followed by a gravestone of the most ordinary kind, in the simplest earthly place.”  

Meanwhile, the Maxwells continued to research and develop new theories.  Tragically, in 1877, Maxwell was working on a new version of his electromagnetic papers when he began to have terrible heartburn that was eventually diagnosed as stomach cancer. 

James Maxwell died on November 5, 1879 when he was just 48 years old.  His local doctor ended his medical report on Maxwell by adding, “I must say he is one of the best men I have ever met,” a sentiment shared by most of the people who knew him.  Katherine died 7 years later and little is known of her activities after her husband’s death.  She has no statue. 

The same year that Maxwell died, a Maxwellian named Helmholtz at the Academy of Science in Berlin offered a prize of 100 ducats for experiment that would prove “the theory of electrodynamics which was brought forth by Faraday and was mathematically executed by Mr. Maxwell.” 

Helmholtz asked his 22-year-old grad student Heinrich Hertz to attempt it, but Hertz was too intimidated.  However, “in spite of having abandoned the solution at that time, my interest in everything connected with electric oscillations had become keener” Eight years later Hertz got a giant spark on one side of a room to create a tiny spark on the other side!  How Maxwell and Faraday (and Helmhotz) inspired Hertz to discover radio waves is next time on the secret history of electricity!


[1] Feb 20, 1854 Maxwell, J. Clerk to William Thomson found in Origins of Clerk Maxwell’s Electric Ideas Edited by Sir Joseph Larmor 1937 p. 3

[2] Maxwell, J. Clerk A Treatise on Electricity and Magnetism Vol 1, p. viii

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