I. Maxwell’s Equations:

His anticipation of light’s immutable speed in vacuum

This essay covers some pivotal scientists and their breakthroughs that established the laws of electricity and magnetism in the 18th and 19th centuries, with a focus on James Clerk Maxwell. Their discoveries fundamentally changed our understanding and application of electrical power.

In addition to highlighting scientists, especially Maxwell, two parameters will feature prominently:

1. The permeability of free space, μ₀ , quantifies the force between currents flowing in wires. It is exactly 4π x 10^-7 newton/ampere² in rationalized MKS units. This is an assigned value. It sets the magnetic field force vector, the ampere (coulombs/second) and hence the coulomb.
2. The permittivity of free space, ϵ₀, is used in calculating the static force between charged particles. Its value is 8.834 x 10^-12 coul²/(newton meter²). ϵ₀ is set by the value of the coulomb, which is set by the ampere, which is, in turn, set by the value of μ₀. Thus, μ₀ and ϵ₀ are inextricably linked.

These constants jointly determine the speed of light in the vacuum, as was first calculated by Maxwell in the 1860s—a result discussed at this essay’s conclusion. Maxwell’s work unified electricity, magnetism, and electromagnetic radiation, shaping the future of scientific thought in several ways.

James Maxwell (1831-1879) was born into established families of pre-Victorian Scotland. His relatives described an inquisitive child, who, at the young age of three, would ask, “what is the go ‘o that?” If not satisfied, he would ask again, “but what’s the particular go ‘o it?” His mother was his first teacher and, though not able to answer all his questions, taught him how to read and write. He could recite large portions of the Psalms and Milton’s poems by the age of eight. His father, a marginally successful lawyer with a variety of interests, a jovial nature, and an independent income, quite enjoyed James’s company; James was his only surviving child.

His mother, Frances, died after surgery for abdominal cancer when James was eight years old. Young Maxwell’s education was then entrusted to a tutor, who was shortly dismissed for his mistreatment of James, and then briefly taken up by relatives before he was enrolled at the prestigious Edinburgh Academy at the age of 10. He didn’t fit in well at the Academy at first because of his rural upbringing and accent, his tendency to keep to himself at that time of his life, and his placement in a class with older boys.  It was there that he acquired the nickname “Dafty,” which stayed with him throughout much of his early life but never seemed to cause him embarrassment. Relatively soon though, he acquired some lifelong friends who later became notable scholars. And it was there at Edinburgh Academy that his intellect and interest in geometry began to emerge.

Though he may have been able to attend Cambridge University sooner, his father wanted him to stay closer to home and to pursue a general curriculum of study at the University of Edinburgh, thinking that he might become interested in the legal profession like himself someday. During this period, James was more drawn to math and science. He developed, among other interests, a fascination with polarized light and color perception…interest that would become significant for him later.

After three years in Edinburgh, he spent six years in Cambridge. He graduated in four years with a degree in mathematics and, as one of their best students, was invited to continue as a fellow.

In 1856, when a position arose at Marischal College in Aberdeen, he was urged to apply by one of his teachers at the University of Edinburgh, Professor Forbes, a mentor who became his lifelong friend and advocate. Before receiving a response from Marischal, his father died. Shortly thereafter, Marischal extended an offer to make Maxwell the head of the Department of Natural Philosophy, and he accepted.

While at Marischal, in response to a posted competition, he wrote a mathematically sophisticated essay describing the rings of Saturn, for which he won the Adams Prize from Cambridge University. He explained how the rings must be composed of particles. He termed them “brickbats,” and described how gravity and momentum organized the brickbats into the ribbon-like features that could be seen around that planet.

At Marischal College he met and married his wife Katherine, who was the daughter of the principal. Katherine helped with some of his laboratory studies, and he credited her with saving his life when he acquired a serious smallpox infection and later when he developed an infection following a head injury from a riding accident.

When Marischal College merged with King’s College of Aberdeen, his job was eliminated, and it was shortly thereafter that he was stricken with smallpox. He was without prospects for a time, but in 1860, Maxwell accepted a position as head of the Department of Natural Philosophy at Kings College, London.

In London at the time was the much older and internationally renowned Michael Faraday, then head of the Royal Institution of London. Faraday had become world famous for his descriptions of electricity and magnetism, which had by then become a topic of interest for Maxwell, too. Though Maxwell and Faraday communicated and were good friends, they never collaborated, as Faraday by this time had begun his declining years. The King’s College period, though, became one of the most productive of Maxwell’s scientific life. In his last years there, he introduced the terms curl, divergence, and gradient for describing the fields of electricity and magnetism, which were representative of Faraday’s “lines of force.” Also, and very importantly, he calculated the speed of light from his theory using the two known fixed constants μ₀ and ϵ₀ mentioned earlier.

In 1865, two years before Faraday’s death and after the older scientist had pulled back from his many scientific obligations due to health concerns, Maxwell resigned his position at King’s College, London, and returned to his family estate in Scotland so that he could concentrate on his experimental and theoretical research. He introduced statistical mechanics into his work on heat, and he continued his theoretical examination of the velocity distribution of molecules within a gas.

In 1871, while still in Scotland, he was offered the first Cavendish Chair at the University of Cambridge and was asked to inaugurate the now famous Cavendish Laboratory using funds from a wealthy benefactor. After some hesitation (he was their third choice), he accepted and soon proved that he was the right individual for this job. He designed and oversaw the building of the laboratory, and he set up its first research program, which concentrated on measurements of physical properties. The Cavendish Laboratory became successful beyond anyone’s expectations producing 31 Nobel Laurates. While at the Cavendish, Maxwell published, among other works, his Treatise on Electricity and Magnetism in 1873, his book Matter and Motion in 1876; and he edited the works of the benefactor’s great uncle, Henry Cavendish, who had been one of the founders of the Royal Institution of London, which Humphry Davy, and later Faraday, directed.

Maxwell died at nearly the same age and of the same disease as his mother, eight years after taking the helm of the Cavendish and a few weeks after publication of his final book, the Electrical Researches of the Honourable Henry Cavendish.

Several characteristics emerge about Maxwell. He had a brilliant and inquisitive mind; he worked constantly but with good humor; and he had a gentle nature. Near his death at 48, he expressed that his life had been privileged and that he had been gently treated both by people and circumstances. It is true that he had personal setbacks, but he apparently didn’t dwell on these near the end. He should rank along with Newton and Einstein among the most influential physicists of all time.

Maxwell is the central character of this essay. However, as is the case for all great scientists, there were predecessors who laid the foundation. This story begins earlier with an aristocratic Italian scientist, Alessandro Giuseppe Antonio Anastasio Volta (1745-1827), who was a contemporary of Benjamin Franklin. Volta demonstrated how to construct a practical battery, making possible much of the rapid development and experimentation with electricity that soon followed. Batteries rely on electrochemistry at the surfaces of stacked metal plates. Batteries were described as a “pile” of alternating stacked metal plates, which were immersed in a saltwater solution or separated by saltwater-soaked cardboard.

Volta also described capacitors, which have a superficial resemblance to batteries; capacitors use two identical metal plates. They accept the flow of, and store, small amounts of charge; no charge may directly cross from one capacitor plate to the other due to their separation by a space gap or layer of electrical insulation. Capacitors have a positive and a negative side, and the stored charge on one plate is counterbalanced by an identical amount of opposite charge on the other. Giuseppe Volta determined that the voltage, 𝑉, across the capacitor’s plates and the charge stored on the plates, Q , are proportional.  All students of physics know to define capacitance as C = Q/V.

It will be useful later in discussing Maxwell’s equation IV to consider the “displacement current” of a simple capacitor with only space between the plates. No current is allowed to cross directly from one capacitor plate to the other, but, as charge builds and is stored on the plates, the electric field,

between them increases.  This buildup of electric field is described as equivalent to a virtual current, or as Maxwell termed it, a displacement current flux:

Equation (1) describes the displacement current flux in a capacitor having a space gap and no insulating or dielectric material between the plates. It requires the permittivity of free space, ϵ₀ , one of the parameters given at the beginning of the essay. One of Maxwell’s great insights was that a changing electric field was fully equivalent to a current flux.

The electric field had already been defined by Coulomb.

Charles-Augustin de Coulomb (1736-1806) was an engineer and physicist. His scientific studies began in the military and continued after resigning his army commission. Though he spent most of his career in uniform, he fortunately resigned his commission at the start of the French Revolution. Coulomb is credited with discovering the inverse square law of electrostatic force, which resembles Newton’s law of gravitational attraction. Coulomb’s Law can be written in modern notation as

where the electric field is defined by

The electric field of equation (3) is defined by the force F on a test charge, q , the magnitude of the source charge Q and the distance between source and test charges, r. Coulomb’s torsion balance made possible the accurate force measurements used to establish his law.

Maxwell, and later Heaviside, changed the terminology, but not the science, of Coulomb’s law. Using the divergence theorem of vector calculus, also known as Gauss’s Law,

Combining this with Coulomb’s Law gives

which is Maxwell’s first equation. Though equation (I) is equivalent to Coulomb’s Law, neither Coulomb nor Gauss knew that they would be given the credit for the first of Maxwell’s four equations.

Johan Carl Friedrich Gauss (1777-1855) was a German mathematician and physicist born to illiterate working-class parents. Very early in his life he was recognized as a child prodigy. A brilliant and prolific mathematician, he was also a perfectionist, as he didn’t publish all his many discoveries until he had a sense of their completeness — he’d reveal them only to his students or leave them as unpublished manuscripts. What physicists now describe as Gauss’s law was discovered by him in the first half of the 19^th century, and even earlier by Lagrange (1736-1813). Gauss’s manuscript on this subject wasn’t published until 1867, twelve years after his death.

Maxwell may not have known of all of Gauss’s work when he published his paper, “A Dynamical Theory of the Electromagnetic Field” in 1865, although he certainly understood the equation we know as Gauss’s Law. It may be expressed as equation (4) for any vector field. The symbol, ∇ ·, signifies the divergence (of the field), a mathematical term which Maxwell introduced and defined. 

Sum all these projections along the circumference.  This summation will be proportional to the total current that flows through the area circumscribed by the closed line made of the infinitesimal segments dl. The proportionality constant is μ₀ . This law can most easily be demonstrated using a magnetic field that loops around a long straight wire carrying a current. Sum the magnetic induction field as represented in equation (5) using any complete loop around the wire. No matter how this loop is shaped or oriented, the summation of the induction field vector of equation (5), will give the same value, and this value will be proportional to the amount of current, I, flowing through the wire.

The fact that the magnetic field vector can be described this way, that is, as loops without beginning or end, is significant.  The divergence of such a field vector must be zero.  This is required by Gauss’s law, which for a magnetic field is expressed as

Any surface area fully enclosing a volume of space containing such a vector field must have the same amount of field lines entering that volume of space as leaving it.  Therefore, it is required that

Equation (6) then also requires that

This is Gauss’s law for any field with no source particle. The magnetic field exists only because of the movement of charged particles. For Maxwell’s equation II there are no magnetic field source particles. Equation (II) is the second of Maxwell’s four equations.

This brings us to an especially important person in electromagnetism. Michael Faraday (1791-1867) came from humble beginnings but had many accomplishments during his lifetime. He advanced the understanding of electromagnetism and electrochemistry; he demonstrated electromagnetic rotary devices (motors and dynamos); and he was the first to use the term “lines of force,” which were later adopted by Maxwell and became his electric and magnetic fields ^{(Footnote 1)}.

In his teen years, Faraday apprenticed as an indentured servant to a bookbinder in London. This gave him the opportunity to read, which he did voraciously. After his indentured period ended, he briefly worked for salary for another bookbinder. Faraday had much higher ambitions, and in his spare time and driven by his desire for knowledge, he attended scientific lectures in London, including those given by the celebrated chemist/inventor Humphry Davy. Faraday carefully wrote notes of Davy’s presentations. At the suggestion of an advocate who frequented the bookbinder shop and knew both Faraday and Davy, the older scientist temporarily hired Faraday after an accident in his laboratory left him in need of assistance. Though hired on a temporary basis, Faraday asked for a permanent job and sent Davy a copy of the lecture notes he had taken at Davy’s presentations in London. After some delay, when a position became available, he was hired as a bottle washer. He soon proved his worth far beyond Davy’s expectations.

Much later, as director of the Royal Institution, Faraday followed Davy’s tradition of providing public lectures. He was especially renowned for his Christmas Day lectures for children, which he always gave himself. In his lectures, he demonstrated his many significant discoveries, including electromagnetic induction, which may be demonstrated by the production of an electric current caused by moving a magnet through wire loops. To the delighted children in the Christmas day audience, this must have seemed like a presentation of magic.

To demonstrate magnetic fields, he wrapped coils of wire around an iron ring or rod and attached the wire to a battery; he used the alignment of a compass needle or the alignment of shavings of magnetite (small flakes of a magnet) to demonstrate the forces within a magnetic field. He described the magnitude and the direction of the force between two wires carrying current, confirming the work of Ampère, his friend, and a trusted correspondent. He also demonstrated that electric charge must exist only on the exterior surface of a conductor, and he invented the Faraday cage, which is used today for protecting delicate electronic instruments. These many results, which were given as parts of his public lectures, were published in his three-volume treatise, Experimental Researches in Electricity ^(Faraday).

During his lifetime, Faraday refused any honors or titles from England on philosophical and religious grounds, and in his Will, he stipulated that he be buried in the family plot (not in Westminster Abby).

Faraday did not have the mathematical background of Maxwell, but his detailed and clear descriptions of magnetic phenomena helped Maxwell put these into the language we use today. To do so, Maxwell introduced and defined terms: divergence, curl, and gradient. An example of Maxwell’s descriptions of electromagnetism is his fourth equation, expressed using the modern notation and assuming a dielectric constant and relative permeability of one:

The curl,

preserves the directional relationship between the looping magnetic field and the current flux as had been demonstrated in Faraday’s and Ampère’s experiments.  Maxwell’s equation (IV) states that any eddies or loops in the magnetic field would correspond to an electrical current flux, J (or a “displacement current flux”, ϵ₀dE/dt).

Perhaps Maxwell, before others, realized that the displacement current flux, ϵ₀dE/dt, like the current flux, J, would generate a magnetic field.

The curl is defined as a line integral making a complete loop around an infinitesimally small area.  The current flux (or the displacement current flux) is represented as a vector pointing from the “eye” of this loop perpendicular to the loop area.  Vector direction is given by the right-hand rule.

Brief description of curl and the right-hand rule.  The curl of any vector field is defined using a surface area, S, and line elements, dl, which is a small part of a complete loop around the edge (circumference) of the surface, S.  The curl vector, which is normal (n) to the surface (S) is symbolized as

The right side of the equation represents a summation of “inner” products of the vector field, A, onto a closed path of the line elements, dl, around the surface area, S. The right-hand rule has placement of the thumb of the right hand pointing in a direction normal to the plane of this surface.  The fingers of the right-hand curve around the perimeter of the surface in counterclockwise direction; the sign of the line integral determines which of the two possible directions the curl vector (thumb) points. 

Maxwell’s third equation relates the changing magnetic field to the curl of the electric field

This is the symmetrical counterpart of the equation IV, illustrating the inherent symmetry between electric and magnetic fields. The third of Maxwell’s four equations states that the line integral around a surface within an electric field corresponds to a changing magnetic field.

By comparison, Maxwell’s equation (IV) for the curl of the magnetic field includes a term for the current flux, μ_oJ .  A current flux requires a “source particle”. Equation (III) has no equivalent term because there are no “source particles” from which a magnetic field diverges, so there can be no corresponding “magnetic current” term. This property is evident in Maxwell’s equation II.

Oliver Heaviside (1850-1925) was a brilliant and largely self-taught (after high school) English physicist and mathematician. Heaviside caught Maxwell’s favorable attention because of an article he had published on the best arrangement for the Wheatstone bridge, a device for accurately measuring electrical resistance. At the time, Heaviside was employed as chief telegraph operator in Newcastle. Maxwell, then chief of the Cavendish Laboratory in Cambridge, put this into his Treatise on Electricity and Magnetism with attribution to Heaviside. Heaviside was only 29 when Maxwell, whom he always called “good old Maxwell,” died of intestinal cancer at 48.

In part validated by Maxwell’s attention and because he was denied a pay raise, Heaviside quit his job at the telegraph company, moved in with his parents, and devoted all his time to understanding Maxwell’s equations. Eventually he simplified and combined Maxwell’s original 20 equations to the four that we know today and, in this process, developed methods of vector analysis. He had many other scientific accomplishments.

Whether because of his partial deafness (the result of scarlet fever), a period of family poverty, or just his general nature, he claimed an unhappy childhood ^{(Basil Mahon, “Heaviside”)}. He subsequently became a difficult adult, known for his “prickly” nature. He was, on occasion, criticized by some in his field and in the press by adversaries. He was not opposed to attacking first or counterattacking with biting sarcasm. Heaviside received only delayed recognition for his many scientific accomplishments, one of which is described below.

With his brother Arthur, he proposed a method for using induction coils to reduce signal distortion in long-distance transmission lines. This method was written for publication as a detailed and technical description for how to balance capacitance, inductance, line resistance, and current leak to optimize and sharpen long distance signals over transmission lines. Publication was halted, though, by Arthur’s superior, who was a powerful government official with a different theory of long-distance transmission. Later, though, with persistence, some help from an indulgent editor and a well-placed and supportive scientist, Heaviside published the method anyway, first in the Electrician and later in the Philosophical Magazine. Thus, after 1887, this came into the public domain. His work was ignored by many at the time but well studied by a few, and it was perhaps minimally improved by scientists at AT&T and a professor at Columbia University in the US. The U.S. patent was awarded to the professor from Columbia. AT&T, rather than spending time and money fighting this decision in court, purchased the patent from the professor, and shortly thereafter, used this process — effectively and quite profitably —for eliminating distortion in their long-distance signaling lines. AT&T earned billions of dollars from this invention. A group of representatives of U.S. signal transmission companies under the umbrella of AT&T later offered Heaviside a stipend from their U.S. profits, but he refused to accept it unless they would agree to give him full credit for the invention. He remained poor for his entire life, relying on family, a small pension from the UK, meager money from his publications, and occasionally a friend (though he resisted most financial help, which he distastefully regarded as charity). He unsuccessfully and persistently insisted on receiving from AT&T what he felt was rightfully his — full credit for his invention.

In other scientific work, he predicted the presence of an ionic layer in the atmosphere now known as the Heaviside layer, which reflects radio waves back toward Earth. This allows signals of certain frequencies to follow the curved surface of the earth and to travel long distances around the world — a phenomenon that ham radio operators know well.

Later in his life, the significance of Heaviside’s work was recognized to a greater extent. He was elected to the Royal Society of London in 1891; however, he never lost the resentment of his earlier mistreatment. He lived the last years of his life in a house owned by the sister of a brother’s wife. He and his brother’s sister-in-law remained there until her health failed and she was no longer able to tolerate the difficult, demanding, and eccentric Heaviside; she moved away to stay with relatives. Alone in the house and visited by only a few, among them George Searle, his friend and a lecturer in physics at Cambridge University, his health declined. He died after a short but apparently comfortable stay in a nursing home where he had consented to be taken by George Searle and Heaviside’s family members.

Electromagnetic radiation and Maxwell’s equations

We have now come full circle with James Clerk Maxwell and electromagnetism.  Although Maxwell had many other scientific accomplishments, he may be best known for his derivation of the propagation of electromagnetic radiation in a vacuum.  In the vacuum, Maxwell dispensed with any properties of matter in his equations. This eliminated charge density,  ρ ; charge movement, J , and any electric dipoles or magnetic moments.  In the vacuum of free space, Maxwell’s equations take a reduced form:

Maxwell realized that, when combined, equations IIIa and IVa represent a traveling wave of electric and magnetic fields ^{(Footnote 2)}.

Also,

The symbol

defines the Laplacian of a vector point-function, shown here in cartesian coordinates. These equations describe coupled waves of E and B that travel with a fixed velocity of

C= 1/√μ₀ϵ₀ ,

which Maxwell realized was very close to the speed of light. At his home in Scotland, when deriving equations (8) and (9), he had to wait impatiently until he could return to London at summer’s end to obtain from published data the values for μ₀, ϵ₀, and (at that time) the approximately known speed of light.  Understanding the speed of light being dependent only on these fixed parameters, he may have been disappointed that the significance of this finding was not more fully appreciated by others. However, appreciation soon came by the end of the 19^th and beginning of the 20^th century — two decades after Maxwell’s death.

To those scientists who paid attention, it seemed an outstanding and surprising conclusion: that there could be a wave of energy (electromagnetic energy) propagating in the vacuum without a “medium.”  Even after the experiments of Michaelson and Morley and the theoretical work of Einstein became known, many good scientists held to the view that a mysterious substance termed “aether” permeated all space, including the vacuum, and they believed that this aether was required for the propagation of light through space.

There were far-reaching implications for the fixed light speed, which became significant in the emerging field of cosmology. Lorentz, Poncaré, and especially Einstein, at the beginning of the 20^th century, came to realize that our understanding of simultaneity, time, distance, and the nature of the universe itself had to be revised.  This came about with the recognition that the speed of electromagnetic radiation, which could be calculated from Maxwell’s equations using the constants μ₀ and ϵ₀, must have the same value for any observer unaffected by the observer’s motion.  As an unimagined result of the fixed speed of light, atomic power and weapons were invented.  Though we may take many of these discoveries for granted, our lives would be drastically different without them.

Footnotes

^{(Footnote 1)} Faraday’s electric and magnetic lines of force became Maxwell’s electric and magnetic-induction fields, E and B.  Faraday reasoned that a magnetic field forced a compass needle into alignment.  For Maxwell, there was a fundamental relationship between the electric field, produced by the charge, and the magnetic-induction field, produced by the charge movement.

^{(Footnote 2)} The step ∇ x ∇ x B = ∇ ²B is not a trivial one.  It may have been easier for Maxwell, though, not only because of his genius, but because he had placed his original twenty equations in a more explicit differential form.  Oliver Heaviside abbreviated these twenty equations to the four expressions we use today. 

Bibliography

Basil Mahon, The Man Who Changed Everything, The Life of James Clerk Maxwell, John Wiley & Sons Ltd., 2004

Nancy Forbes and Basil Mahon, Faraday, Maxwell, and the Electromagnetic Field, How Two Men Revolutionized Physics, Prometheus Books, 2019

Basil Mahon, The Forgotten Genius of Oliver Heaviside, a Maverick of Electrical Science, Prometheus Books, 2017

David Halliday and Robert Resnick, Physics, Parts I and II, John Wiley & Sons, Inc., 1962

Dale Corson and Paul Lorrain, Introduction to Electromagnetic Fields and Waves, W. H. Freeman, and Company, 1962

Grant R. Fowles, Introduction to Modern Optics, Holt, Rinehart and Winston, Inc., 1968

John David Jackson, Classical Electrodynamics, John Wiley & Sons, Inc., 1967

Michelson, Albert A. and Edward W. Morley, “On the Relative Motion of the Earth and the Luminiferous Ether”. American Journal of Science. 34 (203): 333–345, 1887

Einstein, Albert “On a Heuristic Point of View about the Creation and Conversion of Light”, Annalen der Physik (in German) 17 (6): 132–148 (1905)

J.S.R. Chisholm and Rosa M. Morris, Mathematical Methods in Physics, W. B. Saunders Company, 1965

For some biographical information, Wikipedia (2020)