by Neil Turok
After three years at Edinburgh University, Maxwell moved to Cambridge. One of his professors at Edinburgh commented in his recommendation letter, “He is not a little uncouth in manners, but withal one of the most original young men I have ever met with and with an extraordinary aptitude for physical enquiries.”23 Whereas the education at Edinburgh had been free-thinking and broad, Cambridge was far more competitive and intense, and much of his time was spent cramming for exams. After coming second in the university in his final exams, Maxwell was appointed as a Trinity College Fellow at the age of twenty-three. This gave him time to investigate a variety of phenomena, from fish-eye lenses to the flight of falling pieces of paper, and even the ability of cats to right themselves if dropped. He also demonstrated, using coloured spinning tops, that white light is a mixture of red, green, and blue.
Just a year later, in 1856, Maxwell moved to Aberdeen to take up a chair of natural philosophy. He spent five years there before moving to King’s College, London. During this period he contributed to many different fields, applying in each case a deft combination of physical insight and mathematical skill. He showed that Saturn’s rings were composed of particles, a theory confirmed by the Voyager flybys of the 1980s. He developed models of elasticity and discovered relations in the theory of heat, both of which are still used by engineers. Later on in his career, he worked out the statistical properties of molecules in a gas and he demonstrated the first-ever colour slide. But the feat that unquestionably trumps them all began in 1854, when he tried to clean up a bunch of messy equations having to do with electricity and magnetism.24
Michael Faraday, by contrast, was the son of a South London blacksmith and left school at thirteen to become a bookbinder’s apprentice. He had no formal scientific education and no mathematics, but he had a deep curiosity about the world, an alertness to it, and marvellous physical intuition.
On reading an article on electricity in an encyclopedia he was binding, Faraday was captivated. One of the bookbinder’s customers, perceiving the lad’s evident intelligence and thirst for knowledge, gave him tickets to lectures by Sir Humphrey Davy, one of the great scientists of the day, at the Royal Institution. Having attended the lectures, Faraday copied out his copious notes, which amounted to a virtual transcription of the lectures, and presented them, beautifully bound, to the great man. This led to a job, first as a bottle washer in Davy’s lab and, soon enough, as his right-hand man. Eventually he succeeded Davy as the director of the Royal Institution. Despite its walls of inequity and injustice, the Victorian age sometimes let in chinks of light, such as its workingmen’s colleges and public lectures bringing science to the general populace.
As a mature scientist, Faraday was indefatigable and responsible for a staggering range of discoveries. But what fascinated him above all was electricity and magnetism, and he was by no means alone in this. Although electricity had been observed for millennia in certain shocking fish and in lightning, by the nineteenth century its magical properties were beginning to be widely appreciated, though they were not understood. Its spark and sizzle were lifelike — it galvanized the age, you might say. Mary Shelley’s Frankenstein; or, The Modern Prometheus was inspired by electrical experiments, often carried out in public, on living and dead creatures in early nineteenth-century London. Its title compared the modern scientist to the ancient Greek hero Prometheus, a lesser god who became a champion of mankind. He stole fire from the king of the gods, Zeus, and gave it to man. Shelley’s book was a cautionary tale: for his crime, Prometheus was condemned by Zeus to be chained to a rock and have his liver eaten out by an eagle every day, only for it to grow back every night.
Faraday came to know electricity better than anyone, and his work was far ahead of its time. He showed that chemical bonds are electrical, discovering the laws of electrolysis and electrical deposition of one metal onto another. Faraday had a genius for discovering new phenomena using simple experiments. He investigated the magnetic properties of bismuth, iodine, plaster of Paris, even blood and liver. He blew soap bubbles filled with various gases — oxygen, nitrogen, hydrogen — through a magnetized region. He found that an oxygen-filled bubble got stuck in the magnetized region because oxygen is paramagnetic. (The explanation had to wait another ninety years, for the invention of quantum mechanics.)
Faraday also demonstrated the process of electromagnetic induction: how you can seemingly pull electricity out of a magnet by moving a wire past it. Faraday employed this in his invention of dynamos and transformers, now used to generate and distribute electricity all over the world. He even discovered superionic conduction, the basic mechanism of modern fuel cells.25
Faraday also showed that when a metal container is electrically charged, the charge moves onto the outer surface. He sat in a square cage, twelve feet on a side, while his assistant charged it to 150,000 volts. Sparks flew wildly everywhere. His hair flared out in a halo, but he was unharmed — the charge was all on the outside. The next time you fly through a lightning storm in a plane, thank Michael Faraday for showing that it would be safe!
Faraday did much, much more besides, but for our story, the most vital contribution he made was to formulate, for the very first time, a strange, slippery concept that is central to modern physics: a field. Instead of electric charges attracting or repelling one another from a distance, Faraday believed there must be an intermediary that carried the influence of one charge to another.
Faraday was not mathematical, and he could develop the idea only through slow and difficult experiments. It would fall to theory to make the next breakthrough. The eccentric young Maxwell built Faraday’s intuition into the most beautiful and powerful mathematical framework in physics, and in so doing solved one of the greatest enigmas of all time.
THE SIMPLEST FACT ABOUT electricity is that like charges repel and unlike charges attract. You can easily see this by taking a roll of brown plastic packing tape and sticking two long strips, sticky side down, side by side on a tabletop. Holding the end of one strip with your left hand and the other with your right hand, rip both strips off the table at once and allow them to hang vertically downwards. If you bring your hands together gently, the two strips will swing away from each other because of the electrical force of repulsion between them.
The opposite effect is seen with a small tweak of the experiment. Instead of laying the strips on the table separately, put one strip down first and lay the other directly on top of it. Now pull them off the table together, as a double strip. Neutralize the double strip by running your fingers gently down the tape (this has the effect of cancelling any net electric charge). And now, using both hands, rip the two strips apart from one end. Because there was no charge on the pair of strips to begin with, if one strip gains a charge, the other must have lost some. So if one is positive, the other must be negative. Bring the two strips together, one hanging down from each hand, and you will find they draw together.
Magnets, too, have ends that attract and repel: a positive end called the north pole and a negative end called the south pole. Two north poles or two south poles repel but a north and a south pole attract. The name comes about because Earth itself is a giant magnet, which causes the north pole of any magnet — like the needle of a compass — to point in a direction close to that of the Earth’s north pole.
Faraday had come to picture electric and magnetic forces as operating through “lines of force.” He had a real bee in his bonnet about these lines. He believed that forces could not be “felt at a distance,” as Newton had described in the case of gravity. Newton’s picture would mean that if you jiggled the sun up and down, then Earth, 93 million miles away from it, would have to jiggle in perfect synchrony. And other objects would have to jiggle, all the way to infinity.
For Faraday, the idea that forces were transmitted right across space instantaneously between two objects was ridiculous. Instead, he thought, there must be something that actually carries the forces through
the space that separates them. Whatever that something is, it must surround a mass or a charge or a magnet at all times, even when there isn’t an object for it to push. Faraday’s brilliant insight was in fact the first glimpse of the concept of a “force field.”
You are familiar with the north–south and east–west grid lines on a street map. Visualize, if you can, such a grid spreading through three-dimensional space, with its grid lines going north–south, east–west, and up–down, and with every grid line separated by the same distance from its neighbouring parallel lines. The grid is a convenient way of measuring length, width, and depth. Where any three grid lines cross, we have a grid point, labelled by its coordinate on each of the grid lines passing through it. We can make our grid finer and finer by making the spacing between the parallel lines as small as we like. For physicists, this picture of a grid is the way we convert our mental picture of space — as a three-dimensional entity — into numbers, so that each different point of space is associated with three numbers.26
To picture a force field, imagine attaching a little arrow to every grid point (click to see image). Each arrow can point in any direction. The arrows represent the force field; the length and direction of an arrow indicate the strength and direction of the field at that point in space. Any charged particle placed in this force field will feel a force. For example, an electron feels a force given by its electric charge times the electric field.
When Maxwell was only twenty-five years old and working as a College Fellow in Cambridge, he wrote to Faraday, then one of the most famous scientists of the day and director of the Royal Institution. Maxwell enclosed a paper he had just written titled “On Faraday’s Lines of Force,” giving a mathematical description of the effects that Faraday had reported in his experiments. Faraday, who had never studied mathematics, later noted, “I was at first almost frightened, when I saw such mathematical force made to bear upon the subject.”27 Yet Maxwell wrote modestly in his paper, “By the method which I adopt, I hope to render it evident that I am not attempting to establish any physical theory of a science in which I have hardly made a single experiment, and that the limit of my design is to shew how, by a strict application of the ideas and methods of Faraday, the connection of very different orders of phenomena which he has discovered may be placed before the mathematical mind.”28 Their interaction and mutual respect illustrate beautifully the interplay between theory and experiment in its ideal form.
Faraday, encouraged by Maxwell’s work, returned to his laboratory at the Royal Institution in London with renewed vigour. His goal now was to show that electric and magnetic fields take time to move through space. Already sixty-six years old and exhausted after many years of arduous experimentation, he did not succeed (it took three more decades before German physicist Heinrich Hertz finally did). It was time for Maxwell’s “mathematical mind” to come to the rescue.
By the time Maxwell moved to King’s College, London, in 1861, he was ready to begin making sense of electricity and magnetism in earnest. His goal was to describe mathematically the laws governing the “lines of force” envisaged by Faraday. In several stages, Maxwell built this intuition into a full-fledged theory of “fields,” a concept that would dominate fundamental physics in the twentieth century.
Electric charges spew out electric fields, magnets spew out magnetic fields, and masses spew out gravitational fields so that all three kinds of fields are present everywhere in the universe. In modern terms, we represent electric and magnetic fields as a sea of little arrows filling space. The trick in describing all the equations of electricity and magnetism is to figure out how the arrows located at each point in space influence their neighbours. The rules are complicated, and Maxwell had to work them all out for the first time. Newton had invented calculus as the mathematics for describing motion. Maxwell had to extend this idea to describe how force fields change in space and in time. Using a grid of points in space, like the one we described, Maxwell developed the theory of partial differential equations to describe force fields. The mathematics he invented is used throughout science to describe fluids, the flow of air, or even the propagation of disease.
The way in which Maxwell found his equations for the electric and magnetic fields was at first sight surprising. He envisaged a machine whose moving parts represented the fields. In his first attempt, the lines of force of the electric field were represented by “tubes” carrying a fluid out of positive electric charges and into negative electric charges. But gradually the model became more sophisticated, and the fluid-filled “tubes” were replaced by microscopic “rollers” and “spinning wheels” representing both electric and magnetic fields, turning the whole of space into a gargantuan factory (click to see image).
With guidance from William Thomson (Lord Kelvin), Maxwell laid out all the different phenomena and all the laws of electricity and magnetism known at the time — a veritable alphabet of laws: Ampère’s, Biot-Savart’s, Coulomb’s, Faraday’s, Franklin’s, Gauss’s, Kirchhoff’s, Lenz’s, Ohm’s laws and more, built up over the course of the previous century. His goal was to fit all the pieces together into a single consistent mechanical framework.
Maxwell was able to incorporate each of the known laws of electricity and magnetism into his conceptual mechanism — except one. Benjamin Franklin had proposed that an electric charge could be neither created nor destroyed. Maxwell formulated Franklin’s law mathematically, showing how the electric charge in a region of space is changed by the flow of electric current into or out of the region. Carl Friedrich Gauss had described how electric charges give rise to electric fields, and André-Marie Ampère had described how electric currents create magnetic fields. But when Maxwell put the three laws together, he found a contradiction: they were incompatible! The only way he could restore consistency was to change Ampère’s law by adding a new term, according to which a changing electric field would also cause a magnetic field. This was, he realized, similar to the way in which a changing magnetic field generated an electric field, as Michael Faraday had demonstrated.
But wait: a changing electric field can now create a changing magnetic field which, by Faraday’s law, can create a changing electric field. So electric and magnetic fields can create one another, without any electric charges or currents or magnets being present. When Maxwell analyzed his equations carefully, he found that magnetic and electric fields can travel across space together, like an undulating pattern moving across the grass in a meadow. This electromagnetic wave was just the kind of effect Faraday had been anticipating.
While summering in his ancestral home of Glenlair, in 1861, Maxwell made his discovery. Using the best experimental measurements to date, he worked out the speed at which the electromagnetic waves would travel. In a letter to Faraday he wrote, “The result is 193,088 miles per second (deduced from electrical and magnetic experiments). [French physicist Hippolyte] Fizeau has determined the velocity of light = 193,118 miles per second by direct experiment.” And then, with lovely understatement, he added, “This coincidence is not merely numerical.”29
Maxwell, using purely mathematical arguments, had not only predicted the speed of light, he had explained light’s nature. Simply by piecing together known facts and insisting on mathematical consistency, he had revealed one of the most basic properties of the universe.
Once he had reached his conclusion, Maxwell swiftly set about scrapping his mechanical model. Now that he had the right equations, he no longer needed the visual machinery. The equations were the theory: one needed nothing more and nothing less.
Whenever I teach electromagnetism, Maxwell’s discovery is the highlight of the course. There is a moment of sheer magic when the students suddenly see how all the pieces fit together and light has seemingly popped out of nowhere. “If you are ever in doubt,” I tell them, “remember this moment. Perseverance leads to enlightenment!” And the truth is more beautiful than your wildest dreams.
IT WOULD
BE HARD to overstate the importance of Maxwell’s discovery in unifying electricity, magnetism, and light. All at once, this provided a simple, precise description of a vast array of phenomena: the spark from a brass knob on a cold morning; the signals that traverse our nerves or make our muscles move; lightning strikes and candlelight; the swing of a compass needle and the spin of an electric turbine.
The direct impact on technology would soon be felt in radio, television, and radar. But the long-term effect on basic physics was even greater. Maxwell’s breakthrough opened the door to twentieth-century physics: to relativity, quantum theory, and particle physics, our most fundamental descriptions of reality.
One of the theory’s most important predictions was that electromagnetic waves could have any wavelength, from zero to infinity. Just a tiny portion of this spectrum — from two-fifths to three-quarters of a micron (a millionth of a metre) — explains all of visible light: red, yellow, green, blue, violet. Maxwell’s discovery widened the rainbow, predicting the existence of electromagnetic waves with wavelengths ranging from a thousandth the size of an atomic nucleus (the gamma rays produced in the Large Hadron Collider) to thousands of kilometres (the ultra-low-frequency waves used for communication with submarines). In between are X-rays, used for medical imaging; infrared waves, used for night vision; microwaves, used for cooking; and radio waves, used for everything from cellphones to telescopes probing neutron stars and black holes. Maxwell’s equations describe every single one of these waves in exactly the same way. They are just stretched-out or shrunken-down versions of one another.
