Maxwell Equations Quotes

That feeling in your heart: it’s called mono no aware. It is a sense of the transience of all things in life. The sun, the dandelion, the cicada, the Hammer, and all of us: we are all subject to the equations of James Clerk Maxwell, and we are all ephemeral patterns destined to eventually fade, whether in a second or an eon.

The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations - then so much the worse for Maxwell's equations. If it is found to be contradicted by observation - well, these experimentalists do bungle things sometimes. But if your theory is found to be against the Second Law of Thermodynamics I can give you no hope; there is nothing for it to collapse in deepest humiliation.

Maxwell's equations didn't just change the world. They opened up a new one.

That small word “Force,” they make a barber's block, Ready to put on Meanings most strange and various, fit to shock Pupils of Newton.... The phrases of last century in this Linger to play tricks— Vis viva and Vis Mortua and Vis Acceleratrix:— Those long-nebbed words that to our text books still Cling by their titles, And from them creep, as entozoa will, Into our vitals. But see! Tait writes in lucid symbols clear One small equation; And Force becomes of Energy a mere Space-variation.

Commenting on the importance of Maxwell's equations, Einstein wrote that they are "the most profound and the most fruitful that physics has experienced since the time of Newton.

Newton’s mechanics, Maxwell’s equations, quantum mechanics, and so on, tell us how events happen, not how things are.

Accordingly, we find Euler and D'Alembert devoting their talent and their patience to the establishment of the laws of rotation of the solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsot has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligent propositions supersede equations.

In electrodynamics the continuous field appears side by side with the material particle as the representative of physical reality. This dualism, though disturbing to any systematic mind, has today not yet disappeared...The successful physical systems that have been set up since then represent rather a compromise between these two programs, and it is precisely this character of compromise that stamps them as temporary and logically incomplete...I incline to the belief that physicists will...be brought back to the attempt to realize that program which may suitably be called Maxwell's: the description of physical reality by fields which satisfy...a set of partial differential equations.

One of the various theories proposed to explain the negative result of the famous Michelson-Morley experiment with light waves (conceived to measure the absolute space), was based on the ballistic hypothesis, i.e. on postulating that the speed of light predicted by Maxwell's equations was not given as relative to the medium but as relative to the transmitter (firearm). Had that been the case, the experiment negative results would have not caused such perplexity and frustration (as we shall see in forthcoming sections).

Maxwell’s equations also apply to the electrical currents that are used by the nerves in your body to make your hands move on demand or your heart beat or your brain think.

So in the same way that Kepler and Galileo laid the foundation for Newtonian physics, Faraday paved the way for Maxwell's equations.

Clearly, invisibility is a property that arises at the atomic level, via Maxwell's equations, and hence would be exceedingly difficult, if not impossible, to duplicate using ordinary means. To make Harry Potter invisible, one would have to liquefy him, boil him to create steam, crystallize him, heat him again, and then cool him, all of which would be quite difficult to accomplish, even for a wizard.

Quantum fields fill all all space, and the quantum electric and magnetic fields obey Maxwell's equations. Nevertheless, when you observe the quantum fields, you find their energy packaged in discrete units: photons.

The law of conservation of energy, reborn as the law of conservation of mass/energy, has established itself as one of the few unshakable theoretical guideposts in the wilderness of the world of our sense experiences. In scope and generality it surpasses Newton's laws of motion, Maxwell's equations for electricity and magnetism, and even Einstein's potent little E=mc². It comes as close to an absolute truth as our uncertain age will permit.

No physicist started out impatient with common-sense notions, eager to replace them with some mathematical abstraction that could be understood only by rarified theoretical physics. Instead, they began, as we all do, with comfortable, standard, common-sense notions. The trouble is that Nature does not comply. If we no longer insist on our notions of how Nature ought to behave, but instead stand before Nature with an open and receptive mind, we find that common sense often doesn't work. Why not? Because our notions, both hereditary and learned, of how Nature works were forged in the millions of years our ancestors were hunters and gatherers. In this case common sense is a faithless guide because no hunter-gatherer's life ever depended on understanding time-variable electric and magnetic fields. There were no evolutionary penalties for ignorance of Maxwell's equations. In our time it's different.

If we now take a Kaluza-Klein theory defined in 4+N dimensions and then curl up N dimensions, we will find that the equations split into two pieces. The first piece is Einstein's usual equations, which we retrieve as expected. But the second piece will not be the theory of Maxwell. We find that the remainder is precisely the Yang-Mills theory, which forms the basis of all subatomic physics! This is the key to turning the symmetries of wood into the symmetries of marble.

Michael Faraday and James Clerk Maxwell's explanation of electricity and magnetism paved the way for the illumination of our cities and gave us powerful electric motors and generators as well as instantaneous communication via TV and radio.

Things in electronics don’t get interesting until there is change: A signal changes amplitude, a frequency changes, a voltage level changes from a logical one to a logical zero. And the mathematics of change is calculus. Most people don’t like calculus, and that includes a lot of engineers!

If we no longer insist on our notions of how Nature ought to behave, but instead stand before Nature with an open and receptive mind, we find that common sense often doesn't work. Why not? Because our notions, both hereditary and learned, of how Nature works were forged in the millions of years our ancestors were hunters and gatherers. In this case common sense is a faithless guide because no hunter-gatherer's life ever depended on understanding time-variable electric and magnetic fields. There were no evolutionary penalties for ignorance of Maxwell's equations. In our time it's different.

When you listen to the beautiful sounds of stereo music, remember that you are listening to the rhythms of trillions of electrons obeying this and other bizarre laws of quantum mechanics. But if quantum mechanics were incorrect, then all of electronics including television sets, computers, radios, stereo, and so on, would cease to function. (In fact, if quantum theory were incorrect, the atoms in our bodies would collapse, and we would instantly disintegrate. According to Maxwell's equations, the electrons spinning in an atom should lose their energy within a microsecond and plunge into the nucleus. This sudden collapse is prevented by quantum theory. Thus the fact that we exist is living proof of the correctness of quantum mechanics.) This also means that there is a finite, calculable probability that "impossible" events will occur. For example, I can calculate the probability that I will unexpectedly disappear and tunnel through the earth and reappear in Hawaii. (The time we would have to wait for such an event to occur, it should be pointed out, is longer than the lifetime of the universe. So we cannot use quantum mechanics to tunnel to vacation spots around the world.)

Occasionally someone will ask me about how ego fits into the leadership equation. They’ll want to know what keeps a leader from having a huge ego. I think the answer lies in each leader’s pathway to leadership. If people paid their dues and gave their best in obscurity, ego is usually not a problem.

A full explanation of this is beyond the scope of this book, suffice to say that Einstein was forced into this bold move primarily because Maxwell’s equations for electricity and magnetism were incompatible with Newton’s 200-year-old laws of motion. Einstein abandoned the Newtonian ideas of space and time as separate entities and merged them. In Einstein’s theory there is a special speed built into the structure of spacetime itself that everyone must agree on, irrespective of how they are moving relative to each other. This special speed is a universal constant of nature that will always be measured as precisely 299,792,458 metres (983,571,503 feet) per second, at all times and all places in the Universe, no matter what they are doing. This

Newton had invented the calculus, which was expressed in the language of "differential equations," which describe how objects smoothly undergo infinitesimal changes in space and time. The motion of ocean waves, fluids, gases, and cannon balls could all be expressed in the language of differential equations. Maxwell set out with a clear goal, to express the revolutionary findings of Faraday and his force fields through precise differential equations. Maxwell began with Faraday's discovery that electric fields could turn into magnetic fields and vice versa. He took Faraday's depictions of force fields and rewrote them in the precise language of differential equations, producing one of the most important series of equations in modern science. They are a series of eight fierce-looking differential equations. Every physicist and engineer in the world has to sweat over them when mastering electromagnetism in graduate school. Next, Maxwell asked himself the fateful question: if magnetic fields can turn into electric fields and vice versa, what happens if they are constantly turning into each other in a never-ending pattern? Maxwell found that these electric-magnetic fields would create a wave, much like an ocean wave. To his astonishment, he calculated the speed of these waves and found it to be the speed of light! In 1864, upon discovering this fact, he wrote prophetically: "This velocity is so nearly that of light that it seems we have strong reason to conclude that light itself...is an electromagnetic disturbance.

The bushy-bearded Scottish physicist James Clerk Maxwell (1831–1879) subsequently devised wonderful equations that specified, among other things, how changing electric fields create magnetic fields and how changing magnetic fields create electrical ones. A changing electric field could, in fact, produce a changing magnetic field that could, in turn, produce a changing electric field, and so on. The result of this coupling was an electromagnetic wave.

Imagine that light is shining out from a flashlight. According to common sense, if we run fast enough we could in principle catch up with the front of the beam of light as it advances forward. Common sense might even suggest that we could jog alongside the front of the beam if we managed to run at the speed of light. But if we are to follow Maxwell’s equations to the letter, then no matter how fast we run, the beam still recedes away from us at a speed of 299,792,458 meters per second.

Each time scientists have unraveled a new force, it has changed the course of civilization and altered the destiny of humanity. For example, Newton’s discovery of the laws of motion and gravity laid the groundwork for the machine age and the Industrial Revolution. Michael Faraday and James Clerk Maxwell’s explanation of electricity and magnetism paved the way for the illumination of our cities and gave us powerful electric motors and generators as well as instantaneous communication via TV and radio. Einstein’s E = mc2 explained the power of the stars and helped to unravel the nuclear force. When Erwin Schrödinger, Werner Heisenberg, and others unlocked the secrets of the quantum theory, they gave us the high-tech revolution of today, with supercomputers, lasers, the internet, and all the fabulous gadgets in our living rooms. Ultimately, all the wonders of modern technology owe their origin to the scientists who gradually discovered the fundamental forces of the world.

These fields, which govern the interaction of all subatomic particles, are now called Yang-Mills fields. However, the puzzle that has stumped physicists within this century is why the subatomic field equations look so vastly different from the field equations of Einstein-that is, why the nuclear force seems so different from gravity. Some of the greatest minds in physics have tackled this problem, only to fail. Perhaps the reason for their failure is that they were trapped by common sense. Confined to three or four dimensions, the field equations of the subatomic world and gravitation are difficult to unify. The advantage of the hyperspace theory is that the Yang-Mills field, Maxwell's field, and Einstein's field can all be placed comfortably within the hyperspace field. We see that these fields fit together precisely within the hyperspace field like pieces in a jig-saw puzzle. The other advantage of field theory is that it allows us to calculate the precise energies at which we can expect space and time to foem wormholes.

But historically the fourth dimension has been considered a mere curiosity by physicists. No evidence has ever been found for higher dimensions. This began to change in 1919 when physicist Theodor Kaluza wrote a highly controversial paper that hinted at the presence of higher dimensions. He started with Einstein's theory of general relativity, but placed it in five dimensions (one dimension of time and four dimensions of space; since time is the fourth space-time dimension, physicists now refer to the fourth spatial dimension as the fifth dimension). If the fifth dimension were made smaller and smaller, the equations magically split into two pieces. One piece describes Einstein's standard theory of relativity, but the other piece becomes Maxwell's theory of light! This was a stunning revelation. Perhaps the secret of light lies in the fifth dimension! Einstein himself was shocked by this solution, which seemed to provide an elegant unification of light and gravity. (Einstein was so shaken by Kaluza's proposal that he mulled it over for two years before finally agreeing to have this paper published.) Einstein wrote to Kaluza, "The idea of achieving [a unified theory] by means of a five-dimensional cylinder world never dawned on me...At first glance, I like your idea enormously...The formal unity of your theory is startling.

Therein lies the key, I think, to Einstein’s brilliance and the lessons of his life. As a young student he never did well with rote learning. And later, as a theorist, his success came not from the brute strength of his mental processing power but from his imagination and creativity. He could construct complex equations, but more important, he knew that math is the language nature uses to describe her wonders. So he could visualize how equations were reflected in realities—how the electromagnetic field equations discovered by James Clerk Maxwell, for example, would manifest themselves to a boy riding alongside a light beam. As he once declared, “Imagination is more important than knowledge.”6

Mathematical theories have sometimes been used to predict phenomena that were not confirmed until years later. For example, Maxwell's equations, named after physicist James Clerk Maxwell, predicted radio waves. Einstein's field equations suggested that gravity would bend light and that the universe is expanding. Physicist Paul Dirac once noted that the abstract mathematics we study now gives us a glimpse of physics in the future. In fact, his equations predicted the existence of antimatter, which was subsequently discovered. Similarly, mathematician Nikolai Lobachevsky said that "there is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world.

To Einstein, this insight was monumental. Either Newton or Maxwell was correct. The other had to be wrong. But how could it be that you could never catch up to light? At the patent office, he had plenty of time to ponder this question. One day, in the spring of 1905, it struck him while riding the train in Bern. "A storm broke loose in my mind," he would recall. His brilliant insight was that since the speed of light is measured by clocks and metersticks, and since the speed of light is constant no matter how fast you move, space and time must be distorted in order to keep the speed of light constant! It meant that if you are on fast-moving spaceship, then clocks inside the ship beat slower than clocks on the Earth. Time slows down the faster you move--this phenomenon is described by Einstein's special relativity.

If we shuffle three colored quarks and the equations remain the same, then we say that the equations possess something called SU(3) symmetry. The 3 represents the fact that we have three types of colors, and the SU stands for a specific mathematical property of the symmetry. We say that there are three quarks in a multiplet. The quarks in a multiplet can be shuffled among one another without changing the physics of the theory. Similarly, the weak force governs the properties of two particles, the electron and the neutrino. The symmetry that interchanges these particles, yet leaves the equation the same, is called SU(2). This means that a multiplet of the weak force contains an electron and a neutrino, which can be rotated into each other. Finally, the electromagnetic force has U(1) symmetry, which rotates the components of the Maxwell field into itself. Each of these symmetries is simple and elegant. However, the most controversial aspect of the Standard Model is that it "unifies" the three fundamental forces by simply splicing all three theories into one large symmetry. SU(3) X SU(2) X U(1), which is just the product of the symmetries of the individual forces. (This can be compared to assembling a jigsaw puzzle. If we have three jigsaw pieces that don't quite fit, we can always take Scotch tape and splice them together by hand. This is how the Standard Model is formed, by taping three distinct multiplets together. This may not be aesthetically pleasing, but at least the three jigsaw puzzles now hang together by tape.) Ideally, one might have expected that "the ultimate theory" would have all the particles inside just a single multiplet. Unfortunately, the Standard Model has three distinct multiplets, which cannot be rotated among one another.

Religious creeds are a great obstacle to any full sympathy between the outlook of the scientist and the outlook which religion is so often supposed to require ... The spirit of seeking which animates us refuses to regard any kind of creed as its goal. It would be a shock to come across a university where it was the practice of the students to recite adherence to Newton's laws of motion, to Maxwell's equations and to the electromagnetic theory of light. We should not deplore it the less if our own pet theory happened to be included, or if the list were brought up to date every few years. We should say that the students cannot possibly realise the intention of scientific training if they are taught to look on these results as things to be recited and subscribed to. Science may fall short of its ideal, and although the peril scarcely takes this extreme form, it is not always easy, particularly in popular science, to maintain our stand against creed and dogma.

James Clerk Maxwell helped to enshrine this wave theory when he successfully conjectured a connection between light, electricity, and magnetism. He came up with equations that described the behavior of electric and magnetic fields, and when they were combined they predicted electromagnetic waves. Maxwell found that these electromagnetic waves had to travel at a certain speed: approximately 186,000 miles per second.* That was the speed that scientists had already measured for light, and it was obviously not a mere coincidence.4 It became clear that light was the visible manifestation of a whole spectrum of electromagnetic waves. This includes what we now call AM radio signals (with a wavelength of 300 yards), FM radio signals (3 yards), and microwaves (3 inches). As the wavelengths get shorter (and the frequency of the wave cycles thus increases), they produce the spectrum of visible light, ranging from red (25 millionths of an inch) to violet (14 millionths of an inch). Even shorter wavelengths produce ultraviolet rays, X-rays, and gamma rays. When we speak of “light” and the “speed of light,” we mean all electromagnetic waves, not just the ones that are visible to our eyes.

Once again, he was deducing a theory from principles and postulates, not trying to explain the empirical data that experimental physicists studying cathode rays had begun to gather about the relation of mass to the velocity of particles. Coupling Maxwell’s theory with the relativity theory, he began (not surprisingly) with a thought experiment. He calculated the properties of two light pulses emitted in opposite directions by a body at rest. He then calculated the properties of these light pulses when observed from a moving frame of reference. From this he came up with equations regarding the relationship between speed and mass. The result was an elegant conclusion: mass and energy are different manifestations of the same thing. There is a fundamental interchangeability between the two. As he put it in his paper, “The mass of a body is a measure of its energy content.” The formula he used to describe this relationship was also strikingly simple: “If a body emits the energy L in the form of radiation, its mass decreases by L/V 2.” Or, to express the same equation in a different manner: L=mV 2. Einstein used the letter L to represent energy until 1912, when he crossed it out in a manuscript and replaced it with the more common E. He also used V to represent the velocity of light, before changing to the more common c. So, using the letters that soon became standard, Einstein had come up with his memorable equation: E=mc2

To prove to an indignant questioner on the spur of the moment that the work I do was useful seemed a thankless task and I gave it up. I turned to him with a smile and finished, 'To tell you the truth we don't do it because it is useful but because it's amusing.' The answer was thought of and given in a moment: it came from deep down in my mind, and the results were as admirable from my point of view as unexpected. My audience was clearly on my side. Prolonged and hearty applause greeted my confession. My questioner retired shaking his head over my wickedness and the newspapers next day, with obvious approval, came out with headlines 'Scientist Does It Because It's Amusing!' And if that is not the best reason why a scientist should do his work, I want to know what is. Would it be any good to ask a mother what practical use her baby is? That, as I say, was the first evening I ever spent in the United States and from that moment I felt at home. I realised that all talk about science purely for its practical and wealth-producing results is as idle in this country as in England. Practical results will follow right enough. No real knowledge is sterile. The most useless investigation may prove to have the most startling practical importance: Wireless telegraphy might not yet have come if Clerk Maxwell had been drawn away from his obviously 'useless' equations to do something of more practical importance. Large branches of chemistry would have remained obscure had Willard Gibbs not spent his time at mathematical calculations which only about two men of his generation could understand. With this trust in the ultimate usefulness of all real knowledge a man may proceed to devote himself to a study of first causes without apology, and without hope of immediate return.

With the introduction of radio, we now had a superfast. convenient, and wireless way of communicating over long distances. Historically, the lack of a fast and reliable communication system was one of the great obstacles to the march of history. (In 490 BCE, after the Battle of Marathon between the Greeks and the Persians, a poor runner was ordered to spread the news of the Greek victory as fast as he could. Bravely, he ran 26 miles to Athens after previously running 147 miles to Sparta, and then, according to legend, dropped dead of sheer exhaustion. His heroism, in the age before telecommunication, is now celebrated in the modern marathon.) Today, we take for granted that we can send messages and information effortlessly across the globe, utilizing the fact that energy can be transformed in many ways. For example, when speaking on a cell phone, the energy of the sound of your voice converts to mechanical energy in a vibrating diaphragm. The diaphragm is attached to a magnet that relies on the interchangeability of electricity and magnetism to create an electrical impulse, the kind that can be transported and read by a computer. This electrical impulse is then translated into electromagnetic waves that are picked up by a nearby microwave tower. There, the message is amplified and sent across the globe. But Maxwell's equations not only gave us nearly instantaneous communication via radio, cell phone, and fiber-optic cables, they also opened up the entire electromagnetic spectrum, of which visible light and radio were just two members. In the 166os, Newton had shown that white light, when sent through a prism, can be broken up into the colors of the rainbow. In 1800, William Herschel had asked himself a simple question: What lies beyond the colors of the rainbow, which extend from red to violet? He took a prism, which created a rainbow in his lab, and placed a thermometer below the color red, where there was no color at all. Much to his surprise, the temperature of this blank area began to rise. In other words, there was a "color" below red that was invisible to the naked eye but contained energy. It was called infrared light. Today, we realize that there is an entire spectrum of electromagnetic radiation, most of which is invisible, and each has a distinct wavelength. The wavelength of radio and TV, for example, is longer than that of visible light. The wavelength of the colors of the rainbow, in turn, is longer than that of ultraviolet and X-rays. This also meant that the reality we see all around us is only the tiniest sliver of the complete EM spectrum, the smallest approximation of a much larger universe

But so far, we have only discussed applying quantum mechanics to the matter that moves within the gravity fields of Einstein’s theory. We have not discussed a much more difficult question: applying quantum mechanics to gravity itself in the form of gravitons. And this is where we encounter the biggest question of all: finding a quantum theory of gravity, which has frustrated the world’s great physicists for decades. So let us review what we have learned so far. We recall that when we apply the quantum theory to light, we introduce the photon, a particle of light. As this photon moves, it is surrounded by electric and magnetic fields that oscillate and permeate space and obey Maxwell’s equations. This is the reason why light has both particle-like and wavelike properties. The power of Maxwell’s equations lies in their symmetries—that is, the ability to turn electric and magnetic fields into each other. When the photon bumps into electrons, the equation that describes this interaction yields results that are infinite. However, using the bag of tricks devised by Feynman, Schwinger, Tomonaga, and many others, we are able to hide all the infinities. The resulting theory is called QED. Next, we applied this method to the nuclear force. We replaced the original Maxwell field with the Yang-Mills field, and replaced the electron with a series of quarks, neutrinos, etc. Then we introduced a new bag of tricks devised by ’t Hooft and his colleagues to eliminate all the infinities once again. So three of the four forces of the universe could now be unified into a single theory, the Standard Model. The resulting theory was not very pretty, since it was created by cobbling together the symmetries of the strong, weak, and electromagnetic forces, but it worked. But when we apply this tried-and-true method to gravity, we have problems. In theory, a particle of gravity should be called the graviton. Similar to the photon, it is a point particle, and as it moves at the speed of light, it is surrounded by waves of gravity that obey Einstein’s equations. So far, so good. The problem occurs when the graviton bumps into other gravitons and also atoms. The resulting collision creates infinite answers. When one tries to apply the bag of tricks painfully formulated over the last seventy years, we find that they all fail. The greatest minds of the century have tried to solve this problem, but no one has been successful. Clearly, an entirely new approach must be used, since all the easy ideas have been investigated and discarded. We need something truly fresh and original. And that leads us to perhaps the most controversial theory in physics, string theory, which might just be crazy enough to be the theory of everything.

To a theoretician, all these criticisms are troublesome but not fatal. But what does cause problems for a theoretician is that the model seems to predict a multiverse of parallel universes, many of which are crazier than those in the imagination of a Hollywood scriptwriter. String theory has an infinite number of solutions, each describing a perfectly well-behaved finite theory of gravity, which do not resemble our universe at all. In many of these parallel universes, the proton is not stable, so it would decay into a vast cloud of electrons and neutrinos. In these universes, complex matter as we know it (atoms and molecules) cannot exist. They only consist of a gas of subatomic particles. (Some might argue that these alternate universes are only mathematical possibilities and are not real. But the problem is that the theory lacks predictive power, since it cannot tell you which of these alternate universes is the real one.) This problem is actually not unique to string theory. For example, how many solutions are there to Newton’s or Maxwell’s equations? There are an infinite number, depending on what you are studying. If you start with a light bulb or a laser and you solve Maxwell’s equations, you find a unique solution for each instrument. So Maxwell’s or Newton’s theories also have an infinite number of solutions, depending on the initial conditions—that is, the situation you start with. This problem is likely to exist for any theory of everything. Any theory of everything will have an infinite number of solutions depending on the initial conditions. But how do you determine the initial conditions of the entire universe? This means you have to input the conditions of the Big Bang from the outside, by hand. To many physicists this seems like cheating. Ideally, you want the theory itself to tell you the conditions that gave rise to the Big Bang. You want the theory to tell you everything, including the temperature, density, and composition of the original Big Bang. A theory of everything should somehow contain its own initial conditions, all by itself. In other words, you want a unique prediction for the beginning of the universe. So string theory has an embarrassment of riches. Can it predict our universe? Yes. That is a sensational claim, the goal of physicists for almost a century. But can it predict just one universe? Probably not. This is called the landscape problem. There are several possible solutions to this problem, none of them widely accepted. The first is the anthropic principle, which says that our universe is special because we, as conscious beings, are here to discuss this question in the first place. In other words, there might be an infinite number of universes, but our universe is the one that has the conditions that make intelligent life possible. The initial conditions of the Big Bang are fixed at the beginning of time so that intelligent life can exist today. The other universes might have no conscious life in them.

Or think of the tale of the blind men who encounter an elephant for the first time. One wise man, touching the ear of the elephant, declares the elephant is flat and two-dimensional like a fan. Another wise man touches the tail and assumes the elephant is like rope or a one-dimensional string. Another, touching a leg, concludes the elephant is a three-dimensional drum or a cylinder. But actually, if we step back and rise into the third dimension, we can see the elephant as a three-dimensional animal. In the same way, the five different string theories are like the ear, tail, and leg, but we still have yet to reveal the full elephant, M-theory. Holographic Universe As we mentioned, with time new layers have been uncovered in string theory. Soon after M-theory was proposed in 1995, another astonishing discovery was made by Juan Maldacena in 1997. He jolted the entire physics community by showing something that was once considered impossible: that a supersymmetric Yang-Mills theory, which describes the behavior of subatomic particles in four dimensions, was dual, or mathematically equivalent, to a certain string theory in ten dimensions. This sent the physics world into a tizzy. By 2015, there were ten thousand papers that referred to this paper, making it by far the most influential paper in high-energy physics. (Symmetry and duality are related but different. Symmetry arises when we rearrange the components of a single equation and it remains the same. Duality arises when we show that two entirely different theories are actually mathematically equivalent. Remarkably, string theory has both of these highly nontrivial features.) As we saw, Maxwell’s equations have a duality between electric and magnetic fields—that is, the equations remain the same if we reverse the two fields, turning electric fields into magnetic fields. (We can see this mathematically, because the EM equations often contain terms like E2 + B2, which remain the same when we rotate the two fields into each other, like in the Pythagorean theorem). Similarly, there are five distinct string theories in ten dimensions, which can be proven to be dual to each other, so they are really a single eleven-dimensional M-theory in disguise. So remarkably, duality shows that two different theories are actually two aspects of the same theory. Maldacena, however, showed that there was yet another duality between strings in ten dimensions and Yang-Mills theory in four dimensions. This was a totally unexpected development but one that has profound implications. It meant that there were deep, unexpected connections between the gravitational force and the nuclear force defined in totally different dimensions. Usually, dualities can be found between strings in the same dimension. By rearranging the terms describing those strings, for example, we can often change one string theory into another. This creates a web of dualities between different string theories, all defined in the same dimension. But a duality between two objects defined in different dimensions was unheard of.

Maxwell’s equations apply to all electrical systems, not merely to a specialized and idealized class of electrical circuits.

Maxwell’s equations are more general than network theory,

In fact, it was by demanding a local version of special relativity that Einstein got the equations for the metric field that are the core of general relativity! And it is by demanding local versions of rotations in property spaces that C.N. Yang and Robert Mills found the equations that bear their name and govern the weak and strong fluids. Yang and Mills built on the work of Herman Weyl, who showed that Maxwell's equations for the electromagnetic fluid can be derived in that way.

As Maxwell recognized, if atoms and molecules operated on the same principles as the Solar System, the world would be very different. Every atom would be different from every other, and every atom would change over time. Such a world wouldn't have chemistry as we know it, with definite substances and fixed rules. It is not immediately obvious what makes atomic systems behave so differently. In both cases we have a massive central body attracting several small ones. The forces in play, gravitational or electrical, are broadly similar-both decrease as the square of the distance. But there are three factors which make the physical outcome very different, giving us stereotyped atoms but individualized solar systems: 1. Whereas planets differ from one another (as do stars), all electrons have exactly the same properties (as do all nuclei of a given element, or more precisely a given isotope). 2. Atoms obey the rules of quantum mechanics. 3. Atoms are starved for energy. The first item in this explanation begs the question, of course. We're trying to explain why atoms can be the same as each other, and we start off by asserting that some other things, electrons, are all the same as each other! We'll come back to that later. But having the same parts doesn't guarantee the same outcome, by any means. Even if all planets were the same as one another, and all stars were the same as one another, there would still be many possible designs for solar systems, and they'd all be subject to change. We've seen how quantum mechanics brings discreteness, and fixed patterns, into the description of continuous objects that obey dynamical equations. It's the story you'll recall, that unfolds in figures 24 (page 172), 25 (page 174), and 26 (page 187), and plate CC. To close the loop, we need to understand why the electrons in atoms are usually found in just one among their infinite variety of patterns. That's where our third item comes in. The pattern with lowest energy-the so-called ground state-is the one we generally find, because atoms are starved for energy. Why are atoms starved for energy? Ultimately, it is because the Universe is big, cold, and expanding. Atoms can pass from one pattern to another by emitting light, and losing energy, or absorbing light, and gaining energy. If emission and absorption were balanced, many patterns would be in play. That's what would happen in a hot, closed system. Light emitted at one time would be absorbed later, and a balanced equilibrium would set in. But in a big, cold, expanding Universe, emitted light leaks into vast interstellar spaces, carrying away energy that is not returned. In this way we find that dynamical equations, which by themselves cannot impose structure, do so through jujitsu (gentle skill), focusing the power of other principles. They guide the constraining powers of quantum mechanics and cosmology. Cosmology explains their poverty of energy, and quantum mechanics shows how poverty of energy imposes structure.

In fact, Einstein proceeded to turn the argument around, by showing that one could derive the complete system of four Maxwell equations from one of them, by making Galilean transformations to recover the general case. (By putting charge in motion, you get currents, and by putting electric fields in motion, you get magnetic fields. Thus the law governing how unmoving electric charges generate electric fields, after Galilean transformations, gives the general case.) That profound trick was a taste of the future. Symmetry, rather than a deduction from given laws, became a primary principle, with a life of its own. One can constrain the laws by requiring them to have symmetry.

In 1862, the Scottish mathematician James Clerk Maxwell developed a set of fundamental equations that unified electricity and magnetism. On his deathbed, he coughed up a strange sort of confession, declaring that “something within him” discovered the famous equations, not he. He admitted he had no idea how ideas actually came to him—they simply came to him. William Blake related a similar experience, reporting of his long narrative poem Milton: “I have written this poem from immediate dictation twelve or sometimes twenty lines at a time without premeditation and even against my will.” Johann Wolfgang von Goethe claimed to have written his novella The Sorrows of Young Werther with practically no conscious input, as though he were holding a pen that moved on its own.

Faraday's self-education was deficient in one significant respect: he had learned no mathematics. For him, Ampère's equations might as well have been written in Egyptian hieroglyphics. We shall never know what Faraday would have achieved had he mastered mathematics, but, paradoxically, his ignorance may have been an advantage. It led him to derive his theories entirely from experimental observation rather than to deduce them from mathematical models.

Coulomb believed that each type of force was distinct: for example, electrical forces required a different type of fluid from magnetic forces, so it was impossible that one could be converted to the other. From our distant and privileged viewpoint, it is evident that Kant and Schelling were right, at least in broad terms, and that Coulomb was wrong. But things looked different in the early 1800s. Coulomb's equations were clear, elegant, and gave exact answers, while the ideas of Kant and Schelling were speculative and vague, even metaphysical.

When Maxwell’s equations didn’t pick out any aether or any physical medium whatsoever for light it was because there is no physical medium! Light isn’t in the physical world. It’s exactly what isn’t physical.

The relationships (Maxwell's equations) are the epitome of mathematical consieness, compact enough to fit on a T-shirt yet powerful enough to describe all manner of electromagnetism.

As I have watched effective communicators inspire people, I have come to the conclusion that there is a kind of formula, which I call the Inspiration Equation, that comes into play. It works like this: What They Know + What They See + What They Feel = Inspiration

electricity and magnetism into a set of four elegant mathematical equations. On seeing them, Ludwig Boltzmann immediately recognised the magnitude of Maxwell’s achievement and could only quote Goethe in admiration: ‘Was it a God that wrote these signs?

The move from a particle description to a field description will be especially fruitful if the fields obey simple equations, so that we can calculate the future values of fields from the values they have now...Maxwell's theory of electromagnetism, general relativity, and QCD [quantum chromodynamics] all have this property. Evidently, Nature has taken the opportunity to keep things relatively simple by using fields.

The fact that space and time are intimately connected, as in figure 3.2, implies a subtle restructuring of Newton's mechanics, which Einstein rapidly completes in 1905 and 1906. A first result of this restructuring is that as space and time fuse together in a single concept of spacetime, so the electric field and the magnetic fields fuse together in the same way, merging into a single entity that today we call the "electromagnetic field." The complicated equations written by Maxwell for the two fields become simple when written in this new language.

The notions of fields and particles, separated by Faraday and Maxwell, end up merging in quantum mechanics. The way this happens in the theory is elegant: the equation of Dirac determines the values a variable can take. Applied to the energy of Faraday's lines, they tell us that this energy can take on only certain values and not others. Since the energy of the electromagnetic field can take on only certain values, the field behaves line a set of packets of energy. These are precisely the quanta of energy introduced by Planck and Einstein thirty years later. The circle closes, and the story is complete. The equations of the theory, written by Dirac, account for the granular nature of light, which Planck and Einstein had intuited.

The electromagnetic waves are vibrations of Faraday's lines, but also, at small scale, swarms of photons. When they interact with something else, as in the photoelectric effect, they manifest themselves as particles: on our eyes, light rains in separate droplets, in single photons. Photons are the "quanta" of the electromagnetic field. But the electrons and all the other particles of which the world is made are equally "quanta" of a field! A "quantum field" similar to Faraday and Maxwell's, subject to granularity and to quantum probability. Dirac writes the equations for the field of the electrons and of the other elementary particles. The sharp distinction between fields and particles introduced by Faraday vanishes.

According to Carl Sagan: "Maxwell's equations have had a greater impact on human history than any ten presidents.

So when Trump, in his defense of the Robert E. Lee statue in Charlottesville, Virginia, claims that there is “literally no difference” between Lee and George Washington—both “owned slaves,” both “rebelled against the ruling government,” “both were great men, great Americans, great commanders,” and both “saved America”—he is equating Confederate nationalism and American patriotism. He is equating the defense of slavery with the revolutionary cause of independence and scolding the media for not getting the parallel.

Everything passes, Hiroto,” Dad said. “That feeling in your heart: It’s called mono no aware. It is a sense of the transience of all things in life. The sun, the dandelion, the cicada, the Hammer, and all of us: We are all subject to the equations of James Clerk Maxwell and we are all ephemeral patterns destined to eventually fade, whether in a second or an eon.

His system of equations worked with jewelled precision. Its construction had been an immense feat of sustained creative effort in three stages spread over 9 years. The whole route was paved with inspired innovations but from a historical perspective one crucial step stands out-the idea that electric currents exist in empty space. It is these displacement currents that give the equations their symmetry and make the waves possible. Without them the term @E/@t in equation (4) becomes zero and the whole edifice crumbles.

Within the modified equations, Kaluza found the ones Einstein had already used successfully to describe gravity in the familiar three dimensions of space and one of time. But because his new formulation included an additional dimension of space, Kaluza found an additional equation. Lo and behold, when Kaluza derived this equation he recognized it as the very one Maxwell had discovered half a century earlier to describe the electromagnetic field.

These electric and magnetic fields can be elegantly unified into what's known as the electromagnetic field, represented by six numbers at each point in spacetime. As we discussed in Chapter 7, light is simply a wave rippling through the electromagnetic field, so if our physical world is a mathematical structure, then all the light in our Universe (which feels quite physical) corresponds to six numbers at each point in spacetime (which feels quite mathematical). These numbers obey the mathematical relations that we know as Maxwell's equations, shown in Figure 10.4.

It behooves us ... to consider the fateful formula E = mc2, which almost everyone in the world attributes to Albert Einstein's theory [of relativity]. Despite the fact, however, that Einstein did derive this formula from his special theory of relativity, it stems actually from [the] classical part [of the theory]: i.e., from the Maxwell equations for electromagnetic fields, which goes back to 1865. The famous formula has consequently no bearing whatsoever on relativistic physics, a fact Einstein himself admitted in 1950. Obviously, however, in the interim that fateful formula came to be viewed worldwide as the consummate vindication of Einstein's theory: what indeed could be more convincing than the explosion of an atom bomb?

In three dimensions where the invariance of the speed of light is not readily manifest, the Maxwell equations are pieces (shadows) of a four-dimensional object with manifest invariance of the speed of light. This is like the shadow of an upright bicycle wheel cast on the ground, which can look like a line-the circular symmetry is no longer manifest.

This last figure, the White Magician, symbolizes the self-transcending element in the scientist's motivational drive and emotional make-up; his humble immersion into the mysteries of nature, his quest for the harmony of the spheres, the origin of life, the equations of a unified field theory. The conquistadorial urge is derived from a sense of power, the participatory urge from a sense of oceanic wonder. 'Men were first led to the study of natural philosophy', wrote Aristotle, 'as indeed they are today, by wonder.' Maxwell's earliest memory was 'lying on the grass, looking at the sun, and wondering'. Einstein struck the same chord when he wrote that whoever is devoid of the capacity to wonder, 'whoever remains unmoved, whoever cannot contemplate or know the deep shudder of the soul in enchantment, might just as well be dead for he has already closed his eyes upon life'. This oceanic feeling of wonder is the common source of religious mysticism, of pure science and art for art's sake; it is their common denominator and emotional bond.

It’s enormously easier (as it turns out) to write a computer program that simulates Maxwell’s equations, compared to a computer program that simulates an intelligent emotional mind like Thor.

The Unified Field (Theory) of Everything A force feeding all the “energy and matter” in the Universe, creating the unified field, is the Universal Mind (Universal Source). Michio Kaku said that “an equation an inch long would allow us to read the mind of God.” Clerk Maxwell created the first field theory (electromagnetism) in the mid-19th century. In the 20th century, Einstein’s general relativity theory, treating gravitation, was the second field theory. Einstein and some other scientists tried to create a theory where electromagnetism and gravity would be different aspects of the same field, but they failed. With quantum physics, things became even more complicated. Einstein tried to reconcile the theory of relativity and quantum theory with no success. The standard model theory describes electromagnetism and strong and weak nuclear force, yet gravitation stays out of the picture. I must reiterate that gravitation is not a force and cannot fit any equation. The “force” affecting all matter and energy, as described by the law of gravitation, is not the consequence of mass but of motion. The “force” of motion of larger masses is larger than the “force” of motion of smaller masses. The mass does not cause gravitation and is not the cause of attraction and curvature of space but motion. The Universal Mind in action is the Ultimate Unifying Force, the Universal Source of all information needed to create energy and matter, the laws of physics, and the Unified Field of Everything. The Primordial Immaterial Indestructible Energy is the Universal Source powering the Unified Field of Everything.