Maxwell's Equations Were Discovered [video]

**TL;DR:** The video tells the story of how Maxwell's equations were discovered: Faraday laid the foundation of electromagnetism through experiments, and Maxwell unified them into four equations using mathematics, ultimately transforming the world profoundly. ## From Faraday to Maxwell: How Four Equations Reshaped the World From life-saving technologies to auroras, from deep-space objects to communicating with astronauts, and to the light on your screen—everything is thanks to what we now call Maxwell's equations. These four equations, primarily discovered by two scientists in the mid-19th century, led Einstein to discover special relativity, profoundly changed the world, and their importance ranks as the most significant event of the entire 19th century. On the wall of Einstein's office hung three portraits: Newton, Faraday, and Maxwell. Maxwell's position was the most prominent. When asked if he stood on Newton's shoulders, Einstein replied, "No, I stand on Maxwell's shoulders." He also wrote, "The greatest change in our perception of reality since Newton came from the work of Faraday and Maxwell on electromagnetic phenomena." Maxwell stood on Faraday's shoulders. To understand Maxwell's epochal discovery, we must first understand Faraday's scientific research. ## Michael Faraday: From Bookbinder's Apprentice to Experimental Master Michael Faraday was born in 1791 in a London suburb. He received only a very limited formal education—he described it himself as merely a smattering of reading, writing, and arithmetic. He left school to become a bookbinder's apprentice, and it was here that he gained three qualities that would greatly benefit him in the future: dexterous hands, keen observation skills, and most importantly, an obsessive reading habit. He spent all his free time reading any book he could find. One book had a profound influence on Faraday's thinking: Isaac Watts's *The Improvement of the Mind*. Watts taught Faraday to use precise language, always be guided by observed facts, and not to rush into building universal theories based on only a few observations or experiments. Faraday took these principles to heart as a guide for the rest of his life. From Watts, he also learned the importance of having a good teacher, so he began attending lectures by a scientist named John Tatum, who lived nearby and spoke on the subject of electricity. In these lectures, Faraday learned about a recent invention—the battery. Created by the Italian scientist Alessandro Volta ten years earlier, it was sweeping through the scientific world. For the first time, scientists could obtain a continuous, steady current and explore matter in entirely new ways. Making a battery was remarkably simple: alternately stack copper plates and metal plates, placing cloth or cardboard soaked in salt water between each layer, and connecting the top and bottom of the stack with a wire to generate a current. Faraday made one himself—his first experiment ever, which also led him to find his true lifelong pursuit. Fortunately, Sir Humphry Davy—then the most prominent chemist in London—held public lectures. Faraday dreamed of attending but could not afford tickets. Then a regular customer at his bookstore, impressed by his talent, bought tickets for all four lectures and gave them to him. Faraday took very detailed notes, bound them into a book, sent it to Davy, and asked if he could become his assistant. Initially unsuccessful, Davy eventually hired Faraday, paving his way to becoming a scientist. ## Faraday's Key Discoveries: Electric Motor, Electromagnetic Induction, Generator Working for Davy brought Faraday many opportunities. Over the next 15 years or so, he focused his experimental research on exploring the nature of electricity and magnetism, making the most important discoveries of his career. ### The Birth of the Electric Motor In 1820, Faraday learned of the accidental discovery by Danish physicist Hans Christian Ørsted: while setting up a Volta battery, Ørsted placed a compass nearby and observed the needle deflect—the first time humans observed a connection between electricity and magnetism. Ørsted repeated the experiment, placing more magnets around the wire, and found that the magnets formed circles around the wire. He concluded that the electric current moving along the wire somehow produced a circular force around it. Upon reading this discovery, Faraday decided to experiment himself. He believed that, conversely, a magnet could produce a circular force on a current-carrying wire. His device was ingenious: he placed a bar magnet in a basin of melted wax, let the wax solidify to fix the magnet, filled it with mercury, hung a wire from a stand, and then connected one terminal of a battery to the top of the wire and the other terminal to the mercury, forming a closed circuit. The wire indeed moved—the magnet produced a circular force on the wire. He then modified the device so that the magnet floated in the mercury, loosely tied to the bottom of the basin, while replacing the hanging wire with a fixed one. This time he observed the magnet moving around the wire—the world's first electric motor was born. ### The Law of Electromagnetic Induction Ten years passed before Faraday's next major breakthrough. French physicist André-Marie Ampère had already shown that an electric current could not only affect a magnetic needle but also another current-carrying wire. Faraday took these discoveries further. He noticed that if you wind a wire into many coils, the magnet becomes stronger, so he decided to experiment again. He first made a ring out of soft iron and wound two sets of coils around it, only one set connected to the battery. When current passed through the primary coil, it would magnetize the iron ring, and he wanted to see if this could induce a current in the secondary coil. He tested by bringing a magnetic needle near the wire of the secondary coil: if there was current in the secondary coil, the needle would deflect. Nothing happened. However, the moment he disconnected the current from the primary coil, the magnetic needle deflected—there must have been a momentary current flow. When he connected the current again, the needle deflected in the opposite direction. Faraday discovered that when the current is switched on, the magnetic field changes, inducing a current in the secondary coil; but when the current is steady and the magnetic field constant, there is no current; when the current is switched off, the magnetic field changes again, inducing another secondary current. In other words, **a changing magnetic field produces an electric field**. This is Faraday's law of electromagnetic induction. Faraday had successfully used magnetism to produce electricity—but using electricity he had generated to create the magnetism. His next idea was whether the same principle applied to ordinary permanent magnets. He achieved this with the following setup: the circuit had no power source, but was connected to a device called a galvanometer to detect any current. Then Faraday took a permanent magnet and thrust it through the coil. As the magnet entered, the needle deflected in one direction; as it left, it deflected in the opposite direction. The faster the magnet moved, the larger the current produced. By repeatedly moving the magnet in and out, Faraday successfully generated alternating current. He had found a way to produce electricity purely from magnetism. Finally, by placing a rotating metal disk between two magnetic bars, he found a way to produce a continuous direct current—the first generator was born, a decade after he invented the first electric motor. ## Faraday's Theory of Lines of Force: A Qualitative Framework These two discoveries laid the foundation for the modern technological world. But for Faraday and everyone at the time, the nature of electricity and magnetism remained a mystery. Scientists knew that objects could be positively or negatively charged and followed Coulomb's law, but electrons and protons had not yet been discovered. Faraday described the general agnosticism of the scientific community at the time, writing: "By electric current, I mean any ongoing motion, whether it be of an electric fluid, two fluids flowing in opposite directions, or merely vibrations, or more generally, a force in progress." Over the next decade or so, Faraday continued experimenting and thinking deeply. He eventually sketched out a rough framework for a theory: the universe is filled with lines of force—electric, magnetic, and perhaps even gravitational. Where these lines of force intersect is what we perceive as matter. When these lines of force are disturbed, they vibrate, allowing energy waves to propagate through space. He even proposed that light is a form of these energy waves. However, due to his lack of mathematical education, his theory remained qualitative. A quantitative framework to place Faraday's theory on a solid mathematical foundation required another genius. ## James Clerk Maxwell: The Mathematical Genius Arrives James Clerk Maxwell was born in 1831 in Edinburgh. Unlike Faraday, he came from a wealthy family and enjoyed an excellent educational environment. Although he lost his mother at age 8 and was sent to live with an aunt at age 10, he attended one of the best schools in Scotland at the time—Edinburgh Academy. He published his first paper at age 14, entered the University of Edinburgh at 16 and published two more papers, then completed his undergraduate studies at Cambridge University. After graduation, he stayed at Cambridge as a fellow, allowing him plenty of time to pursue his own research interests. He had two particular areas of interest in nature: color vision and the relationship between electricity and magnetism. In color vision, he discovered that by mixing appropriate amounts of red, blue, and green light, he could create almost any color. He built an ingenious device—a spinning top with red, blue, and green paper discs—and by precisely measuring the amount of each color and then spinning the top, he demonstrated that he could perfectly reproduce every color. This process is still essentially used to generate color today. While studying color vision, Maxwell's interest in electricity and magnetism grew steadily. Eventually he devoted his full attention to it, trying to understand the significance of Faraday's experimental work. Although many in the mathematical and physics communities at the time thought Faraday's ideas about lines of force were wrong, Maxwell was convinced Faraday was right. He set himself the task of expressing Faraday's ideas in mathematical language. He wrote in a letter to a friend: "I am again immersed in the study of electricity. I hope to see through the essence of the problem and obtain a decent theory that can be understood." ### The 1855 Paper: *On Faraday's Lines of Force* His first breakthrough came in 1855, when he published a paper titled *On Faraday's Lines of Force*. He sought a suitable analogy for these lines of force and eventually found one in the steady flow of an incompressible fluid—the most typical example being water. The power of this analogy lies in the fact that the fluid itself has a 1/r² property: the velocity of fluid particles flowing out from a source decays as 1/r² with increasing distance; for a sink, the flow is inward. At the time, electric and magnetic forces were known to follow similar laws, so the velocity of this flow represented the strength of the force. Maxwell found a way to mathematically express Faraday's lines of force. This was the origin of Maxwell's first two equations: - The divergence of the electric field equals charge divided by a constant. - Since there are no magnetic monopoles, the divergence of the magnetic field is zero. Through this analogy, Maxwell successfully explained static electric and magnetic fields as well as steady currents, but it did not apply to changing fields or currents. He decided to set this work aside temporarily and spend a few years studying other problems. ### The 1861 Paper: *On Physical Lines of Force* Six years later, he published a masterpiece—a multi-part paper titled *On Physical Lines of Force*. He began by telling the reader not to take it too seriously: "I present this model, not to say that the connections in nature are exactly like this. But from a mechanical point of view, it is conceivable." He knew there must be some kind of rotation. So he proposed the following model: imagine tightly packed small units that can rotate. When each unit rotates, its poles flatten and its equator widens. The combined effect resembles a vortex, representing the magnetic field. The faster the rotation, the stronger the magnetic field. Then he added another element to the model: tiny particles placed between these units. The transcription does not fully elaborate this part, but it means that Maxwell continued to derive other equations using the mechanical model, eventually forming the complete set of Maxwell's equations. ## Conclusion The combined work of Faraday and Maxwell allowed humanity, for the first time, to theoretically unify electricity, magnetism, and light. These four equations not only laid the foundation for almost all modern technology—from power generation and radio to computers—but also profoundly changed our understanding of the universe. As Einstein said, he stood on Maxwell's shoulders. **Source:** [Maxwell's Equations Were Discovered [video]](https://www.youtube.com/watch?v=-hua8RWopfw)