4 Equations That Built The Modern World

Added: 2026-05-21

[00:00:00]This is a video about four equations that revolutionized the modern world.

[00:00:05]From life-saving technology to breathtaking auroras. From the discovery of deep space objects through communicating with astronauts on their way to the moon to the very light on your screen right now. All of this is possible because of these four equations known today as Maxwell's equations. They were primarily discovered by two scientists in the mid 1800s led Einstein to discover his special theory of relativity and eventually changed the world so profoundly that their discovery is arguably the most significant event of the entire 19th century.

[00:00:49]On a wall in Einstein's office hung three portraits, Isaac Newton, Michael Faraday, and James Clark Maxwell. Though Einstein had tremendous respect for all three, Maxwell held pride of place. When asked if he stood on Newton's shoulders, Einstein replied, "No, I stood on the shoulders of Maxwell." Elsewhere he wrote the greatest change in our conception of reality since Newton was brought about by Faraday and Maxwell's work on electromagnetic phenomena. So just as Einstein stood on the shoulders of Maxwell, it is Faraday whose shoulders Maxwell stood on. To get a deep appreciation for Maxwell's monumental discovery, then we must turn to the scientific work of his forerunner. Michael Faraday was born in 1791 just outside of London. His life story is truly inspirational. After very little formal education, which he himself described as consisting of little more than the rudiments of reading, writing, and arithmetic, Faraday dropped out of school and began working as a bookbinding apprentice. It was here that he acquired three qualities that would help him immensely later in life. skillful hands, good observation skills, and most importantly, an obsessive habit of reading. He spent all his spare time reading whatever he could get his hands on across seemingly every subject. There was one book in particular that especially influenced Faraday's thinking, The Improvement of the Mind by Isaac Watts, a sort of intellectual self-help book. Watts taught Faraday to use precise language and the importance of always being guided by observed facts. He said one should be not too hasty to erect general theories from a few particular observations, appearances or experiments. Principles that Faraday took to heart and used as a guide for the rest of his life. It is from Watts 2 that Faraday learned the importance of having a good teacher to learn from alongside reading books. Inspired by this lesson, Faraday began attending nearby lectures from a scientist named John Tatum. The topic electricity. In these lectures, Faraday learned of a recent invention, the battery. Invented just 10 years prior by an Italian scientist named Alessandro Volta. The battery was taking the scientific world by storm. For the first time in history, scientists had access to a continuous flow of electricity, which they could use to probe matter in entirely novel ways. The best part, it was incredibly easy to make, and Volulta provided detailed instructions.

[00:03:44]Make a stack of alternating copper and metal plates. In between each layer, place a piece of cloth or cardboard.

[00:03:52]soaked in brine. And just like that, an electric force was generated. By connecting the top and bottom of the stack with a wire, an electric current would be produced. For Faraday, it wasn't enough to just learn about this.

[00:04:09]After saving enough money, he built one himself. The first experiment he ever performed and the moment where he realized his true calling in life. From this moment on, Faraday would do everything in his power to become a scientist. And fortunately for him, a series of events occurred that opened up a path for him to turn this dream into a reality. Enter Sir Humphrey Davyy, London's most prominent chemist at the time. Davyy was giving a set of four public lectures aimed primarily for London's wealthy elite. Faraday dreamed of attending but could not afford tickets. Then one day, a regular customer at the bookshop who admired Faraday's brilliance decided to buy tickets to all four lectures and gave them to him. Overjoyed at the opportunity given to him, Faraday took incredibly detailed notes while attending these lectures. He eventually bound them up in a book, sent them to Davey, and asked if there was any possibility of working for him as an assistant, though initially he couldn't.

[00:05:16]Davey eventually hired Faraday and created a path for him to be a scientist. If you're anything like Faraday, scientific exploration is one of your passions. And I have just the cool gadget for you. Radiocode is a gadget developed and produced in Europe, made for all scientific enthusiasts throughout the world. It's a smart portable radiation detector and spectrometer that can measure radiation up to 40 times faster than a traditional Geiger counter. I love just carrying mine around and seeing what I can detect in unexpected places. It's always informative. With Radio Code, not only can you locate a radiation source, but you can also find out which isotopes are present. And their free iOS and Android app let you access the radiation level, daily, monthly, or yearly dose rate, spectrum, spectrogram, mapping, and many other features. So, it's a great tool for anyone with a scientifically curious mind. It's even got tons of cool accessories. To learn more about Radio Code, check out radiocode.com or click on the link in the description and use my code to get 5% off. Working for Davey opened up many opportunities for Faraday. He got to work every day on the science that he loved and was able to meet with a lot of influential scientists of the time. Over the next 15 years or so, Faraday focused his experimental research efforts on probing the nature of electricity and magnetism, making some of the most important discoveries of his career. The first of which was the electric motor. In 1820, Faraday learned about a recent accidental discovery by the Danish physicist Hans Christian.

[00:07:06]After setting up a volta battery, Orstead placed a compass nearby and observed the needle moved. This was the first connection between electricity and magnetism ever observed. Urstead then repeated similar experiments many times and upon placing more magnets around the wire saw that the magnets formed a circle around it. He concluded that the current must somehow be spiraling down the wire. Keeping with his initial training, Faraday read about the discovery and decided he had to do the experiment himself. Upon doing so, he came to his own conclusions. The current is moving down the wire, but it somehow produces a circular force around it. He also became convinced that a magnet could in turn produce a circular force on a current carrying wire. And he was determined to show this experimentally.

[00:08:01]His setup was ingenious. He placed an iron bar magnet in a basin that contained melted wax. After the wax hardened and kept the magnet fixed in place, he then filled the basin with mercury. Next, he dangled a wire attached to a stand. Finally, he connected one end of a battery to the top of the wire and the other end to the mercury, forming a closed circuit. The circuit would remain closed even if the wire moved. And move it did. He was right. A magnet does produce a circular force on a wire. But he wasn't done yet.

[00:08:38]He then modified his setup and let the magnet float in the mercury while being loosely tied to the bottom of the basin.

[00:08:46]He also replaced the hanging wire with a fixed one. This time he observed the magnet moving around the wire. He had just created the world's first electric motor. 10 years would pass before Faraday had another significant breakthrough. At this point, the French physicist Andre Marie Aair had shown that a current can not only affect a magnetic needle, it can even affect another current carrying wire. He also showed that if you bent a current carrying wire, it would effectively act like a magnet. Faraday took these discoveries a step further. He noticed that if he wound the wire many times, the magnet became stronger and he was convinced that this could give more insight into the circular force he observed 10 years earlier. He decided to do another experiment. This time he started with a piece of soft iron that was shaped into a ring. He then wound two sets of wires around it. Only one set was connected to a battery. when a current would pass through the primary coil, this would cause the ring to be magnetized. He wanted to see if this could induce a current in the secondary coil and tested it by placing a magnetic needle near the wires of the secondary.

[00:10:04]If a current flowed in the secondary, then the needle should move. Nothing happened. No current flowed. However, as soon as he shut the current off in the primary, he noticed the needle moved.

[00:10:19]So, a current must have flowed for an instant. When he switched it back on, the needle moved in the opposite direction. Faraday had just discovered that when the current switched on, the magnetism changed and induced a current in the secondary. But when the current was flowing steadily and the magnetism was constant, there was no current.

[00:10:42]Similarly, when he shut the current off, the magnetism once again changed and induced the secondary current. In other words, changing magnetism produces electricity. This was a truly brilliant discovery which we now rightly refer to as Faraday's law of induction. Faraday had successfully produced electricity from magnetism. But it was only with magnetism which he had in turn produced from electricity. His next idea was to see if the same would work with an ordinary permanent magnet. He did it with the following setup. This time he had a circuit that had no source of power, so nothing flowed. He attached something called a galvanometer to it to detect any current flow. Then Faraday grabbed a permanent magnet and moved it through the coil. The needle moved in one direction as it went in and in the other as it went out. The faster the magnet moved, the greater the current was. By repeatedly moving it in and out, Faraday was able to produce an alternating current. He had found a way to produce electricity purely out of magnetism. Just like it was earlier known that magnetism could be produced by electricity. He eventually found a way to also make a continuous direct current by placing a rotating metal disc between two magnet bars. The first dynamo was created 10 years after Faraday created the first electric motor. Two discoveries which have made our modern technological world possible.

[00:12:22]Now to make things abundantly clear at this point Faraday had made some significant discoveries but it was still unknown to him and to anyone else what exactly electricity and magnetism were.

[00:12:37]Scientists knew that objects could be positively or negatively charged and behaved according to Kulom's law but the electron and proton were not discovered yet. Faraday captured the general scientific agnosticism of the day when he wrote, "By current, I mean anything progressive, whether it be fluid of electricity or two fluids moving in opposite directions or merely vibrations or speaking still more generally, progressive forces."

[00:13:08]Over the next decade or so, Faraday continued doing experiments and thought deeply about what this all meant. He eventually gave a rough outline of a theory, one in which the universe was filled with lines of force, electric, magnetic, and maybe even gravitational.

[00:13:27]The point where the lines meet are what we perceive as matter. When these lines of force were disturbed, they could vibrate and allow waves of energy to travel through space. He even suggested that light was one of these waves of energy. But because of his lack of mathematical education, his theory was only qualitative. The quantitative framework that put Faraday's theory on firm mathematical ground would require another genius.

[00:13:58]James Clark Maxwell was born in 1831 in Edinburgh. Unlike Faraday, his family was well off and eventually settled down in the nearby countryside. Maxwell also had a privileged educational upbringing, though he did experience his own suffering, losing his mother at just 8 years old. At 10, he was sent off to live with his aunt and attend one of the best schools in Scotland at the time, Edinburghough Academy. By 14, he already published his first paper. And at 16, he went to Edinburgh University. There he published two more papers and eventually finished his undergraduate studies at Cambridge. After graduation, he remained there as a fellow where he could devote plenty of time to his own research interests. There were two aspects of nature that he was particularly interested in exploring, color vision and the relationship between electricity and magnetism. The first he solved rather quickly by discovering that he could create almost any color by mixing the appropriate amount of red, blue, and green. He showed this by creating a clever device, a spinning top with red, blue, and green paper. He could precisely measure how much of each color was used and then would spin the top. By comparing to various colored pieces of paper he had, he showed that he could duplicate every color perfectly. a process which in essence is still used to produce colors today. While Maxwell was working on this, over time he developed more and more of an interest in electricity and magnetism, eventually focusing all his efforts on it and trying to understand the significance of Faraday's experimental work. Despite the fact that many in the math and physics community at the time believed Faraday's ideas about lines of force were wrongheaded, Maxwell was convinced Faraday was right. He made it his task to express Faraday's ideas in mathematical language. As he wrote in a letter to a friend, I am working away at electricity again. I hope to see my way through the subject and arrive at something intelligible in the way of a theory. His first breakthrough came in 1855 with the publication of a paper titled on Faraday's lines of force. He wanted an appropriate analogy for these lines of force and found it in the steady flow of an incompressible fluid meaning the fluid cannot be compressed into a smaller volume. The textbook example being water. The strength of this analogy was in the fact that the fluid had its own 1 / r 2 property. The speed of a particle of fluid flowed outward from a source and fell off as 1 / r^ 2. For a sink, it flowed inward. It was known at the time that electric and magnetic forces behaved according to a similar type of law. So the speed of this flow represented the intensity of the forces and Maxwell had found a way to mathematically express Faraday's lines of force.

[00:17:10]This is the origin of Maxwell's first two equations. The divergence of the electric field is equal to the charge divided by a constant. And since there are no magnetic charges, the divergence of the magnetic field is zero. With this analogy, Maxwell was able to successfully explain static electric and magnetic fields as well as steady currents. But it did not work for changing fields or currents. He decided to set his work aside and work on other problems for a few years. 6 years later, he would publish a masterpiece, a multi-part paper titled on physical lines of force. He began by telling readers not to take the model too seriously. I do not bring it forward as a mode of connection existing in nature.

[00:18:00]It is however mechanistically conceivable. He knew that something rotational must be going on. So he proposed the following model. Consider tightly packed small cells that could rotate.

[00:18:14]As each cell spinned, it would flatten at its poles and get wider at its equator.

[00:18:20]The combined effect would be like a vortex, which represented the magnetic field. The greater the rotation, the stronger the magnetic field.

[00:18:31]He then added one more thing to his model.

[00:18:34]Little particles were placed in between these cells to act like ball bearings.

[00:18:39]Essentially doing two things. Make the cells spin and prevent them from rubbing against one another, losing energy and stopping due to friction. Finally, he had a genius idea. What if these small particles were particles of electricity?

[00:18:57]If they moved along channels around the cells, they would affect the rotation of the cells and also create an electric current. Conductive materials allowed these particles to move freely while insulated materials prevented them from moving. And the density of cells corresponded to the magnetic properties of a material. According to this model, electric and magnetic forces not only existed in material objects, but also had their own form of energy familiar to scientists at the time. Magnetic energy was like potential energy. An electric energy was like kinetic energy. These energies were necessarily linked. A change in one corresponded to a change in the other, which gave rise to Maxwell's third equation. Now, Maxwell thought that these cells existed throughout all of space. And since electric currents corresponded to the motion of these small particles, Maxwell deduced that even in empty space or in a material that was a perfect insulator, it was possible to have brief electric currents, a concept he called the displacement current, which eventually gave rise to the fourth Maxwell equation. But Maxwell wasn't done. There was still an even more impressive achievement. According to his model, the vortex medium was elastic and elastic substances can transmit waves.

[00:20:29]He noticed that even in empty space when the displacement current would be brief, the fluctuations of magnetic and electric forces could propagate through the cells.

[00:20:41]It wouldn't be instantaneous.

[00:20:43]He calculated that the speed would be the ratio of the electromagnetic and electrostatic unit of charge, units that were experimentally known at the time.

[00:20:54]Plugging these values in gave the speed of light, a truly stunning result.

[00:21:07]Maxwell concluded, "We can scarcely avoid the inference that light consists in the transverse undulations of the same medium, which is the cause of electric and magnetic phenomena. Maxwell had discovered that light was an electromagnetic wave, a discovery that would revolutionize the scientific community and the world at large. He eventually refined his theory so that it didn't make use of this vortex model.

[00:21:35]And with the subsequent work of Oliver Heiside, we got Maxwell's theory of electromagnetism as we know it today.

[00:21:46]If you like this video and want to help me make even better videos, join my Patreon by clicking right here. And thank you for watching.