Nobody Explained Maxwell's Equations Like THIS!
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For most of my life, Maxwell's equations were just wallpaper. Four lines of symbols on a physics-building wall and a t-shirt I never understood, and every time I saw them I nodded and kept walking. So I finally sat down and forced myself to figure out what they actually say. What I found was one of the best detective stories in the history of science.
CHAPTERS 0:00 The four lines everyone walks past 2:01 Two sciences and one accident: Orsted and Faraday 5:50 Maxwell's four equations (and the one that was broken) 10:43 The wave that turned out to be light 13:40 The ghost, Maxwell's death, and Hertz 15:52 Every size of light, and the door to relativity
Next time we walk through the door these four equations opened: the speed of light is fixed for everyone, and something has to bend to keep it that way. That something is time. That is where relativity begins. Subscribe so you catch it.
If I got a detail wrong, tell me in the comments. I read them, and I would rather fix it than defend it.
REFERENCES - Maxwell, J. C. (1865). A Dynamical Theory of the Electromagnetic Field. Philosophical Transactions of the Royal Society. - Faraday, M. (1839-1855). Experimental Researches in Electricity. - Orsted, H. C. (1820). Experiments on the Effect of a Current of Electricity on the Magnetic Needle. - Hertz, H. (1893). Electric Waves. - Michelson, A. A., and Morley, E. W. (1887). On the Relative Motion of the Earth and the Luminiferous Ether. American Journal of Science. - Feynman, R. P. The Feynman Lectures on Physics, Vol. II, Ch. 1 (source of the "ten thousand years from now" remark). - Forbes, N., and Mahon, B. (2014). Faraday, Maxwell, and the Electromagnetic Field. - Griffiths, D. J. Introduction to Electrodynamics.
ANIMATION AND COPYRIGHT All 2D and 3D animations in this video were designed and built from scratch by the Animated Physics team. No stock footage or third-party clips were used. (c) Animated Physics. All rights reserved. Reuse, re-upload, or redistribution without written permission is not allowed.
Covered in this video: Maxwell's equations explained, electromagnetism, Faraday's law and induction, Ampere's law, Gauss's law, the displacement current, electromagnetic waves, the speed of light, the luminiferous aether, the Michelson-Morley experiment, Heinrich Hertz and radio, the electromagnetic spectrum, and the road to special relativity.
#MaxwellsEquations #Physics #Electromagnetism #SpeedOfLight #AnimatedPhysics
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Maxwell's four equations reveal that electricity and magnetism are two aspects of a single phenomenon—electromagnetic waves—whose derivation accidentally discovered what light physically is, fundamentally reshaping physics and enabling modern technology.
- Faraday's experimental discoveries (changing magnetic fields create electric fields) combined with Maxwell's mathematical insight (changing electric fields create magnetic fields) generate self-sustaining waves in empty space
- Maxwell's wave equation yields the speed of light when calculated from independently measured electrical and magnetic constants, proving light is electromagnetic
- The fixed speed of light embedded in Maxwell's equations became the foundation for Einstein's relativity and reshaped 20th-century physics
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The speed of light is not a property of light itself but rather a fundamental property of space and time—it is determined by the electric permittivity and magnetic permeability of empty space, making it a constraint built into the structure of the universe rather than something specific to electromagnetic phenomena.
“Remember those two numbers Maxwell plugged in? The ones that handed him the speed of light? They are properties of empty space itself, not of light, of the vacuum. Which means the speed of light is not really a fact about light at all. It is stitched into the structure of space.”
Maxwell had no experimental evidence for the displacement current concept and introduced it solely because without it, his equations were mathematically inconsistent, demonstrating a bold willingness to modify fundamental laws based on mathematical consistency rather than empirical observation.
“And I really want you to sit with how bold this was. There was basically no evidence for it. He added a brand new term to the fundamental laws of physics purely because without it, the equations were broken and ugly. He trusted the math over the total lack of proof.”
Faraday's discovery of electromagnetic induction is the physical principle underlying all electrical power generation: every power plant on Earth—coal, nuclear, wind, hydro—operates by moving a magnet past a coil of wire to induce electric current.
“That discovery is the only reason you have electricity in your house at all. Every power plant on Earth, coal, nuclear, wind, hydro, every single one, is just a fancy way of moving a magnet past a coil of wire. Faraday, the bookbinder who could not do algebra, quietly powered the modern world.”
The invisible electromagnetic waves that Hertz dismissed as having no practical application are now the basis for all wireless technologies: every phone call, every Wi-Fi signal, and every radio station on the planet transmits information using electromagnetic waves.
“That useless invisible wave he shrugged at is now every phone call, every Wi-Fi signal, and every radio station on the planet.”
The Michelson-Morley experiment's failure to detect the ether—the 'missing ghost'—became a crucial clue that ultimately led to the development of Einstein's special relativity, as it suggested that light's constancy did not depend on motion through a medium.
“That failure, that missing ghost, became one of the biggest clues in the history of science.”
Heinrich Hertz, a young German physicist, experimentally confirmed Maxwell's prediction of electromagnetic waves by building apparatus to generate and detect invisible electromagnetic waves in his laboratory approximately eight years after Maxwell's death, effectively inventing radio without initially recognizing its practical potential.
“Then, about eight years after Maxwell was gone, a young German physicist named Heinrich Hertz built an apparatus in his lab, sparked a circuit on one side of the room and detected an invisible wave arriving at a little loop of wire on the other side. He had created and caught an electromagnetic wave out of nothing but Maxwell's math. Hertz had basically just invented radio, and when somebody asked him what it was good for, he honestly answered, 'Nothing.'”
In 1831, Faraday discovered electromagnetic induction: a moving magnetic field generates electric current in a nearby wire, whereas a stationary magnetic field produces no current—the key insight being that it is the change in the magnetic field, not the magnetic field itself, that induces electricity.
“Faraday asks the obvious mirror image question, can magnetism make electricity? So he grabs a magnet and a coil of wire, and he just holds the magnet still near the coil. Nothing happens. No current at all. But the instant he moves the magnet, current flows through the wire. He stops moving it, the current stops. And that is the crucial piece. It is not the magnet that makes electricity. It is the change. A moving, changing magnetic field pushes electricity through the wire.”
Maxwell's fourth equation (relating currents to magnetic fields) contains a logical inconsistency when applied to charging capacitors: in the gap between capacitor plates, no current flows, yet electric charge accumulates there creating an electric field; by the equation, a magnetic field should exist near the current in the wire but not across the capacitor gap, creating a discontinuity that is physically impossible.
“That fourth equation, the one about currents making magnetism, had a hole in it...While it is charging, current is flowing through the wire into the plate. And a current makes a magnetic field around itself, equation four, no problem. But now look right at the gap between the two plates. The current flows into one plate and out of the other, but it never actually jumps across the gap. In the gap, there is just empty space and a growing electric field as the charge builds up. So, here is the contradiction. Right up next to the plate in the wire, there is a current, so there is a magnetic field. But in the gap, there is no current. So, by equation four, the magnetic field should just stop. Blink out of existence in the middle and pop back on the far side. And that is nonsense. Magnetic fields do not have holes punched in the middle of them. The equation was literally contradicting itself.”
An electromagnetic wave can propagate through empty space without any material medium, requiring only the self-sustaining cycle of changing electric and magnetic fields generating each other as formalized in Maxwell's equations.
“It does not need anything to travel through. The two fields just keep regenerating each other out of raw empty space forever. That right there is a wave.”
Maxwell's equations logically entail that electromagnetic waves do not require any material medium—the electric and magnetic fields regenerate each other, holding each other up—meaning the luminiferous ether was unnecessary despite widespread scientific belief in its existence.
“But here is the strange truth buried inside Maxwell's own equations. The wave does not need a medium. The electric and magnetic fields are the only things doing the waving, and they hold each other up. There is no ether.”
Radio waves carrying music, gamma rays from dying stars, and visible light from a sunset are all the same physical phenomenon—electromagnetic waves—differing only in wavelength, demonstrating the fundamental unity of apparently diverse physical phenomena under Maxwell's framework.
“These are all the exact same thing. The radio waves carrying your music, the gamma rays screaming out of a dying star, and the soft red of a sunset are all Maxwell's wave. Electricity and magnetism leapfrogging through space, just at different sizes. The rainbow your eyes can catch is one little octave on a piano that stretches for miles in both directions. All of it, one single phenomenon.”
The discovery that light is electromagnetic and travels at a speed fixed by space's intrinsic properties—not by any reference frame—became the loose thread that Einstein pulled to unravel the nature of time itself and develop special relativity.
“And that speed, the one that just fell out of nowhere, the same for absolutely everyone, is the loose thread that unravels time itself.”
A single oscillation or 'shake' of an electric charge creates a self-sustaining electromagnetic wave: the charge motion creates a changing electric field, which generates a magnetic field, the growth of which generates a new electric field further out, which generates another magnetic field further still, with this process continuing indefinitely, with the fields leapfrogging through space without requiring any medium.
“Imagine you reach out and grab an electric charge, an electron, and you give it one little shake. That shake is a changing electric field. And a changing electric field, thanks to Maxwell, creates a magnetic field right next to it. But that magnetic field had to grow up from nothing, so it is changing, too. And a changing magnetic field, thanks to Faraday, creates a fresh electric field a little further out, which is also changing. So, it makes another magnetic field further still. Do you feel what is happening here? Each field, as it rises and falls, hands the baton to the next one. Electric, magnetic, electric, magnetic, leapfrogging over each other, sprinting away from that first little wiggle, and racing out into space.”
Since light's speed is constant for all observers but ordinary mechanics requires that velocities add, something fundamental must give: Einstein resolved this by proposing that time itself is not absolute but must stretch and contract to keep light's speed constant—an insight that leads to time dilation, length contraction, and the equivalence of mass and energy (E = mc²).
“So, something else has to bend to keep that speed locked in place. And a 26-year-old patent clerk named Albert Einstein looked at Maxwell's equations, took that fixed speed of light dead seriously, and realized the thing that has to bend is time itself. Special relativity, time dilation, E = mc squared, all of it comes pouring out of taking Maxwell's speed of light seriously.”
Maxwell's four equations and the speed of light they imply unlocked the entire 20th century of physics: special relativity, time dilation, general relativity, and all subsequent physics derives from taking Maxwell's constant speed of light seriously as a fundamental universal constraint.
“Maxwell's four little equations did not just explain light. They were the unlocked door to the entire 20th century.”
Maxwell resolved the inconsistency in his fourth equation by introducing the concept of 'displacement current'—a changing electric field acts on magnetic fields in the same way a conventional current does, even in the absence of moving charges, so that magnetic fields remain continuous across capacitor gaps.
“He says, 'What if it is not only currents that make magnetic fields? What if a changing electric field also makes a magnetic field all on its own with no charges moving at all? In the gap of the capacitor, the electric field is growing, changing every instant. And Maxwell says that changing electric field is doing the exact same job a current would. It is making the magnetic field. He called it the displacement current.'”
When the displacement current term is added to Maxwell's equations, the two 'bridge' equations (Faraday's law and the modified fourth equation) create a self-sustaining feedback loop: a changing magnetic field generates an electric field, which then changes and generates a new magnetic field, which changes again and generates another electric field, and so on indefinitely.
“Faraday's law says a changing magnetic field creates an electric field. Maxwell's new term says a changing electric field creates a magnetic field. Do you see it? They point straight at each other. Each one, just by changing, gives birth to the other.”
When Maxwell calculated the electromagnetic wave speed from electrical and magnetic constants, he obtained approximately 300,000 km/second, which is identical to the measured speed of light, providing the first theoretical evidence that light is an electromagnetic wave.
“Maxwell takes those two numbers, drops them into his little formula, works out the square root, and out falls a speed about 300,000 km per second. And Maxwell freezes because he knows that number. Everybody knows that number. It is the speed of light. People have been carefully measuring how fast light travels for 200 years with no idea whatsoever what light was actually made of. And here that same exact number just came tumbling out of two constants about electricity and magnetism.”
Because the speed of light is a universal constant determined by space's properties, it must be the same for all observers regardless of their motion: an observer running toward light, running away from light, or moving perpendicular to it always measures the same speed, violating classical mechanical intuition where velocities add together.
“If the speed of light is fixed by space itself, then it is the same for everyone, no matter how fast they happen to be moving. Run toward a beam of light, run away from it, sprint alongside it, does not matter, you always measure the same speed. And that is impossible in the ordinary world. If I run toward you, our speeds add together. But light refuses to play along.”
Maxwell's four equations reveal that electricity and magnetism are two manifestations of a single unified phenomenon; shaking a charge generates electromagnetic waves that propagate indefinitely through empty space at a speed determined by space's intrinsic properties; and light is such a wave.
“Here is what they actually say in one breath. Electricity and magnetism are two halves of one single thing. Shake a charge, and they leapfrog out through empty space as a wave, and that wave is light. Every color, every radio signal, every X-ray, the warmth of the sun, all of it is the same ripple, moving at a speed baked into the fabric of space.”
Maxwell's equations allow calculation of the wave speed from fundamental constants: the speed of an electromagnetic wave equals 1 divided by the square root of the product of two measured physical constants—the electric permittivity and magnetic permeability of free space—both already known from independent electrical and magnetic experiments.
“How fast do you go? And his equations hand him an exact answer. The speed of the wave equals 1 divided by the square root of two particular numbers. And this is where I got genuine goosebumps. Because those two numbers were already sitting in the textbooks. They had nothing to do with light or waves or any of this. The first number you measure by rubbing charges together and studying the electric force between them. The second you measure with wires and magnets studying the magnetic force. Two ordinary constants measured on a workbench with batteries and coils that any physicist of the day could just look up.”
Light is fundamentally an electromagnetic wave consisting of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation, not a mysterious substance or particle phenomenon but rather the self-sustaining field pattern predicted by Maxwell's equations.
“Light is an electromagnetic wave. His leapfrogging fields electricity and magnetism dancing out of empty space, that was light itself. He did not set out to explain light. He sat down to fix a broken equation about capacitors, and he accidentally discovered what light is.”
In the early 1800s, electricity and magnetism were understood as completely separate scientific domains with no apparent connection—electricity concerned charge phenomena like sparks and lightning, while magnetism concerned iron lodestones and compass needles.
“in the early 1800s when electricity and magnetism were two totally separate sciences. Two clubs that did not talk to each other. Over here you had electricity. Sparks, batteries, lightning, the little zap you get off a doorknob in winter. And over there you had magnetism. Loadstones, compasses, the needle that always swings north. And as far as anyone could tell, these two had absolutely nothing to do with each other.”
Michael Faraday was a self-taught scientist who began as a bookbinding apprentice, taught himself science by reading the books he was binding, and became one of history's greatest experimentalists despite having no formal mathematical training or calculus knowledge.
“Faraday grew up poor in London with almost no schooling. And as a teenager he got a job as an apprentice to a bookbinder. So, he is binding these books all day and he starts actually reading them. He teaches himself science out of the very books he is supposed to be gluing shut. He talks his way into a job washing bottles for a famous chemist. And through nothing but obsession, he becomes maybe the greatest experimentalist who ever lived. But here is the thing about Faraday that matters for our whole story. He could not do math, not really. No calculus, no equations, none of it.”
Faraday proposed that magnets act through invisible 'lines of force' or fields filling the space around them—a concept that was initially dismissed by mathematical physicists as poetic rather than rigorous, since Faraday could not defend it with mathematical equations.
“To explain how a magnet could reach across empty space and push on something it never touched, he pictured invisible lines coming out of the magnet. Lines of force, filling the space all around it, bending and stretching and pushing. He called this a field, and the proper mathematical physicists of the day thought this was adorable, and frankly, a little embarrassing. Real science was equations and forces between particles, not some self-taught bookbinder drawing poetic invisible lines.”
Physicists and scientists of Maxwell's era were convinced that all waves must propagate through a material medium, so they hypothesized the existence of the 'luminiferous ether'—an invisible substance filling all of space through which light waves traveled, despite having no empirical evidence for it.
“Maxwell had his wave, but he and basically everyone else could not swallow one thing about it. A wave in what? Every wave they had ever met waved through something. Sound waves wave through air. Ocean waves wave through water. So light, being a wave, simply had to be waving through some invisible substance that filled all of space. They named it the luminiferous ether, and they were dead certain it existed, because a wave with nothing to wave in felt impossible.”
Maxwell's third equation is Faraday's law: a changing magnetic field generates an electric field, which is the physical principle behind electromagnetic induction.
“Number three is Faraday's discovery, the one we just talked about. A changing magnetic field creates an electric field. Move the magnet, make electricity.”
Maxwell's fourth equation states that electric currents generate magnetic fields, which was known from Ørsted's discovery that current-carrying wires deflect compass needles.
“And number four is the one everybody already knew going all the way back to Ørsted's twitching compass. An electric current creates a magnetic field. Push current down a wire and magnetism wraps itself around the wire.”
In 1820, Danish physicist Hans Christian Ørsted discovered that electric current flowing through a wire produces a magnetic field, as evidenced by a compass needle deflecting when placed near a current-carrying wire during a lecture demonstration.
“The first crack in that wall came in 1820. And it was almost an accident. A Danish professor named Ørsted was running a lecture demo and he happened to leave a compass sitting near a wire. When he switched the current on, the compass needle twitched. It swung sideways. Like there was a magnet nearby. But there was no magnet. There was just electricity moving through a wire somehow shoving a compass needle around. Electricity was making magnetism.”
James Clerk Maxwell was a Scottish mathematician of exceptional ability who took Faraday's field concept seriously and translated his physical intuitions into rigorous mathematical form, formulating equations that unified electricity and magnetism.
“Enter James Clerk Maxwell. Scottish, quietly brilliant, and unlike Faraday, an absolute wizard with mathematics. And Maxwell did the thing almost nobody else was willing to do. He took Faraday's childish invisible lines of force seriously. He looked at them and thought, 'No, the old bookbinder is right. This field is real and I am going to write it in the language of math.' And when he did, he ended up with a set of equations that captured everything anyone knew about electricity and magnetism.”
Maxwell's first equation states that electric charges are sources of electric fields: positive charges emit field lines pointing outward, negative charges emit field lines pointing inward, and electric field lines originate from and terminate at charges.
“Number one is about electric charge. Take any charge and it fills the space around it with an electric field pointing outward if it is positive, inward if it is negative. Charges are the sources. They are where the field lines are born. That is the whole first equation.”
Maxwell's second equation states that magnetic fields have no sources (no magnetic charges exist): magnetic field lines never begin or end, always form closed loops, and consequently isolated magnetic poles cannot exist—breaking a magnet always produces new poles on each piece.
“Number two is the same idea for magnetism, except it says something strange and very specific. There are no sources. Magnetic field lines never start and never stop. They always curl back around into loops, which is Maxwell's mathematical way of saying you can never have a lone north pole all by itself. Snap a magnet in half to try and each little piece instantly grows a brand new south pole. You always get a complete magnet, north and south, no matter how many times you chop. Nobody in all of history has ever found a single isolated magnetic pole. The second equation is that stubborn fact frozen into math.”
Despite vigorous experimental searches, no evidence was ever found for the luminiferous ether; the most famous attempt, the Michelson-Morley experiment designed specifically to detect the ether, completely failed to find any evidence for it.
“For decades, brilliant people hunted for this invisible ghost, and the harder they looked, the more it refused to be there. The most famous experiment in all of physics, done by Michelson and Morley, was built specifically to catch the ether, and it caught absolutely nothing.”
Maxwell died in 1879 at age 48 from cancer before electromagnetic waves predicted by his equations were experimentally confirmed, so he never witnessed the validation of his greatest discovery.
“There is a genuinely sad piece to this story. Maxwell never got to see himself proven right. He died in 1879 of cancer at just 48 years old. His equations predicted these invisible electromagnetic waves flying around at the speed of light, and most of the world still was not convinced.”
Maxwell's prediction that electromagnetic waves travel at the speed of light was correct: Maxwell was vindicated, and Hertz's experimental confirmation demonstrated the profound truth of his equations.
“Maxwell was right all along. He just did not live long enough to hear it.”
Electromagnetic waves can have different wavelengths, forming a continuous spectrum from very long radio waves (meters long) to very short gamma rays, with visible light representing only a narrow band in the middle—infrared, microwaves, ultraviolet, and X-rays are all electromagnetic waves at different wavelengths.
“Once you know that light is an electromagnetic wave, the whole world cracks wide open because a wave can be any size at all. The light your eyes can see is one tiny thin slice. Stretch Maxwell's wave out longer and you get infrared, which you feel as heat, then microwaves, then radio waves meters and meters long. Squeeze it shorter and you get ultraviolet, then x-rays, then gamma rays.”
Maxwell's equations were historically treated as abstract mathematical wallpaper by most people, despite their profound importance, because their deep physical meaning was not intuitively accessible without mathematical sophistication.
“Four lines of symbols on the wall of the physics building on the smug t-shirt in the textbook chapter you quietly skipped. Little upside down triangles, some dots, an E and a B, a couple of Greek letters. And every single time I saw them, I did the exact thing you probably do. I nodded like I got it, and I kept walking.”
Feynman claimed that Maxwell's equations would be remembered 10,000 years in the future as the defining achievement of the present era, even after everything else about our time is forgotten.
“Feynman said that 10,000 years from now, when everything else about our era is forgotten, the one thing people will still remember is Maxwell's equations.”
Faraday possessed extraordinary physical intuition despite lacking mathematical training, which gave him insight into how electromagnetic phenomena work without requiring formal equations.
“What he had instead was the most staggering physical intuition anyone has ever had.”