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0:00
Electricity. Electricity. Okay. Let's start with some basics Part one. Vectors and vector multiplication, dot product and cross product. I'm going to start with the definition of scale and vectors. Scales have value, but no direction. Vectors have both value and direction. For example, your mass is a scalar as it has value, but no direction. Your weight, on the other hand, is how much you pull by gravity. So it is a vector. That is why you say you feel weightless
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when you feel like you're not being pulled by gravity. But you never say you feel massless because you're never massless. Even in zero gravity, as vectors have magnitude and direction, you can represent them by a single arrow pointing in some direction where the size of the vector is represented by the size of the arrow. Or you could represent it by how much it points in the x, y and z direction. Like this vector a is a x and the extra action
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plus a y in the y direction plus easy in the z direction. Now, Maxwell knew about vectors, but what he didn't know was about vector multiplication. Currently, there are two common ways to multiply vectors. The DOT product, which involves multiplying the components in the same direction, and the cross product, which involves multiplying the components that are perpendicular. The best way to understand this concept, I think, is to use vectors with some physical meaning.
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I like to use the example of force and distance. Now if you take the dot product between force and distance, or how much force and distance are pointing in the same direction, you get the work done on an object or the amount of energy that you give to the object. Like those high school physics problems where you have someone pushing a lawnmower and the problem wants you to calculate the work done. You get the work as force thought, distance,
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or how much you push the object in any direction times, how far you push it in that direction. Note that the DOT product produces a scalar or a value with no direction. Just as force and distance are vectors, but the work or the energy given is a scalar as energy has no direction. But the cross product, on the other hand, gives you another vector or something with both value and direction. Let's look at the cross product of the force and the distance,
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or how much the force is perpendicular to the distance. In that case, you get the torque or spinning force, which has a direction depending on what direction the spin goes, like with turning a screw, which direction the screw moves. Notice that the direction of the talk is perpendicular to both the direction of the force and the distance. Now we're spinning the order of operations matters. So let's take the talk of the distance. Cross the force.
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If you take a cross product of a distance in the X direction with a force in the Y, then your object twists out of the page or in the positive Z direction. If you take the cross product with a distance in the wide reaction with a force in the X direction, then your object twists into the page or in the negative z direction. Since the force and the distance could have components in the x, y, and z directions. You have to determine all the possible twists.
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You end up with three equations, one for each direction or a single very long equation. Now, figuring out all the directions to twists can be very difficult. Or at least I find it very difficult. Luckily, I was taught this little trick where you draw an X, Y and Z in a triangle where if you take the cross product in the clockwise direction, you get a positive result. And if you take it in a counterclockwise direction, it's negative. For example, Y Cross
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Z is clockwise, so it results in a torque in a positive direction while z cross y is counterclockwise. So it is in the negative x direction. You can pause the video and check it out. It works every time. What Maxwell was trying to do with his equations was to put Faraday's idea of lines of magnetic and electric force into mathematical four. So he needed to know how they changed over space. He needed to take the derivative of vectors for this.
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He really could have used something that I call the DEL function because it has both derivatives and a direction. Which brings me to part two the DEL function divergence and Gauss's law. See this triangle function? It has three common names Altered Novela and Del. I'm going to use the name Del because I'm American. I'll explain what my birth place has to do with it in later videos. Anyway, the DEL function is a vector that has a derivative in each direction,
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pointing in each direction as the DEL function is a vector. You can take the DOT product operation, which is called the divergence or the cross product operation, which is called the Curl. Let's start with the divergence. Del got a function as it's the easiest. Remember with the DOT product. You multiply both x components plus both y components, plus both z components. Therefore, if you take the divergence of the vector, a, you get the derivative of a x,
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the x plus the derivative of y y plus the derivative of ac dc or how much the vector h changes in all directions as you move in space away from a point. Now, let's take a closer look at the Maxwell's equations as they're commonly given the use, the divergence, both of which are called Gauss's law. The divergence of V equals row over epsilon, not and the divergence of B equals zero. Now, before I go into what they mean, I feel like I need to give a little backstory.
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C Way back in 1784, a French engineer named Charles Auguste and Coulomb published the description of this very sensitive torsional scale that was described as an all too unsteady, twisting machine. The following year, Coulomb used his device to determine that the electric force is dependent on the size of the to charges and inversely dependent on the distance squared. Then the year after that, 1786, Coulomb used his machine to determine the charges only reside
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on the outside of the conducting material and do not penetrate into its interior parts. 51 years later, in 1837, Michael Faraday published his results inspired by Coulomb around the effect of charges on conductive containers, where he ended up building a giant conductive cage big enough to live in. Which is how he discovered that not only did the inner surface of the cage not have any excess charge, but also if electrical experiments were conducted outside the cage, Faraday found that
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the conductive cage shielded him from these forces. Faraday immediately realized that what was going on was due to static electric conduction or charged objects moving other charges without touching. Because of this experiment, Faraday imagined that all charged objects emanated what he called lines of inductive force, or, as he later called it, lines of electric force, similar to how he felt that magnets emanate lines of magnetic force. That is why in late 1855 and early 1856,
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James Clerk Maxwell decided to mathematically model Faraday's idea by using an analogy to water flow. So he had tubes of incompressible fluids where Maxwell imagined how these lines of force would be supplied by a source, i.e. a positive charge, and afterwards be swallowed up by a sink, i.e. a negative charge. Maxwell then used the method that he learned from a German mathematician named Carl Friedrich Gauss, where he made an imaginary closed surface around the charges
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and measured the amount of electrical lines d the pointed out of the surface or took the integral over the surface of the s. Maxwell felt that the number of lines that emerged from a surface have to be dependent on the charge in that surface. Maxwell then used integration by parts to show that this integral was the same as the integral of the sum of the derivatives of D over the volume enclosed. Note that Maxwell didn't know about the divergence,
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but we can recognize that he just proved that the integral of d dots over the surface is the same as the integral of the divergence over the volume enclosed by that surface. In addition, Maxwell knew that the total charge enclosed by a surface is the same as the sum or integral of the excess charge per volume row integrated over that volume, i.e. q enclosed equals the integral of row d, d, y, z, which means that saying that the integral of D that the s equals
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Q enclosed is the same thing as saying that the integral of the divergence over the volume equals the integral of the charge density over the volume, which means that the divergence of d equals row the charge density, which is sometimes how gauss's law is written. Now this D field is not the electric field. Instead, Maxwell called it the displacement field as it displaces the charges in the medium. As these lines of force emerge from charge objects.
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However, Maxwell stated that if the medium is uniform, then the displacement field D and the electric field E are related by simple constant K, which Maxwell called the electric elasticity. Years later, Oliver Heaviside decided that he would prefer to use a constant d called the permit tivity that was defined as the inverse of Maxwell's electric elasticity. If we use the Greek letter epsilon for the permit tivity where epsilon is one over k, then we get the electric field e
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is D over epsilon, which makes gauss's law the divergence of v equals row over epsilon. Note that epsilon is a constant that depends on the electric properties of the media. In the case of a vacuum or air at room temperature and pressure where there is either no electric properties or limited electric properties, you still need a constant epsilon not called the permitted active of free space to make the units work out. This situation is so common that often we are given
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Maxwell's laws for that case as if it was universal. The divergence of the equals rho over epsilon not even though it only works in a vacuum or air at room temperature and pressure. Heck, even in air, this equation isn't 100% accurate as the real equation is. Actually, the divergence of v is row over 1.06 epsilon not the most scientists figure those close enough to epsilon not to not bother with the factor of 1.06. However, don't be distracted by the constant.
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What's much more important is what this equation is say, which is if you have a volume with more positive charges than negative, then the divergence electric field will be positive and more lines of force will merge from the surface around that volume, then enter it. If you have more negative than positive than the divergence is negative and more lines will enter the surface. If the volume has equal amounts of negative and positive, then the total number of lines
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entering equals the lines exiting and the divergences zero. In addition, Maxwell also found from manipulating other equations that no matter what the medium is, the divergence of a magnetic field is always zero, which means that although you can have more positive electric charge the negative or more negative than positive in the volume for magnet, you can't have more north and south or more south and north. So when you follow the magnetic field lines, they can never converge or diverge
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from a point, even on the north part of a bar magnet where the magnetic field lines look like they're merging from the north. Like a point the lines actually sneak back to the north inside the bar magnet and you still get the divergence of B is zero, all of which means since magnetic fields can't converge or diverge from a point, all they can do is curl around in circles. Which brings me to part three. Maxwell's equations and Curl. So let's start with the cross product with Del or the Curl.
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Now I want to remind you how the cross product works between a displacement and a force to get the talk, you end up with three equations one for each direction or a single very long equation. By analogy, if you take curl of a vector, you get three equations for the three directions or a single very long equation. If you think of divergence as measuring how much its diverging from a point, you can think of curl as measuring how much it changes as it curls around a point in a circle.
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Now before we get into this, I'd like to do a little bit of history. Back in the early 1800s, a Danish philosopher named Hans Christian Orsted really wanted to make physics, not a collection of fragments on motion, on heat, on air, on light, on electricity, on magnetism and on who knows what else. But instead, he wanted to determine a way to include the whole universe in one system. Orsted was beyond delighted when, in May of 1820, he found a link between electricity and magnetism.
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When he discovered the current in a wire would move a magnetic compass needle to point around the wire, which he published in July of 1820. As soon as a French polymorphous named Andre Marie Ampere heard about it, he immediately decided that if electricity could move one magnet, maybe all magnets are electric. To validate his claim, Ampere wanted to model a bar magnet with electricity, or, as he put it in 1820. This action that Mr. Orsted discovered led me to look for how the electricity
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might produce all the phenomena presented by magnets. However, Orsted's original experiment with a straight wire didn't act anything like a bar magnet, but Ampere knew wires don't have to be in straight lines. Maybe he could make the wire, make some other orientation that would act like a bar magnet. Ampere thus created a pancake of wires and what is called a plane or spiral two is the light ampere found that this pancake worked like a band where one side of the pancake was in north
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and the other south, depending on the direction of the current as one pancake spiral work like a magnet, Ampere wondered if two would work like magnets with each other, which they did. Ampere then realized that in replacing the magnet with another spiral with its current in the same direction, the same attractions and repulsion occur. It is in this way that I discovered the two electric currents attract each other when they flow in the same direction and repel each other in the other case.
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But that is not all in that same paper Ampere tried wrapping his wire in a helix around the glass tube, and to his shock, the joint actions of the coils made this helix, or, as he called it, a few years later, a solenoid act in a way that was perfectly similar to the action of a magnet. In 1826, Ampere published the collection of his works in a book titled The Theory of Electrodynamics Uniquely Deducted from Experiment, where he found, among many other things,
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that if he had a straight, infinite wire with Current I and he put another current of I prime and length l next to it, then the force between the two wires is proportional to the two current times the length divided by the distance between them in modern units, the force per length for two straight wires is written as force. Per length is mu i1 i2 over two pi r where mu is called the permeability, which is dependent on the magnetic properties of the medium.
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This is leads back to James Clerk Maxwell in late 1855 and early 1856, Maxwell stated that if a magnetic field in a closed curve is constant over time, it can be made to be a measure of the quantity of current which passes through it. Maxwell then imagined a flat or open surface with a perimeter surrounding a current under Maxwell's logic, the integral of a magnetic field h around a current is equal to the current enclosed by that surface.
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So the integral of htl equals I enclosed note that for a line l dl points along the length of the circumference. So as you are talking about the dot product with H, you are measuring how much H points around the circumference, but the current enclosed is the same as the integral of the current area. J Over the area. So Maxwell was saying that the integral of H that d l is the same as the integral of j dot the s for the flat surface. Now Maxwell didn't have vector notation,
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so it did the specific case where the current per area is only in the extraction so that the area is in the Y z direction. Assuming that the flat surface is a tiny square of sides d y de z, Maxwell proved that the integral over the circumference of HTL is the same as the integral over the flat surface of d a, z, d, y, minus d, h, y, z. By analogy, Maxwell derived the current per area in the z and the y direction. Then vector notations is clearly just that the current prime area is the curl of h.
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Note the Maxwell preferred to use this magnetic field h instead of the traditional magnetic field b but that for a uniform medium they're related by a simply ratio. B equals mu h where mu is the permeability, which is the same constant the relates the force and currents divided by the distance used in ampere's force law. So Maxwell was really saying that the curl of B is mu j. Then starting in 1862, Maxwell came up with this idea that there could be another reason that the magnetic field curl
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and not just around moving charges. He thought it could also curl if there was a changing electric field. His logic was that the displacement field D polarizes the molecules and the ether. So Maxwell boldly declared that a variation of displacement is equivalent to a current. That is how Maxwell derived Ampere's law with Maxwell's addition to be the curl of h is j plus d d d t. Once again, sometimes this equation is written just in this way, but if we want to get it in the more traditional form,
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we can multiply both sides by mu to get the curl of mu h equals mu j plus DDT and use the fact that in a uniform medium we can replace mu h with B and replace D with epsilon e to get the curl of B is mu j plus mu epsilon ddt where mu the permeability is a constant that's dependent on the magnetic properties of the medium and epsilon. The permit tivity is the constant dependent on the electric properties of the medium. I am sorry that the permeability and the primitives these sound so similar.
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It's not my fault anyway. Once again, in a vacuum or an air, these constants are still needed for the units and we use the magnetic permeability of free space, new knot and the electric permit tivity of free space epsilon. Not to get the curl of B is not j plus not epsilon, not DDT, which is how this law is often taught to students. Now, once again, don't get bogged down with the constants. What's more important is what this equation is saying
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and what this equation is saying in a fundamental way is that the curl of the magnetic field is either due to a current that the magnetic field is circling, or from a changing electric field that is inducing a perpendicular magnetic field around it. One more equation and one more little short back story. In 1825, an English shoemaker named William Sturgeon accidentally discovered that amperes coils of wire with current in them worked as a much stronger magnet
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if they're wrapped around iron rather than glass where they got the new name of electromagnets. Then in 1831, Michael Faraday was playing with electromagnets, trying to get iron to move electricity from one coil to another remotely. That is how Faraday discovered that he could move electricity remotely without the wires touching, but only when he changed the strength of the electromagnet. In other words, when he changed the current of an electromagnet, a neighboring coil
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would have current induced in it, something he called electromagnetic induction. After Faraday realized that the trick to inducing electricity was changing a magnet, he found that if he moved a bar magnet into or out of a coil, he could also induce current in the wire after this Faraday had the insight to create the idea of magnetic fields, or what he called magnetic lines of force. And to describe the rule of induction is that current is induced in a circle
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when these magnetic lines of force are broken or cut by the wire. Fascinatingly, Maxwell wasn't particularly interested in modeling this phenomenon directly. In fact, he was working on a different problem where his modeling, the work done in a unit of time depended on either the magnetic or the electric field and bumped into it almost by accident and did not give it a name. However, by his famous paper of 1864, Maxwell seemed to acknowledge that it was from Faraday.
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As he said, it was a law deduced from the laws of induction. What Maxwell derived in 1862 is that the curl of E is minus mu DHT, but since B equals mu h maxwell was also driving that the curl of e is negative db d t. This equation is saying that a changing magnetic field will create an electric field that curls around the changing magnetic field, just as Faraday experimentally discovered over 30 years earlier. Let me explain the negative sign. So if the curling electric field is in a conductor,
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it will create its own circling current, which, according to Ampere's law, will make its own curling magnetic field. The negative sign means that the magnetic field from the current will push against the changing external magnetic field if the negative sign wasn't there, you could not only build a perpetual motion machine, you could build a perpetual motion machine that constantly accelerates. Now, although that would be convenient for certain things that would violate conservation of energy.
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So we need the negative sign there so it makes physical sense. And also so the laws of thermodynamics still stand. All reminds me of one of my favorite clips from the TV show The Simpsons. Check it out. And this perpetual motion machine she made today is a joke. It just keeps going faster and faster. Lisa, get in here
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in this house, we obey the laws of thermodynamics. Now that I've gotten to all four equations, I want to take a moment to talk about why they're important, specific. Quickly, I want to talk about why they're important in terms of magnets and light. Which brings me to part four, Maxwell's Laws and Magnets. Let's look at two of Maxwell's laws again in free space, as they say, the first equation GAUSS'S law is showing that if you have an excess of charged particles rho, then
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the electric field will diverge from those charges. The second equation Ampere's law with Maxwell's addition states that you will have a curling magnetic field induced if either you have charges moving J or you have a changing electric field. DDT. Now compare that to the other two Maxwell's laws, where the first equation Gauss's law for magnets is showing they're no magnetic particles. Well, magnet equals zero. And the second equation Faraday's law states that a curling
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electric field can be induced by changing magnetic field, but not by a magnetic current, as there are no magnetic particles. There also are no such thing as a magnetic current. In this way, what Maxwell's laws are doing are supporting Ampere's idea that there are no magnetic particles. According to Maxwell's equations, all magnets and magnetic fields are due to the action of electric particles. Take a moment and stop and think about what a profound statement this is.
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Electromagnetic forces are considered one type of force for a reason. They are both fundamental a type of electric force. Not that thus to stop people from trying to find magnetic monopoles. After all, who doesn't want to be the person to make an amendment to a fundamental equation? Okay. If you feel like your brain is exploding, I have more ready. Part five Maxwell's Laws and Light. Part of the reason that Maxwell added that changing displacement field
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current to his equation for ampere's law with the curling magnetic field in 1862 was because five years earlier, in 1857, a German team named Weber and Carl Roush experimentally measured the ratio of the electric constant K where K equals one over epsilon, not to the magnetic constant in free space Munich and found for the right sets of units that the square root of K of four mu which is the same as one over the square root of mu not epsilon not was 3.1 times ten to the eighth meters
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per second, which seemed close enough to the speed of light to be considered. Well, the speed of light, which is why in free space ampere's equation with Maxwell's addition is sometimes written as curl of B is mu, not j plus one over c squared d edt. Where C is the speed of light? Now let's look at Maxwell's curling equations again. An air or a vacuum in the specific case, when there are no electric currents, j equals zero. A changing magnetic field induces a curling electric field,
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and a changing electric field induces a curling magnetic field, times the speed of light squared. With this, Maxwell felt that he had the mathematical proof to validate Faraday's cry. The idea from 1845 but light or radiation is a vibration of electric and magnetic lines of force, but also, Maxwell proved the Faraday was incorrect, as Faraday thought that light might be a longitudinal wave. As Maxwell's wave only works when the electric field is perpendicular to the magnetic field,
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and both of which are perpendicular to the direction that the wave progresses. Or, as Maxwell put it in 1862, the velocity of transverse undulations in our hypothetical medium, calculated from the electromagnetic experiments of Mr. Corrosion. Weber agrees. So exactly with the velocity of light that we can scarcely avoid the inference that light consists of these transverse undulations of the same medium, which is the cause of electric and magnetic phenomena.
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This is why you can buy a T-shirt that says and God said, then has Maxwell's equations and ends with the line. And there was light. No, I'm not trying to offend anyone with this T-shirt. I'm just trying to explain with the joke in the T-shirt was anyway, Maxwell's mathematical ideas about the nature of light were to have major implications. For example, in September of 1887, a young German scientist named Heinrich Hertz experimentally validated Maxwell's view of light
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when Hertz realized that something called the spark generator would produce bursts of vibrating electricity that vibrated far slower than visible light. Now, Hertz felt that even though it wasn't vibrating fast enough to be seen, it should still make an electromagnetic wave, a light wave, just an invisible one. Therefore, Hertz had an antenna to the generator and caught the spark in a receiver distance away and the term and that these electromagnetic waves did move at the speed of light.
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These hertz and waves were eventually called radio waves and soon people started sending signals wirelessly and the wireless revolution went off. And we are still living in it hurts. However, it didn't see the practical applications of it and instead in 1892 wrote that the object of these experiments was to test the fundamental hypothesis of the Faraday Maxwell theory. And according to Hertz and many other scientists, the discovery of radio waves was a complete validation of Maxwell's laws.
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I can mention so much more, but I would be remiss if I didn't point out one other discovery from Maxwell's equations. A young patent clerk named Albert Einstein wrote a friend that he had amusing and seductive thought about the relativity relations that he had just discovered. Einstein wrote that the relativity principle in association with Maxwell's fundamental equation requires that the mass be a direct measure of the energy contained in the body.
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In other words, Einstein himself said that his equals C squared equation is a direct consequence of Maxwell's law. Amazing. Right? So that was my overview of Maxwell's laws. I hope you found it informative. I finished the script on the amazing life of a mathematical scientist named William Hamilton. How Hamilton created something called Quaternions and how Quaternions work and eventually inspired vector algebra. So make sure to subscribe and hit the bell thingy
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because that's next time on the lightning tamers. So as usual, if you want a script for this video with tons of clickable links so you can read what the original people said. Head on over to my website W WW that Kathy loves physics dot com and you can see all of that you can also get a place where you can download the first three chapters, over 100 pages of my book, The Lightning Tamers, which is coming out in less than three weeks. You can order it on Amazon, you can order from book bookstore,
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you can get a donation for me to do my own audio book. All of that is on my website. Go for it. You can also become a patron and you get videos early and if you donate more than $8 a month, you get a free e-book, which is a pretty good deal if you want to help, but you don't want to spend any money. Lots of ways to help. Give a thumbs up, make a comment, subscribe, share it with friends, with neighbors and real life on social media, all of which really helps me out. Okay.
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Stay safe and curious out there by
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now. Let's take a closer look at the Maxwell's equations that used to diverge. Let Maxwell then use the method that he learned from a German mathematician named Carl. What was his name? Carl Frederick. You can pause the video and check it out. It works every time.
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