 We've seen current carrying loops generate magnetic fields and behave like tiny magnets. And since atoms contain a lot of electrons going around, and electrons are charged particles, meaning atoms also have current loops, and these current loops should also generate magnetic fields, meaning atoms also tend to behave like magnets. So the question we want to try to answer in this video is, what decides the strength of this atomic magnet? And as we answer this, we'll also uncover mysteries like, how did people realize that electrons have a spin? What's that going to do with magnetism? And finally, if everything is made up of atoms, why isn't everything magnetic? So imagine the simplest atom we can think of. One proton and one electron going around it in a perfect circle. So we are looking at it from a side, so we can't see the circle properly, it looks like an oval, but imagine this is a perfect circle. Now we know electrons moving around, since electrons have charged, moving electron produces a current. And because it produces a current, we know that current loops behave like tiny magnets. And so this atom now, immediately you can see, should behave like a magnet. But one question that immediately comes to my mind is, what would be the direction of that magnet? Would this behave like a magnet with the north pole pointing up? Or would it behave like a magnet with the south pole pointing up? How do we figure that out? Well, to quickly recall, we use our right hand thumb rule. So if you clasp your right hand in such a way that the forefingers give you the direction of the current, then the thumb points in the direction of the north pole. So can you use this and pause the video and think about what the direction of the current is and then use your right hand thumb rule to figure out whether the north pole or the south pole points upwards. So can you pause and give it a shot? All right. So in this case, the electron is going in this direction. So immediately what comes to my mind is, hey, if I use my right hand thumb rule such that the encircling finger goes in this direction, then my thumb would point downwards. So maybe the north pole is pointing downwards, right? But we need to be careful. The encircling fingers should represent the direction of the current, not the direction of the motion of the charge. Why the distinction? Because remember, electrons are negatively charged particles, which means if the electrons are going in this direction, then that means the current is in the opposite direction. So the current direction is actually in this way. This is the direction of the current. And therefore, if you use your encircling finger, you should encircle your fingers in this direction, which means the thumb points upwards, which means the north pole should be pointing upwards. So indeed, this behaves like a magnet, but it's not pole pointing upwards. Does that make sense? OK, this now brings us to the main question of the video. What is the strength of this tiny atomic magnet? And just think about what we are doing over here. We're trying to figure out how strongly an atom, which are invisible to our eyes, how strongly it behaves like a magnet using a pen and paper. I mean, come on. Isn't that mind blowing? Just think about how far we have come. All right. So how do we do this? Well, in physics, when we talk about strength of a magnet, we remember we're talking about the quantity magnetic dipole moment. And we've seen the expression for this. It is the product of the current, current and the area of the current loop. And if there are multiple loops, then we can multiply by n. But since here is only one loop, we'll not do that. And this basically means if you have more current, it behaves like a stronger magnet. If it's a larger area, again, it behaves like a stronger magnet. And if you're wondering where this is coming from, we have derived this in our previous videos. Feel free to go back and check that out. So let's go ahead and calculate it. Let's first do the area. I think that's a little simpler. So current times, what's the area of this loop? Well, since we're assuming this to be a perfect circle, if we call the radius of this circle to be r, let's just write that over here, r, then the area of the circle is going to be pi r squared. All right, that leaves us with the question, what's the current? And I think that's a little tricky. Because we just have one electron going around, so it does produce a current. But I think this is, you know, you have to think a little bit about it. So I go back to my basics. What is the definition of current? Well, we can say mathematically current is charged by time. But that's not enough. You have to understand the meaning of this. The way I like to think about it is this basically means at any point, you wait for some time t, and you figure out how much charge q is passing by in that time t. And then that ratio represents the current. How much charge is passing per second? And so we can do the same thing over here. Since the electron is whizzing around, what we can do is we can take some point on this loop, wait for some time t, and figure out how many times the electron whizzes around it, and that will give us the current. For example, if you find that the electron is whizzing around 10 times, then we know the charge is 10e. 10e divided by the time t will give us the current. The question now is how long do I have to wait? How should I wait? How do I still do this? So the trick that we like to do over here is instead of waiting for any time, wait for exactly one time period. A time period is the time it takes for the electrons to make one complete circle. So if you wait for exactly one time period, then the electron will pass through that point just once. Think about it. If the time period was say 10 seconds, if it took 10 seconds for the electrons to go around, then if I were to start my timer right now, then in that 10 seconds, the electron will just complete one circle, which means in that 10 seconds, the electron went through this point once, which means the charge that went through this point is e. And in fact, I can wait for another 10 seconds, and I'll again find the charge that goes through this point is e. And therefore, can you see that if you wait for one time period, then the charge that goes by at any point is e. And therefore, the current would then be e divided by the time period. And so now all we have to figure out is what that time period is. And I think we can do that just by using speed, distance, and time relation. We don't know the speed, so let's throw in some variables. So let's say the speed is v. The speed is v. So I want you to now pause the video and think about this. Can you, given the speed, and you probably can figure out what the distance is, can you figure out the time period? And then can you plug it and figure out what the current is going to be? And while you're at it, feel free to see what the expression for magnetic moment ends up being. So pause and give this a shot. Okay, so the current will be e divided by, how do I calculate the time period? Well, I use speed equals distance by time, so time equals distance by speed. What is the distance that it travels in one time period? That's two pi r, that's a circumference, right? So the distance traveled in one time period is two pi r. And what's the speed? So time equals distance by speed. And speed you're taking is as v. There we have it. That's the current. I can just write that over here now. So that gives you e times the v comes on the numerator divided by two pi r times pi r squared. And we're done with all the substitution. Now we just have to simplify. So r cancels. We have a pi that cancels. And that gives us m equals e times v multiplied by r divided by two. And so this means that the strength of this magnet depends upon two things. How fast the electron is going and what the radius of that orbit is. In fact, it's actually another way of just saying the current and the area if you think about it, right? Okay, now one question I have is instead of thinking in terms of two things, like the speed and the radius of the orbit, can we club this together and come up with one single quantity? And there is such a quantity that we can come up with. Remember, we may have learned in mechanics before that when things are going in circular path, we can say they have an angular momentum. And just to jog your memory, that angular momentum, which we used to represent as L for particles going in circular path, it would just be m v r. So we could say the electron has an angular momentum and its angular momentum would be m e times v times r. And so what I can do now is I can represent that v r, the product of v and r as L divided by m e. And so in doing so, let's see what happens. So we'll now get m equals e divided by two, e divided by two and v r is L divided by m e. L divided by m e, where m e is the mass. And so in doing so, what we have done now is we can now say that the magnetic moment depends on, this is a constant, depends on what? Depends on one thing about the electrons. It depends upon its angular momentum. And that's why we like this expression. We box that. And so this means that the origin of the magnetism, we will now answer the question. The origin of magnetism comes from angular momentum of electrons. More angular momentum means stronger magnetic moments. Now before we wrap up, I wanna dig a couple of steps further. First of all, I wanna write this equation in a vector form because both magnetic moments and angular momentum, they're both vector, right? So what's the direction of the magnetic moment? Well, the direction of the magnetic moment over here is given by the thumb itself, okay? Or you could say whichever direction is the north pole, that direction is the magnetic moment. So in our case, the magnetic moment is pointing upwards. What about the direction of the angular momentum? That's also given by the right hand thumb rule. But this time, the encircling fingers give us the direction of, you know, the direction of which the particle is moving, not the current. Now current doesn't matter. Now notice, since electron is going in the opposite direction, if I were to use my right hand rule to represent the angular momentum, that would be in the opposite direction. Does that make sense? Because the electron is going in the opposite direction. So my encircling finger should be in the opposite direction. Ooh, this means that my angular momentum is pointing downwards. And guess what? This will be always true for electrons. And this means that magnetic moment and angular momentum will always be in the opposite directions. So I can now write this vectorially. Magnetic moment will always be in the opposite direction. And that's why I write negative over here, opposite direction of the angular momentum. You can also imagine that the negative sign is purely coming because the electrons are negatively charged. If this was a positively charged particle, then I hope you agree that both of these would be pointing in the same direction. Finally, here's an interesting question for you. Imagine we just took an electron, which is not moving, not inside an atom, nor bits, nothing. Do you think that would have a magnetic moment? What do you think? Well, if you ask me, I would say, look, magnetic moments require current loops. Forget about loop. There is no current here at all. So it shouldn't have any, right? That's what people thought too. But guess what? Experiments showed otherwise. Experiments showed that even individual electrons themselves behave like tiny magnets. What? You may be wondering, what kind of experiments are these? How do you do this? Well, just like how magnetic needle experiences are deflection in a magnetic field, in a similar manner, we can check if these things experience a deflection. If they do, we can say that they have a magnetic moment. Of course, things may not be that simple, but let's take that for granted that you can experimentally figure this out. The question is, what do you do of this? Why do electrons have magnetic moment? It doesn't make any sense. People were baffled. And this is where some people hypothesized that maybe, just maybe, electrons are spinning around their own axis. And so if you imagine electrons to be a ball of charge spinning around its own axis, then you have an angular momentum. And because of that angular momentum, maybe there is another magnetic moment generated. And this is how we say electrons have a spin, right? I used to always wondered, how did people figure this out that electrons are spinning on their own axis, right? Well, I need to mention, though, that when today, when we say electrons have a spin, we don't really mean they're a ball of charge spinning around in our own axis. Today, we realize that the picture is a little bit more complex, right? The spin is a more quantum mechanical phenomena, but we don't have to worry too much about it. So what we wrote over here is orbital magnetic moment due to orbital angular momentum. Along with that, we will have a spin magnetic moment generated due to spin angular momentum. And it turns out that although we cannot derive it because it requires quantum mechanics, you get a very, very similar relationship. And this means to find out the total magnetic moment of any atom, you need to add up all its orbital magnetic moments and its spin magnetic moments. And in most materials, when you add up all these magnetic moments of all the atoms together, because they're all in random direction, they all cancel out. That's why most materials are not magnetic. Finally, I want to leave you with one question. Just like how electrons have spin magnetic moments, turns out that protons also have a spin magnetic moment, which also obeys a very similar relationship. You could also say that their magnetic moments is due to their spin angular momentum. But it turns out that their magnetic moments of protons and neutrons are way smaller, way, way smaller compared to that of electrons. And we can completely neglect it. Can you guess what could be the reason for that? This equation has the clue for it.