Let There Be Light: Maxwell's Equation EXPLAINED for BEGINNERS

It is my fundamental belief that there are many complicated physics concepts that can be explained to someone with just a basic understanding of high school physics and mathematics. And this video is my attempt at explaining Maxwell's equations, or at least one of them. So, first of all, why am I making this video? Well, last week I did a poll on Instagram asking you guys what video you wanted to see first. A video on Maxwell's equations, or a video about the structure of the atom. Maxwell's equations won out, so here we are. Thank you so much if you voted in the poll. And if you're not already following me on Instagram, then please head over there and follow me. I post one-minute long physics videos, as well as just random stuff I'm doing throughout the day on my stories. And follow me on Twitter if you want to hear the worst physics puns you've ever heard in your life. Anyway, let's get into the video. Maxwell's equations are a set of four equations that brilliantly describe electricity and magnetism. Collectively, this is known as electromagnetism. They are complicated and intricate, so probably deserve about ten videos to be dedicated to them. So, to do justice to Maxwell's equations, in this video I won't be focusing on all four of them. I'll just be focusing on one of them. This one. So, let's get straight into it. What do these symbols even mean? Well, like I've already said, Maxwell's equations focus on electricity and magnetism. Now, B in this case represents the magnetic field that we're studying. Why do we use the letter B to represent a magnetic field? Yeah, don't even ask. Don't ask. The downward-pointing triangle and the dot next to it together represent something known as divergence. So, what is divergence? Right, here we go. Well, the divergence is often applied to something known as a vector field. Okay, Parth, now you're just saying words. What the hell is a vector field? Guys, trust me, bear with me here. It will all make sense very shortly. Now, to understand divergence properly, first we need to understand what a vector field actually is. So, a vector field can be thought of as a region of space where we can assign a vector, or an arrow pointing in a certain direction, to every point in that region of space. Like I said already, by the way, a vector is just an arrow with a particular size and pointing in a particular direction. This vector can be used to represent something in the vector field. A classic example is a vector field showing the direction and speed of wind on a weather map. A lot of us have seen these on TV, for example, where the vectors assigned to every point in this region basically show us the direction and speed of wind. So, for example, at this point the wind is blowing really hard to the east, and at this point the wind is blowing really softly to the south. The larger the vector, the higher the wind speed, and the direction shows the direction of the wind. So, this overall is a vector field. It's basically a field of vectors. The vectors represent something, in this case, the speed of the wind. Now, guys, before I continue, if my explanation of a vector field is not clear enough, then let me know in the comments down below. I've got a couple more examples coming for you guys that should hopefully clear it up, but also, if there's anything in this video that isn't quite clear, then let me know in the comments down below as well. And, I'm gonna be cheeky here, if you're enjoying the video so far, then please leave a thumbs up. But let's get back to it. Here's another example of a vector field. Coming back to electromagnetism, a classic example of a vector field is the magnetic field around a bar magnet. We've often seen these in high school, especially demonstrated with a bar magnet and iron filings. The iron filings are easily magnetized, and so follow the magnetic field around the bar magnet, because the bar magnet's magnetic field exerts a force on each one of these little iron filings, and the magnetic field lines that we draw basically show the direction of force on these iron filings. So, yes, a magnetic field can be represented as a vector field, where this time, a vector in the vector field represents the direction and size of the force experienced by a magnetic object placed in the magnetic field. So, now that we know that a magnetic field around a magnetic object can be represented as a vector field, let's get back to looking at the divergence of a vector field and what that means. Well, when we're trying to find the divergence of a vector field, essentially what we do is we choose a small region of space within that field, and we see how much that vector field either points into that region or out of that region. Again, this is fairly confusing, so let's use an example to demonstrate. Once again, let's put aside magnetic fields and let's consider another vector field. This time, I want you to close your eyes and imagine you're drawing yourself a relaxing bath, because obviously there's no better place to do physics. No, but seriously, imagine that you're running yourself a bath in a bath that has taps at one end and the drain at the other end. Now, this is fairly uncommon. I know that usually the drain is on the same side as the taps, but for demonstration purposes, let's imagine that they're on opposite ends of the bathtub. Now, let's say that in this case, you've forgotten to plug the drain, so you're doing a really bad job of drawing yourself a bath, because the water flows into the bathtub from the taps and flows right back out from the drain. Now, at this point, you're obviously wasting water and damaging the environment, but it's okay, because it's for the sake of physics and because it's only in your head. Now, let's say that we're looking at the bathtub from above. We can represent the flow of water on the floor of the bathtub with a vector field. We know that at the tap end of the bath, the water is flowing down onto the floor of the bath and then spreading outwards. In the middle region of the bathtub, water is flowing away from the tap end and towards the drain end. And at the drain end, all of the water is flowing into the drain and down the plug hole. Now, of course, it's important to realize that this is only the net or overall flow of the water, because, of course, some of the water will reflect of the sides of the bath and be flowing in all different directions. But, overall, the water is flowing away from the tap end and towards the drain end. Now, let's say that the vector field at any point is represented by the vector V. V standing for velocity, unlike B standing for magnetic field, apparently. But, of course, because the vectors are different at every point along the bathtub floor, sometimes they're large, sometimes they're short, sometimes they point towards the right, sometimes towards the left. V, obviously, changes at every point. Okay, so now that we have a vector field, which represents the velocity of the water on the bathtub floor, let's take the divergence of the vector field. Let's start, then, with the middle region of the bathtub floor. When we're taking the divergence, we look at the vector field flowing into that circle and out of that circle. In this case, water is flowing into the circle from the left and is flowing straight out to the right. In other words, if we take the divergence of V in this region, then we say that its divergence is zero. Because, and here's the thing, this means that, overall, there's no flow into the circle or out of the circle. But, hang on, aren't all regions in the vector field like that, where water flows in and flows straight back out again the other side? Nah, fam, check this out. Let's say we now take our divergence at the tap end of the bathtub floor. Now, if we place our circle here, then, clearly, water is flowing out in all directions. So, overall, there is a net flow of water outwards from our divergence circle. Now, because water is all flowing outwards, that means that this is a source of the water. And this is fairly common terminology. If the vector field is, overall, flowing outwards from your region, then that region is known as a source of the vector field. And, more importantly, this region is said to have a positive divergence. In other words, the divergence of the vector field V in this region is positive. Now, conversely, we can take our divergence at the other end of the bathtub floor, this time at the drain end. And we can clearly see in this region that all of the water is flowing in towards our circle. This kind of region, where, overall, the vector field is flowing into our region, is known as a sink. And the sink of the vector field is said to have a negative divergence. And so that is how the divergence of a vector field works in a relatively intuitive way. Of course, there's a little bit more mathematical subtlety and intricacy to it, but that's not really important to us right now. So, let's go back to Maxwell's equation that talks about the divergence of the magnetic field. Now, this equation tells us that the divergence of the magnetic field is zero, always. Because it's not saying that, in some specific regions, the divergence of the magnetic field is zero. No, it's saying that the divergence of any magnetic field of any magnetic object is always zero. This is really important, because it tells us a couple of things. Firstly, it tells us that there are no individual sources or sinks of magnetic field. Compare this with electric fields, by the way, which has sources and sinks. Positive charges are sources, because the electric field goes outwards from a positive charge. And negative charges are sinks, because the electric field points inwards towards a negative charge. This is not true for magnetic fields, according to Maxwell's equation. So, let's check this is true for the simple case of a bar magnet. Let's see if we can find any sources or sinks of the magnetic field on this diagram. Is there any region that we can draw where overall the magnetic field is either flowing in or flowing out? Well, let's start with this region. Just a random region in the magnetic field. Look, magnetic field flows in, and then it flows straight back out again. So, the divergence in this region is zero. But what about the poles of the magnet? This looks fairly promising, because if we draw a sphere around the north pole of the magnet, for example, then we can clearly see that magnetic field is only flowing out of the sphere. There's nothing flowing back in. It's a source, right? Well, no, not really. The only reason it looks like a source is because we haven't drawn the magnetic field lines inside of the bar magnet. But when we do, it looks like this. And there you go. Magnetic field now flowing in and flowing straight back out again. From this, we can deduce something really important. Maxwell's equation is telling us that there's no such thing as a magnetic monopole, or single individual pole. You can't just have a north pole by itself, which gives out a magnetic field, and you can't just have a south pole by itself either. Whenever you have a magnetic substance, that substance will always have a north and a south pole. This is why when you chop a bar magnet in half, you don't just get a separate north pole and a south pole. You now get two little bar magnets, each of which has a north pole and a south pole of its own. However, interestingly, there are some modern theories that predict the existence of magnetic monopoles. So, scientists have been on the search for the existence of these monopoles for a little while now. No evidence has been found as of now, as of when I'm recording this video. There's even an episode of the Big Bang Theory, where Sheldon develops a theory that predicts the existence of magnetic monopoles. To search for evidence of this, the guys go on an expedition, I think to the Antarctic, and, spoiler alert, in the next episode they come back, supposedly successful. They think they've found evidence for the magnetic monopole. And this is a huge deal, because it means that Maxwell's equation is wrong. And therefore, this evidence has started a new era in physics. So, getting back to Maxwell's equations, you see what I mean about them being really complicated and intricate. This video has gone on long enough, and I've only discussed one of the four equations. The equation that we discussed was probably one of the simpler ones to explain, and it's taken me this long-ass video to do it. So, if you liked this video and found it useful, then please leave a thumbs up. I really do appreciate it. But now, it's time for the Weekly Question of the Week. My Weekly Question of the Week for you this week is, What is your favourite thing about physics? It can be a particular area of physics, or it can be anything about physics as well. Like the fact that it's evidence-based, and it's probably the best thing that we have in trying to understand how our universe works. Also, comment down below if you want me to do another video covering another one of Maxwell's equations. And yeah, if there's something that I haven't made quite clear enough, then let me know down below. And if I've made a mistake, then feel free to blast me down in the comments as well. Subscribe if you haven't already for more physics content. I make fun physics videos, though I don't have to try too hard because physics is already fun. And, hit that bell button to be notified every time I upload. Okay, now that all the YouTubey stuff is out the way, guys, thank you so much for watching. I really appreciate it, and I will see you next time. B-b-b-bye!