Here's What Maxwell's Equations ACTUALLY Mean.

Hi there, my name is Parth, and in this video I want to talk a little bit about Maxwell's equations of electromagnetism, as well as the meaning of the terms and the mathematical language used in each one of these. I've already made a few videos breaking down some of the Maxwell equations in detail, which I'll leave linked in the description below, and as we look through the meanings of all of these mathematical terms used in these equations, feel free to use the timestamps somewhere down here to skip through the video if you're familiar with some of them already. First of all, let's briefly talk about what Maxwell's equations are all about. This is a set of four equations that describes essentially how electric fields and magnetic fields behave within our universe. These equations also show how electric and magnetic fields are intricately linked with each other, and can be used to create a new equation that describes the behaviour of electromagnetic waves, such as light, among other things. So with that being said, let's take a look at the first Maxwell equation on our list, and understand the terms used in it. The first of our Maxwell equations looks like this. Downward pointing triangle dot capital B with an arrow on it is equal to zero. This can be read as saying the divergence of any magnetic field that we are studying is equal to zero. Let's begin by looking at the magnetic field term. We use a magnetic field to represent what would happen to an external magnet, such as a compass, in the region of space where our magnetic field appears to be acting. Specifically, the magnet placed in this field will experience forces on it such that its north pole ends up pointing in the direction of the field at every point in space, while the south pole points in the opposite direction. This is because a magnetic field, by convention, points away from a north pole and towards a south pole, and as we know, opposite poles attract. The magnetic field therefore shows us what direction an external magnet will end up pointing in, and the magnitude or size of the field also tells us how strong the forces acting on the external magnet will be. Now this divergence term is used to measure the net flow of magnetic field through any closed region of space. Basically, if we take any volume we like, let's say this cube, and consider the amount of field pointing inwards, then this exactly balances the amount of field pointing outwards, which is why the divergence is zero. Now Maxwell's equation says that this is always true for any magnetic field that is allowed to exist in our universe, as well as for any volume that is closed, meaning we consider all the surfaces around our region of space. For a more detailed description of this, I have a video discussing the downward pointing triangle known as the Nabla operator, which can be used to represent two other functions known as gradient and curl, as well as divergence. Check out that video up here if you're interested. And if you want to know more about this particular Maxwell equation with visual examples and a more detailed description, then please check out this video up here. Now before we go any further, I want to take a moment to thank the sponsor of this video, Wren. 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Huge thanks to Wren once again for sponsoring this video. Let's move on to looking at another equation, and the mathematical language used within it. The second one of our Maxwell equations looks like this. Once again, we see our friend the divergence operator, but this time we're taking the divergence of the electric field, E. And we can see that this quantity is not equal to zero. We'll look at what the right hand side of the equation says in a moment. But first, let's take a quick look at the electric field. Now, similar to a magnetic field, the electric field shows what happens to electrically charged objects placed within the field. If we take a positive charge and place it here, for example, then it will experience a force in the direction of the field lines. And the strength of that force depends on the strength of the field, given by the size of these arrows. And a negatively charged particle will experience forces in the opposite direction. An electric field begins at a positive charge and ends at a negative charge. Therefore, we can see that opposite charges attract in this model, just as we expect. Now it looks like the divergence of our electric field is non-zero. This means that if we choose a region of space, it is possible for the total amount of electric field pointing into that region to be bigger than the amount pointing out, or vice versa. What it depends on is this quantity, rho. Rho represents the charge density, or the charge per unit volume, within the region of space that we're considering. The larger the charge density, the more charge packed into our volume, the larger the divergence. Because charged particles are sources, or sinks, of electric field, right? So, the more of them we have in our region of space, the larger the net effect of the field pointing outwards, or inwards. The signs simply tell us whether we have more outward pointing fields generated by our charges, or more inward pointing ones. Now, this quantity, epsilon-naught, is just a constant. But it's a very important constant. It's known as the permittivity of free space, or simply the electric constant. It basically determines how electric fields behave in a vacuum, or free space. It directly links the strength of an electric field with the size of the electric charge generating it, and how far away from the charge we are. So that's the second Maxwell equation briefly discussed. It's interesting that electric fields and magnetic fields behave slightly differently, so check out this video up here for a more detailed discussion of the second Maxwell equation. Let's move on to the third one, though, and again look at the mathematical language, the terms used in it. The third of these equations looks something like this. We read this as saying that the curl of the electric field is equal to negative the rate of change of the magnetic field. So this equation now links together electric and magnetic fields with each other. The right-hand side is a measure of how quickly the magnetic field we are studying changes over time. The faster it changes, the larger the value of dB by dt, and the negative of this rate of change must be equal to this operator curl applied to the electric field in our system, if the system is to be something we can find in our universe, assuming Maxwell's equations do actually describe our universe accurately. Now this operator curl looks a bit like the divergence operator from earlier, but this time we have a cross here rather than a dot. There's a reason both the divergence and the curl operators use this downward-pointing triangle, this nabla, as part of their notation. Again that's fully explained in my video about the nabla operator, but curl actually measures what is known as the circulation of our electric field. This is difficult to visualize for electric fields, but much easier to understand if we think of a different vector field, which is something that can be described by a vector at every point in a region of space, so that it has both size and direction. The vector field we'll think about is one that describes the flow speed of a fluid. Large vector means fast flow, small vectors mean slow flow, and the direction just gives the direction of the flow. Now the curl of this vector field can be thought of as what would happen to a little fan that we drop into the fluid. At this point for example, the fan would rotate this way and fairly quickly, because this flow is much faster than this flow. So the curl of the field measures this circulation and its direction. Now this is much more tricky to understand in a simple physical way for the electric field, but the curl is a useful indicator of the properties of any electric field we study. How it behaves can be indicated by its curl, and the curl must equal minus dB by dt. Another way to think about this is that an electric field can be generated with these specific properties, with this specific curl, if we have a magnetic field that changes over time with this particular rate. Again, for more details check out my full video on this Maxwell equation, but for now let's take a look at the final one. It looks a bit like this. This time we see the curl of the magnetic field, and how it relates to the electric field in the system that we're studying. We see this term, epsilon nought, from earlier, as well as the rate of change of our electric field, or how quickly the electric field changes over time. But now we have two new terms here, mu nought and J. Mu nought is known as the permeability of free space, or the magnetic constant. So it's basically the magnetic version of epsilon nought in many ways. It determines how magnetic fields can exist within empty space, for magnets of given strengths. And finally, this quantity, J, is Maxwell's addition to an equation that already existed, but wasn't quite accurate enough. It's known as the displacement current, and basically represents the current passing through any surface around the region of space we are studying. Now remember, current is just a flow of charged particles. So in a very simplified manner, without going into too much nuance, this equation says that a magnetic field can be generated by a changing electric field within our region of space, or a flow of charged particles into or out of our region. There is a lot more nuance to this equation though, that I'll cover in a future video, which will be the final one in this Maxwell equation series. Now, here's a fun fact before we finish up. The speed of light in vacuum is directly related to the permittivity and the permeability of free space. Because light is an electromagnetic wave, and these constants tell us how electric and magnetic fields should behave in a vacuum, therefore electromagnetic waves travel at this speed here. And with all of that being said, we've briefly touched on all the mathematical language used in Maxwell's equations of electromagnetism. Thank you for watching. If you enjoyed this video, then please hit the thumbs up button, subscribe, and hit that bell button for more fun physics content. Please check out my merch, linked down below. It features a quantum dice design based on a famous quote from Einstein. And lastly, I'd like to say a huge thanks to all of my Giga patrons, as well as all of my other patrons over on my Patreon page, that's linked down below too if you'd like to support me on there. Thank you so much for watching, and I will see you very soon.