Let There Be Light, Maxwell’s Equations: Explained
Maxwell’s equations are, and I quote Wikipedia on this one
“A set of coupled partial differential equations that, together with the Lorentz force law, form the foundations of classical electromagnetism.”
Well, Wikipedia’s got it right for the most part. They are coupled. They are differential equations. And of course, they do form the foundations of electromagnetism. But what does any of this really mean? Well to kick this off, I’d like to start by saying that even if I could do justice to his four equations in one post, I wouldn’t. Each of Maxwell’s equations are intricate and brilliant in their own right. Even a 100 articles wouldn’t be enough. But, I’ll try.
This article here won’t be focusing on all four of them. Just one. This one:
So, what do these symbols even mean? Well, off the top, we see arrows. Two of them. This means that we are dealing with vectors — quantities that have both magnitude and direction. And well, like we’ve already established, Maxwell’s equations focus on electricity and magnetism. The “B” that we see, represents the magnetic field. Obviously then, here comes a question: “why do we use B to represent a magnetic field?”.
Don’t even ask. Don’t.
The downward pointing triangle and the “dot” next to it, together represent something known as divergence. So, what is divergence? Right, here we go:
The divergence is often applied to something known as a vector field. There’s more to it but to understand divergence properly, we first need to understand what a vector field even is.
Okay, so vector fields.
Well, they’re pretty straightforward. A vector field is a region of space to which we can assign arrows at each point. Each point has a quantity that can be measured in both size and direction. A very prevalent example is a vector field that shows the direction and speed of wind. Direction would be characterized by the orientation of the arrow and speed, well, the size.
A magnetic field, too, is a vector field. Though of course it’s got nothing to do with winds and speeds. The field lines show the direction of force and the magnitude of that force. These field lines, in principle, extend out to infinity. At some point, however, for all practical purpose, we can ignore the effect of magnetism as it becomes really small.
In the image above, the field lines show the direction and the amount of force a magnetic object would experience if it were placed at that point. Now note that these field lines are continuous and they are an infinite number of them. The image does not completely encapsulate the idea but I suppose you get the point. So now that we know the magnetic field is also a vector field, let’s have another shot at divergence.
When we find the divergence of a vector field, what we’re really doing is finding out how much of that vector field either points “inward” or “outward” of a region of space. This may seem awfully handwavy but I hope the bathtub below makes it make sense.
When you look at a bath tub from above, you can represent the flow of water using a vector field. We know that at the “tap” end of the bath, water is flowing downwards and spreading radially outwards. In the middle region, water’s flowing away from the tap and towards the drain. And finally, at the drain end, all of the water is flowing inwards and down the plughole. Now it’s important to realise that this is only the net flow of water because of course, some of the water will be reflected by the walls of the bath. Overall, however, the water is flowing away from the tap end and towards the drain end. You get the idea.
Now, let’s say that the vector field at any point is represented by the vector, v. v as in velocity. Unlike B which somehow stands for magnetic field. Now since the flow of water is continuous, v obviously changes at every point. Near the drain you’d expect the water to be faster and around the walls, much slower. So the arrow’s direction and size are both changing continuously — a function of time. Now that we have a vector field that represents the velocity of the water on the bathtub floor, let’s take its divergence.
When we consider a region of space at the very middle, we find that water flows in from the left and flows out from the right. All the water that flows into that pink O flows out again. In other words, if we take the divergence of v in this region, we find that it is zero. If you haven’t noticed already, this is entirely because there’s zero net flow of water. Everything that comes in, goes out.
Let’s now consider divergence at the tap end of the bathtub floor.
If this is the region we study the divergence in, we find that water is flowing outwards. The net flow is not zero and this must mean that there is some divergence. Essentially, what this translates to is the idea that the tap is the source of the vector field. If a vector field is overall flowing outward from a region, then that region would be considered a source. More importantly, this region is said to have a positive divergence.
Conversely, we can take our divergence at the other end of the bathtub floor. The drain. I don’t suppose it comes as a surprise that the drain’s divergence would be less than zero since it’s a sink and not a source.
Of course, there’re some more mathematical intricacies and subtleties to the idea of divergences and vector fields. The core of it all, however, is what you just spent the last 5 minutes reading. Anyway, now that this has been established, let’s get back to Maxwell’s equation.
Now this equation, if any of the above stuff made sense to you, tells us that the divergence of a magnetic field is always zero. It’s not saying that in some specific regions it’s zero. No. It’s saying that the magnetic field of any object over any region is always zero.
This is really important since it tells us a great deal. Firstly, since the divergence of a magnetic field cannot be any value other than zero, we understand that there cannot, also, be sources or sinks of magnetic fields. Compare this with electric fields by the way, which have both sources and sinks. Positive charges are sources because the electric field radiates outwards and negative charges are sinks since their fields radiate inward.
Secondly, this equation tells us that it’s virtually impossible to have a monopole. You cannot have just a north or just a south. You need both. Because in principle, if either of these were to exist, their divergence would not be zero. Any magnetic substance will always have a north and a south pole.
This article has gone on long enough and I’ve only explored one of Maxwell’s brilliant equations. This is probably one of the simpler ones to explain. If there’s something I’ve not elucidated properly or have gotten wrong, feel free to dismantle me in the comments below. For now, however, I’ll leave you with this. As always, thank you!
