Let There Be… Voltage? Maxwell’s Equation: Explained
Maxwell’s equations are four equations of unparalleled brilliance. Compiled by Maxwell, as the name suggests, these equations are the founding principles of electromagnetism. In my previous article I explored only one out of the four Maxwell equations. We discussed the idea that magnetic monopoles cannot exist and that their divergence must always be zero. If this does not make much sense to you, stick around. I hope it will by the end of this. If you’d like a more thorough understanding of it, however, check my previous article out below:
It’s alright if you’re not familiar with Maxwell’s genius yet. There’s a first time for everything. Including the foundations of electromagnetism. This is the second article in a four-part series of Maxwell’s equations. Here, I’ll be exploring the Maxwell-Faraday equation:
That, however, is its differential form. Rewritten, it doesn’t look nearly as friendly:
We’ll stick with this form of the Law since it’s a whole lot easier to explain. You’ll see why in a minute. For now, however, don’t bother considering why or how we got here. Just take me on blind faith. If I do it all right, you’ll have an understanding of what any of this really means by the end of this article.
Enter: the Maxwell-Faraday Law
The Maxwell-Faraday law states that the total electric flux out of a closed surface is equal to the negative of the rate of change of the magnetic flux through that closed surface. I realize that I’m probably just throwing words at you — “electric flux”, “magnetic flux”, “closed surface”. So, let’s break things down a bit.
First, the two symbols.
Both of these are notations used to indicate integration. If you’re familiar with calculus, you’ll recognise these. If you haven’t, however, seen or understood these before, here’s a quick primer:
First, let’s suppose we’re dealing with a mathematical function. And for some really odd reason, we want to find the area between that function and the x axis. What we can then do, is consider an infinite number of rectangles under the curve and sum their areas all up. Something like this:
What you’d notice, off the top, is that our approximation for the area increases as the number of rectangles do. So, in essence, integrating that function over the region 0 to 1, will give us the value that we’d obtain from adding all these rectangles up. As the number of these rectangles increases, their width would decrease. Integration is simply a convenient way to consider the total area we have here. Written mathematically, it looks something like this:
What this translates to is the idea that integrating a function, f(x), is equal to summing up all the infinitely tiny rectangles you can have. As this number tends to infinity, you get the actual value of the area instead of a mere approximate. The right hand side is fairly straightforward. The f(x)∆x simply indicates the area of the rectangle at some point, x. Since f(x) would be the y value and ∆x, the x value, their product would give us the area of the rectangle. Child’s play. That’s really where the integral sign comes from. It represents the adding up of a really large number of really small pieces. The idea is that you add lots of tiny elements to give you the whole thing.
I don’t want to delve deeper into integration since it would probably overshadow the purpose of this article but if you’d like to know more, check this article out by Maths and Musings:
Right well, back to Maxwell, Faraday and their law.
From here, we see integration appear again. So, with a leap of confidence, we can say that what we’re really doing is adding a lot of really small things up to get something bigger. But what are these things?
Well, to understand what we’ve got here, we need to remember the meaning of B and E. If you’ve read my previous article on the First Maxwell equation, you’d recall that B represents the magnetic field and E, the electric field. Both E and B are vector fields. Each position in space can be assigned a direction, the direction of force, and magnitude, the magnitude of that force. So in that sense, wherever an electric or magnetic field exist, we can assign a vector value to every point in space.
So now know that we’re integrating two things. The B field and the E field. But we’re still missing out on a ton.
dl and dS. These are both the little elements of something over which we’ll be integrating. I know this sounds really vague but before we look at them specifically, we need to realize that both of these quantities are vectors too. They’ve arrows on them. The dots between them and their field doesn’t represent a normal multiplication. It represents a sort of vector multiplication. It’s called the scalar product. The reason this is considered a scalar product is that although you’re multiplying two vectors, the result of this will be a scalar quantity, i.e, a quantity with just magnitude.
So, what’s dl? Well, dl is a very small vector that lies on the perimeter of a surface that we’ll call, S. Now this S will become very important soon so, keep up. To tidy things a bit, let’s also say that our surface, S, is a circle and dl represents a very small value of length on its circumference. In other words, adding all dls will give us the circumference itself. So what we’re doing on the left hand side is multiplying the electric field by the dl vector. What this tells us is how much of the electric field points in the direction of the dl vector.
Add an electric field and this would probably yield a better understanding:
At that point, since the dl vector points in the same direction as the electric field, they’re both aligned. So when we take their dot product, we understand that the contribution of the electric field in that direction is large. But what I’ve drawn is only one dl. What we’re looking at is the contribution of the electric field across all dls. What we’re really doing here is adding up the behaviour of our electric field along the entire length of the circle. If we were, however, to consider a dl which was perpendicular to the electric field, we’d find that its dot product would be zero and therefore, the contribution of the electric field along it would also be zero. I’ll run over this again but let’s move on to dS for a minute.
dS is a similar kind of vector but instead represents an area. Specifically, every single dS is a vector that is perpendicular to a little area inside the circle we’re considering. In other words, if we take a very tiny area, an area which we’ll call dA, then dS is the vector that’s perpendicular to that. So what we’re doing here, now, is multiplying this area vector with the magnetic field. In other words, what we’re looking at is the contribution of the magnetic field within every single dA. Once we add it all up, we get the contribution of the magnetic field through the entire area. Then, what we’re looking at is the total contribution of the magnetic field through our surface.
So now, the idea is that on the left hand side we’ve got the contribution of the electric field along the perimeter of our surface and on the right we’ve got the contribution of the magnetic field through our surface. We still haven’t, however, addressed the d/dt part.
Well, all this means is that we’re finding the rate of change of whatever is inside the integral. Think about it this way: you know that when you find the speed of an object, you’re trying to find the distance it travels divided by the time it takes to travel that distance. In other words, you’re trying to find the change in distance divided by the change in time. Or, dx/dt.
Here, we’re doing the same thing but a little but more calculus-ly. Basically, we’re finding how fast the integral changes over time. And well, the negative sign basically just flips the sign. It’s just the negative value of this rate of change.
Putting it together:
On the right-hand side we’re finding the negative rate of change of the magnetic field through the area of the surface with respect to time. Now, what if I tell you that the perimeter of the circle we were considering, is a loop of wire. And of course, that wire can carry a current. What if I also tell you that the left-hand side of the equation gives us the voltage of that wire. You can see where I’m getting at with this.
The Maxwell-Faraday equation is a very glorified way of representing electromagnetic induction — a process where a changing magnetic field through a conducting material causes a voltage to be generated. In other words, a changing magnetic field through a conducting surface causes a voltage to be generated in that conducting surface.
Maxwell’s Second Law describes how a continuously changing magnetic field across a conducting surface will induce an electromotive force, i.e., a voltage. Similarly, its effects are valid the other way around too. An electric field gives rise to a continuously changing magnetic field. From this, I hope you understand that it’s the rate of change of magnetic flux — the contribution of the magnetic field within that area — which makes a difference. Just putting a magnetic field in a coil of wire won’t get you too far. That field must change.
Of course, there’re a lot more intricate details I haven’t delved into. Partly because they’re not essential to this article and partly because I’m afraid I’m a little rusty with Maxwell’s equations. Hopefully, you now have a better understanding of what any of this really means and its definition does not sound all that daunting anymore.
The Maxwell-Faraday law states that the total electric flux out of a closed surface is equal to the negative of the rate of change of the magnetic flux through that closed surface.
With that, I’ll leave you here. Let me know if there’s something I haven’t entirely explained or have gotten wrong. As always, thank you for reading!
