Let There Be Charge, Maxwell’s Equations: Explained
This is the third part in the Maxwell’s equations series. I think I’m being entirely honest when I say that I’ve dreaded writing this. Not because it’s “complicated” or because it’s “difficult”. But just simply because of the dreadful effort that’ll go behind these visual interpretations. I’m not an artist so excuse my abysmal drawings; I try.
Anyway, If you haven’t read the other two Maxwell equation articles, check them out here:
This article is an overview of the third Maxwell equation: Gauss’ Law for electric fields.
Since both of these are just different formalisms of the same law, we don’t need to necessarily worry about which one we look at since the meaning we derive will be the same. I find that, in general, the differential form is a whole lot easier to work with but I suppose that argument could work both ways.
The electric field vector is the E with the tiny arrow on its head. The fact that it even appears here suggests that what we’re really dealing with is the behavior of charges. Before we delve into the specifics, it’s important to understand that an electric field is a type of vector field. For the uninitiated, a vector field is a region of space where each point can be defined by a vector. In this sense, each position in space can be assigned a direction, the direction of force, and magnitude, the magnitude of that force. So in principle, wherever an electric field exists, we can assign a vector value to every point in space.
If we took a charged particle and placed it in the electric field, these field lines would give us an idea of the direction and magnitude of force exerted on that particle.
From this, it’s fairly straightforward to understand that placing a charge at that point in our electric field would cause it to experience a force that is in the direction of the electric field vector. Judging from the length of the field line, it’s also possible to assert that the magnitude of force experienced by that charge would be relatively higher. But it doesn’t just stop here. What if, instead of placing our charge closer to the source, we were to change its magnitude? A larger charge means a greater force experienced. Instead of 1 Coulomb, we throw in a 2 Coulomb charge. Well either way the governing equation would be F = qE. Where q is the charge, E is the electric field strength and F is the force experienced by that charge.
As always, we find that causality prevails. A charge in an electric field will always experience a force, no matter where it is. In that sense, the E represents the force experienced per unit charge, i.e., the electric field strength. This allows us to quantify the electric field without introducing a dependency on charge. Since the electric field strength would tell us the force experienced per unit charge, when multiplied by the charge, we’re met with the force. Anyway, back to our equation.
After establishing what E is, we only have three more terms to work through. The next logical step would be to introduce the downward triangle. Why? Well, you’ll see in a minute.
For those of you that are not familiar, this represents a quantity known as the divergence operator. The symbol itself, however, is canonically just called “nabla” or “del”. Needless to say, the divergence here operates on the electric field — purely in the mathematical sense. Very handwavy but, we now understand that the operator affects the electric field in a very particular way. A way that results in the Greek stuff on the right.
We’ll come back to the right side in a moment. First, let’s dismantle the left. What does it mean to take the divergence of a field? Well, mathematically, it looks something like this:
Yes. It’s daunting. But as far as we’re concerned, mathematical formalism is irrelevant. The divergence of a vector field can be thought of as the measure of how much of a field is flowing in or out of a region of space. If visualization helps, consider a positive charge.
Now the vector field around this positive charge points outward. In other words, we say that the divergence is positive. There is more of the field flowing out than there is flowing in. Actually, there’s no field flowing in at all. The positive charge is a source. All sources have a positive divergence because field lines act unidirectionally outward. Think of it like water flowing out of a tap. No water goes back in. The field lines act outward. Similarly, if you considered a negative charge, you’d find that the divergence is negative since negative charges behave like sinks.
But there’s also a third case. A more prevalent one, actually. Vectors just pointing in the same direction with the same magnitude. Wherever we put our circle, there are as many vectors going in as there are coming out and so the divergence is zero. For example, in a magnetic field, the divergence is always zero. There isn’t a magnetic source or a magnetic sink. Each point in space has the same quantity of magnetic field flowing in as it does flowing out.
It’s important to note that the divergence takes a numerical value and is calculated by a function. This function is derived by applying the divergence operator to the vector field. If the last 500 words made any sense to you, I trust that you now understand what taking the divergence of a field means. It’s simply a measure of how much of the field flows into or out of a particular region of space.
That’s a perfectly rusty explanation. We can do better than that, I think.
In the images above, we had only a couple of vectors drawn. In reality, however, we’d have infinitely many vectors. Just like how between 1 and 0 there are infinitely many numbers. There are infinitely many vectors because a vector field is defined by a function.
Before taking the divergence of a field, every point in space has a vector associated with it. After taking the divergence, however, this vector is replaced by a value. A value that tells us how much of the field flows inwards or outwards. For example, this equation would render the following vector field:
After taking the divergence and substituting the relevant coordinates, we find the numerical value associated with each point in space.
Our goal here is to understand divergence. The physics matters, the math can come along. So I suppose you don’t need to necessarily worry about how it’s done.
Now, the electric field of a negative charge is simply be represented by:
This equation renders a divergence of zero. Zero at all points except for where the charge is actually located. This is because the amount of electric field passing through a point in space will equal the amount of electric field passing out of that point in space.
What we find, actually, is this:
This is how charges behave. All points, excluding those with charges, have a divergence of zero. It’s only at the position of the charge that the divergence takes a non-zero value. And since we’ve eliminated the idea of zero, the divergence must yield some value. No matter how small or large, the divergence cannot be zero at a charge. Since if it were zero, it wouldn’t be a charge at all.
This is exactly what Gauss’ Law tells us. The divergence is equal to the charge density, ρ, divided by the permittivity of free space, epsilon naught. But this probably doesn’t mean much yet.
Essentially with rho, the idea is that the total charge in a region divided by the total volume of this region will tell us how much of the charge is there per unit volume. The charge density is a measure of how much charge there is in a certain amount of volume. In other words, it’s the charge per unit volume.
Epsilon naught is a fundamental property of empty space. This constant of nature is known as the permittivity of free space. It’s perhaps better suited to save this for the future but for today, it just represents a constant. A very important constant but still, just a constant.
So we now we’ve established that the left side of the equation tells us how much of the electric field flows into or out of a chosen volume or region. On the other hand, the right-hand side (pun intended) is the quotient of two quantities: the charge density and the permittivity of free space. Now, in any region of space that does not have a charge, it goes unsaid that the charge density is zero. Since if charge is zero, so is charge per unit volume.
Dividing 0 by anything would still yield zero. This works exactly as intended. Gauss’ Law is remarkable in a lot of senses. Most importantly, however, it allows us to understand the behavior of electric fields in unprecedented detail.
Similarly, for a charge, finding the divergence becomes very straightforward. All that needs to be done is to compute the charge density. The rest? Well, Gauss’ Law will handle that. The charge divided by its volume will equal rho. And well, divide that with the permittivity of free space and you have the divergence. Child’s play.
Anyway, I’ve dragged this on long enough. I’ll end this here. If I’ve made a mistake, please tear me down in the comments below. And as always, thank you for reading!
