So on Special Relativity: Princeps Mathematicorum

Oh Gauss, may you bless this post. Now, we have gone through a lot of concepts as to how charges interact with each other, how charges interact with electric and magnetic fields, and how magnetic and electric fields interact with each other. Right now, if you’re just starting in college you can say you have a basic notion of how things work, not enough to have a passing grade, but a grade better than zero for sure. For a moment, I thought I should change this chapter to Electromagnetism instead of Special relativity, but believe me, there are many reasons why it is useful for us to go through this. To understand why Special Relativity is so important, it is useful to understand why there was a need for it in the first place. And the connections between classical electrodynamics and special relativity are too much that you can find numerous books on the subject. This is but a dumbed-down version of the books I am basing off.

Now, if you were to say who was the greatest genius of all times, who would you say? Einstein, Newton, you name them, they’re all valid choices. If you were to say Gauss, no one would judge you. In fact, many would agree with you. That’s just how awesome the guy was. Some even call him Princeps mathematicorum or “The greatest mathematician since Antiquity” (and by antiquity, they mean Euclid), which I think is kind of an insult to Euler, but well, that’s just how much Gauss did in mathematics alone. This German mathematician and physicist (yes, he was a physicist too, back then, many mathematicians were part-time physicists) did so many contributions to these and other fields of science that the list would too long for this. And if what he himself did wasn’t amazing enough, he was the doctoral advisor to mathematicians like Riemann, Dedekind and Dirichlet, all notable mathematicians, with Riemann being the founder of differential geometry, toppling down millennia of dominion by Euclid’s geometry (Side note: Riemann’s developments allowed Einstein to develop a mathematical framework for his General Relativity, and also it is said that Gauss discovered it before independently and never published the results). If you are a mathematician, not knowing about Gauss is an offense to him (although that can be forgiven if you end up being a Ramanujan), and if you are a physicist, that’s one of the least things you could do. He wasn’t self-taught as our previous two, but he was a prodigy like no other. At the age of seven, he developed a method to solve numerical series faster than previous methods used (I learned that method in the last years of high-school, to give you the testament of how he was). And after his death, in his notebooks they found mathematical theorems and proofs that were only proven 50 years after he had reached the proofs himself.

Now about what connection does Gauss have with electromagnetism? Well, to begin…

We have talked about many topics. Some of them being the interaction of an electric field with a charge, or the interaction of a magnetic field with a charge, we even know how changes in some of those fields generate the other. But if I were to ask you what is the source of a field, could you answer me that? Now bear with me for a second. I do realize we have already talked about the electric field being generated by a charge, and the magnetic field being generated by a moving charge, but can we prove it, like mathematically. Different from our other cases were most of the mathematical results were due to Maxwell’s work, these questions were answered due to Gauss’s Theorems that Maxwell applied to electric and magnetic fields, and he wasn’t that great with experiments, but he did mathematics like no one else. And these results show the last Maxwell’s Equations that we will see (although they are the first ones in order) in the future. 

This post is different than others, because Gauss’s theorem was made for multivariate vector calculus and field theory, based on initial works of other mathematicians (and part-time physicists) as Poison, Euler and Lagrange. Since electric and magnetic fields are fields, well, you get my point. This is to say that results from his theorems can be expanded to other fields too, allowing for us to look at Maxwell equations from a very neat mathematical perspective.

Disclaimer: since this is for electrodynamics, our field is the electric field. I won’t make any mathematical proofs. Not because I don’t like them, some mathematical proofs are very trivial, but fascinating to look upon; it is mostly because, for simplicity, I don’t want to resume my entire Calculus II semester, and this is just so we can better understand the differential form of Maxwell, because I honestly prefer them to the integral forms. You may not really get the difference now, but when we get to the final expressions, you will notice how concise and simple the final equations become.

The main reason why we call two of the Maxwell’s equations as Gauss’s Laws are not necessarily because he was the one who made them, but because he was the one who came with a mathematical tool that allowed physicists to describe the nature of electric and magnetic fields.

For a start, let’s remember the way we write the electrostatic force between two charges and define electric fields.

And just for a refreshing, let’s consider the expression for the flux of the electric field in a surface

If you need a refresher on electric flux, you can visit the post about Faraday’s law, linked below, or the proper bibliography I will leave at the end.

Now, let’s say, that the surface you wanted to analyze was a closed surface. What is an example of a closed surface? Well, think of something like a balloon. It has a volume inside of it trapped from the outside. You, it’s a hollow 3D thing. And since we are using this surface to calculate the flux, mathematicians have a name for it, a Gaussian surface. While the next thing we are going to talk about is valid for every surface, since we are lazy, we will pick an easy surface, something symmetric so the integral can be neatly done. And that r in our equation above shows that the electric field magnitude (its strength) depends on the radius, so we can use a spherical Gaussian surface S of random radius R. And for the sake of simplicity, let’s assume E to be constant at R. So, the integral would essentially be:

I want you to focus on the last part, that conclusion. On one side you have the flux of a electric field, on the other, you have an electric charge in the interior of the volume, times a constant (if you don’t remember, check the first post of electromagnetism, although that was too long ago, I will leave some links you can check out later in the description). So, you have a flux that is equal to a charge inside of the closed surface. Hummm, that sounds fishy. What if we take the charge out the volume? Well, then you have no charge in the interior and . So, your electric flux is equal to zero. But you still have that area there.

If your flux turns to zero, after removing your charge, and your area is still there, the only thing that could’ve vanished is the field right? So, what this thing is essentially saying is that you only have an electric field in V if there is an electric charge in V. No charge, no field, no flux. “Really, you made us go through all that just for this? Underwhelming!!” I know, I know, and indeed for electric fields it’s very intuitive, because you had Coulomb’s Law before. And you can derive Coulomb’s Law from Gauss’s Law. But, we won’t do that here. Now, the real magic begins. Let’s talk about divergence.

Divergence is a concept that appears a lot in vector and tensor calculus. If I were to say this in a way analogous to flux (or to what I found on Wikipedia), it’s kind of like a flux density per unit volume. You know, it’s sort of like finding a source. It tells the quantity of a field source at each point. If it is zero, you have a source in your volume. If you don’t have a source of a field in your volume, it is equal to zero. The image, I think, gives a good idea of what is divergence. If you think that a field has a “flow”, that flow has a certain direction. The divergence essentially tells you where the flow comes from, and where does it go to. If the divergence is negative, that means the flow is coming into the small volume, and if the flow is positive, that means that the flow is coming from the small volume. But mathematically and physically, they are both sources. And this divergence thing seems to be very useful in us finding a source for our electric field. A source from which it “flows”. But how can we use it? Well, there’s coincidentally a theorem in calculus called divergence theorem, or Gauss’s Theorem, that says that for a random field F in a volume V inside of a surface S

The left part assumes that the divergence is not equal inside of all the volume, so instead of just multiplying it by V, we multiply it by dV, or better said, a small volume inside of V where the divergence is the same. I hope you get this part, because remember, the divergence is the flux density.

So, applying this to the original expression we had above for the electric field, we get

So far, that’s all cool, but that is still boring, because we still have an integral there, we still need a volume. How do we get of that?

Do you remember of mass density? Well, in school, we learn that mass is density times volume. And as such we learn that density is mass per volume. A glass of water has less mass than an whole sea, but they both have the same density, because they’re water. But what if they didn’t? It could happen, specially in the sea, since density depends on things like temperature, which can be very different if you are at the surface or in lower depth. That way, you can’t find an accurate measure for your density. Instead of you just multiplying density by the huge volume V, you would divide your huge volume in small chunks of volume dV that have about the same size, and inside those volumes your density is the same, then you add the different density values times the volumes they’re in, until you get the mass. Now you where this is going, and it is an integral of the density in that volume. So, that would be a general expression for the density of anything.

If you have something “a”, in a volume V, and you know that density of that something is different across that volume, you can just

And in this case our something is charge. You can have a huge volume with all the charge you want, but that doesn’t mean, your charge is spread the same way across that volume, so you can say that your charge is

And if you replace it in the expression above for Gauss’s law, you get a neat result:

It may seem like I just made things harder for you. I didn’t, I promise. But before I show you that, I need to show you just another thing from calculus.

From what I’ve told you, integrals are sophisticated addictions, and one of those properties of addiction is that the sum of addictions

And that property remains in integrals, so that if

Another important thing is that if a function is continuous, and its integral is zero, then the function is probably zero too, unless we are considering a closed integral, in that case, according to a theorem (Cauchy’s Theorem), all integrals of continuous functions should be equal to zero. But outside of this case, the other possibility applies.

So considering all these corollaries that I won’t demonstrate nor prove, we can apply to our previous expression, and get rid of the integrals at once.

Now, we have the same law, but without the hassle of having an integral all the time, and this time you have a direct relation the electric field and the charge density. You could say the source of your electric field is your charge (at least in classical physics).

Now, we know about the sources of electric fields. But what about magnetic fields? What is their source? Do they even have a source? We can find out by using the same logic we previously used  for the electric field.

If you used the definition for the magnetic field we came up with a couple of weeks ago, the Biot-Savart definition

How would we proceed in this case if we were to calculate the flux of a field in a closed surface? Again, we would follow the same logic as we did for the electric field. In this case however, the magnetic field has a direction that is always perpendicular to the radius, different then in the case of the electric field. So:

Something we did talk about was that the inner product has a maximum value when the two vectors have the same direction, like in the case of the electric field. But for the magnetic field it is equal to zero, because the vectors are perpendicular to each other.

What this result implies is that you don’t have a magnetic charge or magnetic monopole as we say. This result is absolutely amazing. You have a field, that you know exists, because you can measure its effects, and yet, the field doesn’t have a source. This is called the Gauss’s Law for Magnetism. And with this I conclude all the individual laws required for me to present to you Maxwell’s equations. Oh yeah, I will use another mathematical tool there so that we can change the integral forms into these intuitive differential forms. Speaking of differential forms, I’ve yet to tell you how to calculate the divergence.