So, on Special Relativity: Far, far, Faraday

Boy, how long since the last time, uh? I mean, spending the summer working on my final project and starting my graduate studies this year has not left me with much time for here, but every now and then, I will try to not leave this as a wasteland.

Where were we again? Boy, still at electromagnetism? Well, if we’re gonna do this we have to go through this all the way to the end. I was supposed to introduce you to Maxwell’s Equations, and they’re honestly my favorite of all parts in classical Electrodynamics. Why? Well, because these equations beautifully bring together concepts that were once considered as having different nature. I don’t know if you realized it, but so far, we have spoken about electric field and magnetic fields. They sound different and they interact in fundamentally different ways with charges. Now two things are missing right now, and those are the interactions between electric and magnetic fields, and how they in the end, will reveal themselves as just being different manifestations of the same field.

The first person we will talk about is this guy which you have probably heard about:

This fellow right is Faraday, Michael Faraday (*cue to the James Bond theme*). Now Faraday was amazing in many ways. For once, he made fundamental discoveries and has a whole list of things named after him. Arguably, his most important contribution to physicists (and I am a physicist, at least, according to my degree, I am in a subfield of physics) being what we will discuss next. Chemists might say that electrochemistry (a field he kickstarted) might be his greatest contribution, and neuroscientists agree with them. And he didn’t have any formal education, although back then, if you had YouTube and Wikipedia, everyone could be a Faraday. He was the only one, however, who had the insight to go further. He was arguably the best experimentalist of his time, and he even became Fullerian professor of chemistry at the Royal Institution. But his mathematical skills were kind of, well, not impressive, not knowing beyond trigonometry and simple algebra. It took the work of another fellow to formalize his results in a mathematical manner. Now, while I could spend the entire day talking about Faraday, his biography is way too interesting and this post would be too long. But I have my commentaries for him in the future.

Now, for the most important thing he discovered (in physics), we first need to look again at the magnetic field. So far one thing that our field has always been is static. It didn’t change once, believe me, I checked. It doesn’t depend on time. So, naturally, one would think, “Hummm, I wonder what happens if this thing started changing somehow”. Now, Faraday was doing experiments with coiled wires (Remember from the last chapter, where we discussed that a current on a wire can generate a magnetic field), and passing the current in one coil, he noticed that another current was generated on the other coil. Now this is known intuitively, we know not to take our electric devices in a MRI room. But this was revolutionary. Essentially, with a magnetic field, you generated a current, you made charges move. Now. That all sounds cool, and I will even use a Wikipedia image to show the apparatus he had.

Now this is all very cool, I mean, it’s something you can do at home. But be careful. All this sounds very cool but if I were to ask you right now, “How does the magnetic field generate electricity?”, you would be able to tell me. And that’s why we have mathematical formulae and equations. Love’em or hate’em, mathematics describes our fundamental laws of reality, and the best way to confront your fears is to face them. That advice is something I should follow in relation to bugs, by the way, but what can I do?

So, a way to start would be to look at some other details that may influence the intensity of the current of potential difference generated. And that is what Faraday did. I told you guys he was a great experimentalist. And the final description he got something like

 or

Now, let me break that down for you. Essentially what he did was to first change the magnetic field, which is easy to do,  he just add to change the current passing through the wire, by changing the resistance, or by introducing a capacitor (we can talk about electric circuits later). The other way is to change the area of the coiled wire. Essentially the expression depended on the magnetic field and on the area this magnetic field crossed, or better said, in the change of these parameters over time. Now something I forgot to tell you back then was that there is a relation between the potential difference, or voltage and the electric field. Essentially if you were to measure the potential difference in two thing different points, what you would find would be equivalent to the electric field in these points and the distance between the two points. Something like

So that the expression below can be described as

Although the proper description is not like this, we still must consider a closed loop, such that b-a is equal to 0, but don’t worry, that does not turn the expression above to zero. And this is a relatively simple way to write Faraday’s Law, if you don’t know integrals or differentials. Since both changes in B and A affect the E generated we can consider the product of both quantities as the parameter that changes over time. That new quantity is the magnetic flux of the magnetic field B in the surface A. If it regular plane we could use the old expression, but more than often we have to consider more complex surfaces with weird shapes.

The proper way to describe this law would be:

 Suppose you have a closed path C. The change of the magnetic flux B in the surface S defined by C generates a potential difference V in C (or generates an electric field E along the path, if you prefer this formulation).

Again, if we have a weird shape, the best approach would be to divide the area in some small areas dS. So, your quantity flux will be defined for each small regular area as being . Now a couple of things you may have forgotten about before. The magnetic field is a vector field, which means it depends on direction, but as we said, we only care about the magnetic field that goes through the area element dS. So the way we express that in vector calculus, is by writing  . That is a inner product between the magnetic field and the normal direction n, which means the direction perpendicular to the area element dS. As you know from high-school math, this value is maxed if both vectors face the same direction and is zeros if the vectors are perpendicular. So this way you can guarantee that all your values are only those of the magnetic field lines crossing the surface. Sometimes, in times, they will join dS which is the area element and the normal direction n in a single quantity  or . That is mostly for notation purposes, but it will show up more. Finally, after having the expression for each individual area element dS, we need to add them together so that we can find the total magnetic flux. Let’s call that  because that name is too long. So we are adding a potentially infinite number of elements inside of an area S or A. We could use the old summation term, but there is a neat trick Newton (or Leibniz, depending on where you stand on the issue) made a few centuries ago called Calculus. And we will use one of the most useful tools it provided us called integral. Essentially it can be written as

And that is essentially the magnetic flux. Usually the first description of a flux is the electric flux, but I changed the orders a bit this time. Anyway, if you want the expression for the electric is the same thing. You just change the B for an E in the expression above and

Great, now we are free from the flux, and we can rewrite our expression above as

But there is a major problem with this expression, because it is not totally correct. We are assuming that the magnetic flux changes instantaneously, but, the change is continuous and smooth. We can divide this change in smaller and smaller chunks of time, like

We are almost reaching the best description for the induction law, but we still must deal with the left term of the equation. When we were talking about the magnetic flux we said that we wanted to generalize the expression for when we have complex shapes. Well, the same thing happens on this other. Sometimes the path described by our expression does not necessarily mean that a path can be easily described by just two points. Surely many of you have seen a wire, and you know, many times, you can’t just use two points to draw it. The same logic applies in this case. A way to approach this would be, like in the case of the magnetic flux, for us to divide the magnetic flux in small chunks of lines, let’s call them dl, l of length. Knowing that, the electric field is a vector field, and that we only are concerned with the electric field in the same direction as the path we can reach the same conclusion as we did with the flux. So this is expression that defines the potential difference

We could replace this in the final expression, but first let me remind you the first part of our formal definition: “(…) closed path C (…)”. Meaning that, this integral we defined needs to be considered in a closed path, where the initial and final points are the same. The final expression would then be

And this is Faraday’s Law of Induction, as defined mathematically by Maxwell in its integral form. This expression is arguably of one the first that shows an intimate connection between magnetism and electricity. But things don’t end here, and the connections are just getting started.