So, on Special Relativity: One Père, Two Père, Ampère

Last time, we were talking about Faraday’s Law and oh boy, was that a wild ride. Now I want to talk about some other guy. Let’s get out of Great Britain for a moment and swim to France, where another physicist (this time, a mathematician as well), André-Marie Ampère, was also working on electromagnetism. Ampère was an autodidact as well, a lot of people were back then, it was one reliable way for you to become a physicist (and surprisingly, a mathematician). But, that’s not true these days. Sure, he was part of a wealthy merchant family and they had a great library, so it wasn’t that hard for him. But he was also self-taught as was Faraday. I mean, the guy did learn philosophy from masters of Enlightenment, but through books he mastered the works of other masters, like Euler and Bernoulli. The guy we briefly mentioned before, Oersted, rediscovered this magnetism phenomenon. And Ampere was excited with the brand-new possibilities that could represent. He wanted to go beyond what Oersted did. And he worked on it. Initially, we found from expressions above that the magnetic force could depend either on the interaction of a moving charge or of a current with a field. His experiment wanted to look at how currents interacted with each other (he believed they could generate magnetic fields partly due to results from Biot and Savart. You know, that Biot-Savart Law:

In the same way we expressed the magnetic force through current we can do the same thing with this law.

Now, I must admit something to you. While Biot-Savart Law can be written in that way, the proper way to describe this law in terms of current to consider that this expression is only applicable in a short distance and as such only an expression of a small portion of the magnetic field, because again the path may be complicated and as such, so the magnetic field. With that in mind, the best approach would be for us to write the expression as

The value  is the small distance crossed by a charge q moving at speed v. The value  is the unitary vector perpendicular to the path crossed by the charge q. And the outer product between the two vectors show that the magnetic field generated is perpendicular to both vectors.

And using the tools we already know, we can indeed write the final generated field as

Now you may have noticed in the images that you have some lines from your magnetic field around your path. Now that is not a coincidence. Those lines are what you could call equipotential lines. They are regions of space where the magnetic field has equal value. And one thing you also notice is that those lines are circular. That is also not by mistake. Because it turns out that the value of B is equal in every point at the same distance from the path, regardless of the direction. So, if you were to integrate the value across a linear path for example:

(Not considering the direction which is ). So, if you now were to integrate across the equipotential value for the magnetic field, you would get the magnetic equivalent of an electric potential. But the result you would get from that would be amazing.

Again, you must consider that linear path, where we have defined everything and got the result, but this time, we will integrate on the left side first.

Now, this result is incredible. Essentially what we are saying is that the magnetic potential of a system only depends on the energy. And you may think that this result only holds in this case, but Ampère (and Gauss, too. He is a mathematical god we will talk about next) proved this result could hold in a loop of any shape. As natural, the expression would then be:

And if you want to share this Ampère’s law (or since Oersted was the first one to observe it experimentally, Oersted’s Law) with someone you know, you could just say that the current passing through a surface S would generate a magnetic field along the path C surrounding said surface. Now, that all sounds amazing, but it is not enough. Because experimental results showed something else was happening. Ampère’s result was incredible, but incomplete. And the reason for that was because of something I have yet to talk about. Usually the idea of current is the idea of charges moving along a certain material, or just moving. And Ampère’s Law is only valid for when you have charges on your system. But I will show you now a case where this doesn’t happen. First, let’s consider a capacitor (it is generally two conducting plates with an insulator sandwiched between them). An interesting property of semiconductors is that while the charges don’t travel from one conductor to another, the accumulation of charges in one conductor generates an electric field that causes an accumulation of charge in the other conductor, and there is a potential difference between the two plates. They are very interesting, but for now, that’s all you need to know. Now consider that capacitor where one side is being charged and keep in note that no charge carriers pass through one plate to the other. Now imagine two imaginary surfaces S1 and S2, bounded by the same path.

According to Ampère’s Law, the magnetic field across that path should be equal to the current  times the constant, but if you calculate the expression for B in the surface S2, the final expression is zero. Which is absurd because you know that there a current right after that surface, equal to the first one. That is a discontinuity and physicists abhor these things. They are our most frustrating nemesis, and we constantly spend multiple weeks or months trying to come up with conditions to avoid them. I can only imagine what they went through when this showed up. It took the works of Maxwell to come up with a solution, by defining what we can the displacement current, defined as

You can find that current through one of Gauss’s laws. But that is another thing we will leave for later. But this expression states that a current can be generated by changes or variations in the flux of the electric field, which is exactly how capacitors work, by using the electric field generated in one side to generate a current on the other side. And now, our updated Ampère’s Law is called Ampère-Maxwell’s Law. Now we have talked a lot about Maxwell, but so far, we never really talked about him and what he did. Let’s just say for now, he did a lot. Like really a lot.

Oh right, I forgot to show you the new law:

Now you can also the current, by considering a current surface density defined as

And if you consider that only the electric field is changing, you can simplify this even more as

You may be asking, why would you need to change this in this form. It doesn’t look that convenient. Believe me, you will later appreciate this change.