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.

Jingly Bells and Big Bangs

Hohoho, Christmas  must be in the air and as always, we have some religious nuts going around complaining about secularism. Now I try to maintain my stance as irreligious in this debate, so I don't usually argue with them, but a really funny question popped up. 

"Why do people still believe in the Big Bang Theory when it's been disproved by the Bible?"

Sorry, but if you in some way agree with that reasoning, I am pretty sure you are not a Catholic. Or you are a really terrible one.

No, not because I think that Catholics are heretic whose only real thing they have going is the fact they are around for pretty much two millennia and for a large portion of that time, were the most powerful corporation in the planet.

And also not because I think not being a catholic means you are an heretical protestant that should receive the wrath of Inquisition, as the good old days of the Holy See as the overseer of the world (as opposed to the modern US, which I am also going to assume you are from)

Now, the reason why I know you’re not a catholic is because, as a former catholic, I know one of the things we would pride ourselves is in the fact that the catholic church and/or catholic individuals were innovators in various fields of science. And every catholic likes to brag about it to their protestant and atheist friends. I used to be like that.

Some of the greatest astronomic discoveries were made by catholic priests like the fact that the Earth revolves around the Sun, by Copernicus, which if I recall, is also in contradiction to the Bible (Actually later, Galileo got in some hot water with the Holy See because of this same heliocentric view). Another catholic priest, Mendel, is known as the father of genetics, which, along with many other things, have been used to prove that evolution is an actual thing (again in contradiction to the Bible). And the topic of your question, the Big Bang theory, was first proposed by a Belgian catholic priest, Lemaitre.

So, you have the biggest christian church in the planet, where its members have made discoveries that well may contradict the Bible, which is like the constitution of Christianity. Now you have a few options on what to believe:

A - You take the catholic approach, that is to say that while it may seem like it contradicts the bible, in reality, it doesn’t contradict it at all, since in essence, even though Bible is the word of god, they may have taken some literary liberty in some descriptions. I mean, if you think about it, while the bible doesn’t say anything about evolution, it doesn’t say anything against it, just like with Big Bang. If anything, the only suspicious “theory” here is that the Earth revolves around the sun, when Genesis clearly says the sun was made as a lighter above the firmaments. Although I do think the Catholics are really good with Jedi Mind tricks and Doublespeech.

B - You take my approach, and assume that a Book written (allegedly) over the span of two millennia with multiple writers none of them with the scientific knowledge we have today, and as such, on a bad trip, assumed to have met a “god” who told them words of wisdom. You know, like when Marduk slayed Tiamat and with her body created the world (you should know about them, they were part of Abraham’s first religion, back when he was just Abram). Fairy tails.

C - Or you take the fun approach, the planet X approach, or the flat-earther approach, believing that the catholic church has somehow allied itself with the scientists and the Muslims (even back when they were killing each other in the crusades) in a massive conspiracy to promote heretical teachings that contradict their own word, for the purpose of… making frogs gay with chemicals in the water? On that note, should we open an inquiry on the subject of heliocentric solar system? It seems like Sun worship to me.

Yeah, this isn’t about the scientific reason, because we both know we wouldn’t understand. The reason why I believe in the Big Bang, is because believing in it wouldn’t really hurt me in any way. And if I believe that the Earth revolves around the sun in an elliptical path, then it’s not much of an stretch to believe that the universe is expanding.

So long, and happy Holidays everyone!!

Parity in Physics

Something that always confused when I was first studying physics was the concept of parity, parity transformations, parity conservation... So I figured that a good way to properly understand the concept of parity and in a way to help others in doubt, is to talk about it in a informal way.

A good analogy for parity would be for you to imagine yourself in a mirror. Your mirror image looks exactly like you, except for the fact it’s inverted. Try raising your right hand, your mirror image raises the left, and so forth. You could that the mirror you is the you who suffered a transformation, where your front is now your back, and your left is your right, so in mathematics, something like

(x,y,z)⇒(−x,−y,z)$(x,y,z)\Rightarrow (-x,-y,z)$

x and y represent respectively your left-right and front-back direction. Now that is almost like parity. If your mirror image is also upside-down, then it is a true parity transformation. In short, a parity transformation is essentially a transformation that flips the sign of your coordinates. Something like r⃗ −→P−r⃗ .$\overrightarrow{r}\stackrel{\mathcal{P}}{\to }-\overrightarrow{r}.$

So yeah, it’s just that in essence. An inversion operation under all spatial coordinates. And yet it is very important in physics. Let’s say you have a physical quantity dependent on spacial coordinates, and let’s say that quantity suffers a parity transformation. Then if

f(−r⃗ )=f(r⃗ )$f(-\overrightarrow{r})=f(\overrightarrow{r})$

the quantity is said to have even parity. Examples of classical quantities with this property are energy, mass, the electric potential, usually scalar quantities. Else if

f(−r⃗ )=−f(r⃗ )$f(-\overrightarrow{r})=-f(\overrightarrow{r})$

the quantity is said to have odd parity. Examples being those like the position (seems rather obvious), force, or the linear momentum.

Now, those things are not really what we care about in quantum mechanics and nuclear physics. But with this part, you hopefully understand what we mean by even or odd parity.

In nuclear physics specially, the concept of even or odd parity is important to describe gamma decay (or isomeric transition) and nuclear stability. The quantum state of each nucleon has either odd or even parity. I am not gonna prove this, but the parity operator can be represented by its eigenvalue as show below:

P|ϕl>=(−1)l|ϕl>$P|{\varphi }_l>=(-1{)}^l|{\varphi }_l>$

That l is a quantum number equivalent to the angular momentum (but not necessarily being an actual angular momentum). So you can pretty much define the parity of a nucleus by the product of the individual nucleons. If both Z and N (number of neutrons) are even, then the parity of the nucleus is even. If (Z+N) is odd, then the parity is determined by the “valence nucleon”, the nucleon at the highest energy level, you could say. If (Z+N) is even, but Z and N respectively are odd, well, there is no way for you to know. The parity of the nucleus determines its stability, so you can see there is importance in this detail.

The parity is also important for isomeric transitions, or gamma decay. But I will come back later to edit it.