So on Special Relativity, Special De-Tour - About lights and waves

So now, I want to talk about light. And more specifically, about electricity, magnetism and how they come together. And I will be honest, I will gloss over a lot of detail, because this is a series on special relativity not electromagnetism, so I won't go in detail. But here are the things you may need to know:

- High school mathematics, specially derivatives

- And hopefully, some pre-calculus, like integrals and such. I would recommend you watch 3blue1brown's series on calculus, because it gives you an intuitive understanding of what a derivative or integral means.

I will use mathematical expressions to summarize the ideas, but I will make an effort to give a written explanation of what they mean, after all, my point for you right now is not for you to understand the math behind it, but to understand that maths give sense to physical conclusions, no matter how unconventional they may seem.
Also, a couple of considerations here. To avoid complexity, I will consider mostly point-like bodies when talking about these equations, because let's be honest, I don't really want to put that much detail into things.

So, the first thing we will talk about today is about Coulomb’s Law, a law that describes the interaction between charged bodies:
 
Now, let me give a quick explanation of what does this expression mean. First, let’s explain the constants, you have your classic π, and then you have this symbol ε_0, which in physics is called electric permittivity of the vacuum. It is a physical constant that expresses how vacuum permits electric fields. Now this permittivity may seem a complicated concept, but it essentially says if a certain space is resistant to electric fields (similar to the concept of electric resistance in an electric circuit) and in this particular case, the vacuum permittivity is a constant (at least, we think it is, but is a whole new discussion), it has a specific value, of about 8.854∙10^(-12)  m^(-3) kg^(-1) s^4 A^2 (with A being ampere, the unit of current). And four is a number, so we usually just consider all constants assembled in a single constant:

, which we call Coulomb’s constant, and which value is 8.99∙10^9  Nm^2 C^(-2). We can simplify the expression as: 

Now, the q_i  ,i=1,2, are the charges of the bodies, as of now you can think of them as a fundamental property of matter like mass. And r is the classic distance. So what this expression says is: If you have two bodies with electric charge, they will exert a force on each other that decreases with the square of the distance. Let’s consider their initial distance is A, and they feel a certain force F. If they are at twice the distance A, the force is four times weaker. If they are at half the distance, the force is now 4 times stronger. As for the arrows, as you now, the force is a vector, so it has a direction. And that’s what the last term represent, it shows that for two bodies 1 and 2, the force goes from 1 to 2, and vice-versa. Now if you pay attention, you will see that the formula we have above is like Newton’s Gravitational law:

 , that describes the interaction between massive bodies. And this is a good way for you to notice a couple of properties for the Coulomb’s force:
First, the intensity of the force does not depend on the direction, but only on distance. Which means it doesn’t matter if you are in any specific point in space, if you are at the same distance, you feel the same strength. That is what we call radial symmetry, and it is something that the gravitational force also presents. Also, you may realize something in the equation, that Newton’s law has a negative sign. Here is something we know in physics (it is more of a convention actually), when forces have negative signs it means they are attractive, while when forces have positive signs they are repulsive. Now, as of right now, we have never found anything with negative mass, so we can be sure Newton’s law is always attractive, but we can’t say the same for Coulomb’s law. We do now there are positive and negative charges, and that means the electrostatic force can be negative and positive depending on the charges, more specifically on the product q_1 q_2. If it is positive (if both charges are positive, or both charges are negative) then the force is repulsive, and they push each other apart. If it’s negative, they have opposite charges, and attract each other.

 And that pretty much describes all of the fundamental interactions between charged particles. By the way, the gravitational constant G has a value of like 6.6⋅10^(-11)  Nm^2 kg^(-2) which means that Coulomb’s force is like 10^20 times stronger than gravitational force. That's 1 with twenty zeros behind it. Which is why we don’t perceive gravitational force in small scales, like the ones we experience daily, and why all our molecules are bound to each through electrostatic interactions.

Also, I would like to point out about electric fields. As you see from the expression above, force can be described as an interaction between two charged bodies. But it can also be described as an interaction between a charged body and the electric field of another body. This description is particularly useful if our body has some weird shape and consequent charge distribution. We don’t really want to deal with complex integrals that much, but only when necessary. So we can define the force as:

This expression is especially good if you know what is the value of the field, but don’t know where it is coming from, or if you don’t know the source. By the way, you can also describe Newton’s gravitational law as a field equation, like this:

 which also has units of acceleration in this case. For Earth’s surface, our gravitational field is approximately 9.8 N/kg or 9.8 m/s^2, the unit of acceleration. So, you can see how useful it is for us to use fields instead. And this is useful if we want to go to the next section. If you want to describe the acceleration caused by an electrostatic force, it would be:


Which means the body would feel an acceleration in the same direction of the electric field. By convention, this is how we show the electric field lines of a charged particle. 

Now that you know about electrostatic forces and electric fields, you know one of the main things in electromagnetism. The other thing you need to know is about magnetism. Now the existence of magnetism was known for a long time, ever since Ancient Greece. In Ancient China, compasses were already used as a reliable mode of orientation. And yet it lacked a way for us to properly describe it. No one knew why certain things would attract metals, or why the compasses would always point to the north. The first expression to describe the magnetic field’s behavior has this one:

Now, we have a couple of different things displayed on this expression. First, now we have the velocity appeared, as v, which means the body 1 is moving, and we have this new thing, B. That B is the magnetic field. A current as we know is a collection of moving charges so, what this expression is saying is that a moving charge inside of a magnetic field feels a force that is proportional to both how fast it is moving, and to the strength of the magnetic field it crosses. Now as you see there is a cross product between the two quantities. I hope you already know what a cross-product is but the short explanation I can give is that if you have two vectors and you make a cross-product of them, what you get is another vector perpendicular to both. What this means is that force vector is perpendicular to both the direction of the magnetic field and of the velocity of the particle.  Visually, it would probably be something like this.

Now, to be truthful, if it was a negative charge it would be pointing in the opposite way, but this is just for you to visualize what a cross-product mean. Which brings me to another point I want to make that the cross-product as its maximum value if v is perpendicular to B, and its minimum value if v is parallel to be. For you to understand this more intuitively, there is the formula for the absolute value of the force:

, where the θ represents an angle between the vectors velocity and magnetic field. If they are parallel to each other, then the angle between the two vectors is zero, which means sin(θ) and by extension, F, is null. But if they are perpendicular, then θ is equal to π/2 which means sin(θ) is equal to 1, and:

, its maximum value. And if you look, the direction where the force points is always perpendicular to the motion of the body, which means that this force changes the direction of motion. 

Remember when I told you about current? Well, let me just explain it real quick. Current is formally defined as being the variation of charge over time, or:

Which means, in some cases, charge can change over time. So in our previous expression:

If instead of you having a constant charge moving a certain displacement l, you had a certain constant path l where charge would change over time (example: a wire), you could write the expression above as:

You could say that expression describes the strength of a magnetic field on a conducting wire, also called Laplace force.

But there is still a mystery left. Where does the B come from? We know that the electric field E comes from a body with a charge q, but we don’t know the source for B yet. As it turns out, there is an answer for that. Two French physicists, Jean-Baptiste Biot and Felix Savart, where the ones to find a first answer to this question, in a law we now call Biot-Savart Law:

Like with the time we were analyzing Coulomb’s equation, let’s break this one down a bit. First, let’s see which one of these are constants: we have our usual 4 and π, and this time, instead of ε_0, we have µ_0, which is called the magnetic permeability (not permittivity) of the vacuum. It is kind of like permittivity, but instead of determining how resistant (or permissive) is a space to electric fields, it determines how permeable is a space to magnetic fields. And you also have that common factor q/r^2 which means the fields is proportional to the charge that is generating it, and to the distance from that source, like the electric field expression. But the main difference here is on that last term. On the electric field expression, we only had the vector direction u_r that meant the field had radial symmetry over the source. But here we have a cross product between the vector v and the unit direction vector. If you have a cross-product between them can only mean that the field generated is perpendicular to the velocity vector and to the direction. It would look something like this:

If it was a positive particle, the magnetic field would be pointing in the opposite direction.

So while it still holds the radial symmetry of the electric field, the magnetic field has some really interesting characteristics of its own. Like, it is generated by a moving charge, which means that in a place where charges aren’t moving you will not find magnetic fields (not exactly true, but not exactly false either, I just don’t wanna explain more). Now this is most you need to know before we talk about Maxwell and you will finally understand why maybe Galilean Relativity was not the best relativity.

Damn it, this was way longer than I expected. But this is what happens when you try to study half a semester worth of Electromagnetism in one day. Luckily this is not my first encounter with it, and honestly I think it is really important for me to talk about this basic concepts in EM, because we will soon talk about Special Relativity, and then the birth of quantum mechanics. I understand many wouldn't feel comfortable with this much content at once, but you don't have to take it at once. This is not a test, so you don't to be very attentive and fast on learning. But it is important for you to learn if you want to understand anything that may come behind this. Anyway, thank you for your time in here and remember to comment or send me an email at this adress rafaelhoppfer97@gmail.com, for critic or any suggestions you may have. 

Some references for you to learn more about this:
- Physics for Students and Engineers (with Modern Physics), Serway & Jewett: Part 4: Electricity and Magnetism, Chapters 23 (23.4-7), 29 (29.1-3), 30(30.1)

For a little more advanced description, we also have this:
- Electromagnetic Fields and Waves, Lorrain & Corson: Chapter 3: Electric Fields(3.1-3)
Chapter 18: Magnetic Fields (18.2)

Also, Wikipedia may give a good introduction on the topic. These books also happen to have a bunch a exercises you can use to pratice, so I highly recommend you to try and solve them, and email me your answers to certain problems, if you are having difficulties.

No special relativity this week, just an opening of heart

Hawking, the challenger!

I woke up Wednesday and when I got to college I saw the news that Prof. Hawking had passed away. It kinda hit me hard, because I think it is no exageration to say he was an inspiration to many young physicists and future physicists.

So this week I decided to give a piece of the influence Hawking had on me and on those who were close to me. Not as a mourning, but mostly as praise, and celebration for the life he lived.

As most on Earth, I heard about him in the usual way, through some news about his opinions on current events on the world at the time. At the time, I was likely a kid, probably going to start middle school or something close to that. And I just saw this guy on television talking about something. The ammount of attention people were giving to him, made me think that perhaps he was an important person. But that was about it. I liked space when I was young and I even had the luxury of having an amateur telescope (it was not good though), so I would constantly be fascinated with everything new I would learn about planets, stars and galaxies, and I would devour the pop-science books who had these contents (sometimes literally, I mean, the pages looked appetizing).

In one of these books I happened to come across Hawking again, probably about black holes. As a pre-teen, you can imagine I would love the ideas of black holes, and knowing someone who worked on them, and was able to explain them, made him into someone I definitely admired. At this time, I had just started memorizing famous scientists names, mostly to look inteligent, because that's how things were in the late 00's. So I already knew of Galileo, Newton and Einstein. And I would look for quotes of them on the internet, and try to make those my personal philosophy (later I found out most of those quotes were misquoted or fake, so it came and bit me in the ass). What would fascinate me most about Hawking was how he was still able to do physics while on a wheel chair. I was the kind of person to believe that just because I would have a bad grade on some subject I wasn't able to do it, and yet you have someone who if he didn't try, no one would judge him, you could even say he was justified to not do physics anymore, and yet he persisted, even when losing his voice. I begged my mother for a copy of  "A brief history of time" (One of the perks of being a kid, you can still get some things from your parents if you beg properly) and I read, and re-read, and re-re-read the book.

I started being fascinated for this new science I wasn't aware of. Physics. I wanted to be a doctor prior to that, so it was kind of a change on paradigm. Where I grew up, no one really knew about physicists, we knew they were people who studied stuff and were super smart (we believed that) and knew all the mysteries of the  universe. However, the closest to physicists we had there were engineers. Which was not bad, but an engineer is not necessarily a physicist. They may know about the physics required to their field, but they are more of an applied physicist than the ones I truly admired. I didn't really admire many engineers at the time, maybe Archimedes. And besides those, we only have physics teachers from high schools, which may be good. But still they didn't give me the feeling I would feel with my "idols". To me, I felt like being a physicist was nothing but a dream, that someone from a small African country shouldn't really think about. But again, I would remember that I wasn't on a wheelchair, so fuck what I thought back then.

 After passing through high school, I already knew english and also, how to use Wikipedia, so I would spend my time there reading biographies and things that I would really understand. Those formulas would end up giving me headaches, which only cemented my idea that physicists were above regular people. But I knew enough about maths to start to notice a particular beauty in the physical laws and formulas I would see in high school. Luckily, I always loved maths, and the unusual apathy that most had for math was unknown to me (Really, one day I should do a post about this crazy idea people have about math), probably because there were some great pop-science books about mathematics I would read. And also because I had great teachers who kept me motivated to learn. So I was trained in trying to see things abstractly, or through just mathematical expressions. But I still was not able to understand the expressions of the scientists of old.

I got to college. My first year was fairly tough. I had to suddenly adjust my schedule from high-school to university, and believe me, college will not be easy (specially if you spend more time in bars than in your bedroom studying, but you should still relax at least once a week). But I still remembered those same ones who inspired to go through this journey. One of the things I thank the most while being here in college, was not only seeing actual physicists who dealt with the various fields of physics and were active on research (many of them inspired me to keep going) but I also got to meet various other genious of before who made key contributions in physics and mathematics, like Maxwell, Boltzmann, Dirac, Schrodinger, Feynmann, Euler, Gauss, Laplace and Ramanujan. And also some contemporary who still live, like Englert, t'Hooft, Gell-Mann and Witten. My personal favourite when it comes to explaining physics is Feynmann, mostly for the way he exposed physics in such a way that both physicists and layman could understand. I aspire to someday explain people as well as he did.

But still Hawking was someone I deeply admired. Maybe it was because he had such a deep presence in my life as a child compared to the others. I mean, he was still alive, and I could daydream of a day we would meet, and I could ask him all sort of questions, while sadly I couldn't do it with Feynmann (I still can with t'Hooft and Englert, at least that). Now I already knew a bit about physics, a tiny bit that could give me a basic notion of what he was trying to say on his equations. Yeah, now, I had the luxury to know a bit about what he actually did for physics, besides bringing it to the eyes of the public. And I was hoping that in the next years, I could muster enough knowledge to finally be able to talk with him. And this fatality happened.

At the beginning I was sad, but later that sadness was replaced with a feeling of admiration. While only having two years to live, he outlived all expectations, reaching the new century and 55 years beyond what he should. And he was able to bring a field like physics to the public. If someone was able to do for maths what he did for physics, we wouldn't have to worry about kids not liking math. Now it is true that many other physicists did relevant contributions, some more than Hawking, (many of the names I have above), but Hawking brought physics to the spotlight, and thanks to him, and others who like him, saw the usefulness in bringing physics to the public, people like me, could become physicists. Now roughly 10 years after I first heard about him, I am going to graduate in an Applied Physics and Engineering course (it's a double major, so I get to have the best of both worlds and I know, this isn't theoretical nor mathematical physics, which were his expertise but those are PhD objectives). But still through his work on physics, and through the exposition of this wonderful field to the masses, he will be remembered.

We may have never met, but the impact he had on me was not trivial. So, instead of farewell, I will say, for all others who like me, got to know the field of physics through him, thank you, Hawking. 

So, on special relativity - Here is a bit of math

Last week, we were talking about the whole concept behind frames, their necessity in physics, and about relativity, you know, the relative motion of bodies to an observer. This week, I decided to put a bit more of emphasis on the mathematics behind it. In high school you likely have already seen it, and if you haven't, well it does not really involve complex mathematics, so you can definitely understand it. This one personally excites more than the other posts because it has maths. But essentially it is just a mathematical summary of the ideas discussed on the last post. 

Here is some considerations before we start:

We will be mostly dealing with frames moving at constant velocity or not moving, or what we call inertial frames, and they would be observational frames of reference, which implies the observer would be at the origin in its own frame, at rest (its velocity is zero in its frame).

For simplicity, we will consider 2 bodies (you could think of them as being points with no dimension), in a 2-dimensional space. And our notation would be (x,y) for the spatial coordinates and t for temporal dimensions.

In this first case, we will consider body A's frame. For convention, this referential is considered as frame S, and for the sake of simplicity, we will consider B as moving in a constant velocity v in the x direction. Now let's make some analysis of what is shown in the picture above. (A hint: A lot of problems in physics requires you to use a bit of intuition, it will make problem-solving easier... Until you reach quantum mechanics, because then, you can throw intuition out of the window). B's initial position is r1 = (x1,y1), and its final position is r2 = (x2,y2), again emphasizing this is relative to A, and how can we get from point (x1,y1) to point (x2,y2)? At this point, it will be good if you know about uniform rectilinear motion where displacement x_final = x_initial + vt (for one dimension). In fact this first part is just your classical physics problem, and since you know about v, you can easily write: r2 = r1 + vt, but remember both r1 and r2 have two components (since we are talking about two dimensional space) so how do the individual coordinates change from r1 to r2?

Let's see the image once again. If you have noticed, it seems that while B is moving, it remains at the same relative height (i.e. y doesn't change), while x changes from a point x1 to another point x2.  So we can simplify r1 = r2 + vt as:

{x2=x1+vty2=y1

As you can see we can describe the movement of B completely with the expression above, and for any t = t', we can predict the position of B relative to A.
Now let's switch the positions. Now B will be the observer and you will see something interesting happening. Remember, the distance between them must be conserved when we switch from A's frame S to B's frame S':

So in this frame, B is not the one moving, and as such we know have A moving in a constant velocity v. One thing to note here though is that we also can describe A's motion relative to B in same way we did before, with the only difference being that for B, A is moving to its left, which in this convention implies the coordinates change negatively. So r'2 = r'1 - vt and it terms of its coordinates:

{x′2=x′1−vty′2=y′1

So what can we take out from all of this? First, we should realize something:

r2−r1=r′1−r′2=vt⇔r′2+r2=r′1+r1⇔{r′=r+vtr=r′−vt
Now that last part would show us a relation between the different frames, as it shows us that we could switch between S and S', and still be able to predict the motion of bodies. Even better, it shows us that if we know the movement of a body in one frame, we can easily switch to another frame and still be able to predict the results. Now a couple things you should know. This is only valid because we assume time to be the same in both frames (is what we call in physics an invariant, something that remains the same when we change frames of reference), and is in fact, referred in classical literature as Galilean Invariance, which states:
A - Time is invariant to any frame transformation in inertial frames
B - Newton's laws of motion are invariant to any frame transformation in inertial frames. (This part was added later, since Newton was born in the year Galileo died (Talk about eerie coincidences, in the same year a genius died, another was born).

In Special Relativity, we will see that is not actually true, but Galilean Relativity is still widely used in many applications of physics, like various engineerings (Since for all purposes, Galilean Relativity is really accurate for cases when we are not moving at velocities close to c).

By the way, the relations we see above allows us to switch between various referentials, in a certain way transforming the frames (or better saying, transforming the coordinates), which is why we usually refer to them as Galilean transformations. If we were considering 3-dimensional space, but with v still only in the x direction, we would have something like this:

⎧⎩⎨⎪⎪⎪⎪x′=x+vty′=yz′=zt′=t⇔⎧⎩⎨⎪⎪⎪⎪x=x′−vty=y′z=z′t=t′

And if you already know about matrices, since we are talking about vectors after all, there is a neat way to write this in a general way, like:

⎛⎝⎜⎜⎜x′y′z′t′⎞⎠⎟⎟⎟=⎡⎣⎢⎢⎢⎢100001000010vxvyvz1⎤⎦⎥⎥⎥⎥∗⎛⎝⎜⎜⎜xyzt⎞⎠⎟⎟⎟

This is general because we are considering that velocity is not only in one direction, but if you set v_y and v_z to zero you get the expression above.

And if you don't know about matrices don't worry about it, but I recommend you watch a series of videos in the youtube channel 3blue1brown, he has a playlist where he gives a good intuition on linear algebra and you can learn more about matrices and vectors and how you can use them for things like describing transformations between frames  (or vector spaces in linear algebra, which is a bit more general). After you have a better intuition of vectors, you will see why in physics, sometimes it is better to write things like this, instead of using the usual way. And it will specially useful in Special Relativity.

Some references:

Physics for Scientists and Engineers, by Serway & Jewett, 9th Edition , Part I: Mechanics, Chapters 3 (if you want to know a bit about vectors in physics) and 4 (if you want to know about motion in 2D, and they also tackle this part about relative motion). They have some exercises you can solve as well to give you a better intuition of Galilean Relativity. I also advise you to look for exercises online, both on vectors and in relative motion, and I will put some links below for you to check out.

(this is not an usual way to quote scientifically, but for a layperson, this is easier to understand, so I will quote it this way)


In this site, you find exercises with solutions to various physical problems, and I recommend you check this sections in specific if you need to look at these sections:
- https://physics.info/frames/
- https://physics.info/vector-addition/
- https://physics.info/vector-components/
- https://physics.info/vector-multiplication/
- https://physics.info/vector-multiplication/
- http://www.problemsphysics.com/vectors.html
- http://www.physicstutorials.org/home/exams/vectors-exams-and-solutions/150-vectors-exam1-and-solutions

Now I believe this will be enough for you to practice a bit. One thing I've learned from all the teachers and professors I've had, is that the best way for you to get a concept, is for you to try it yourself, so I will try to motivate you to constantly do exercises if possible.

{x′2=x′1−vty′2=y

{