Fundamentals of Physics II

PHYS 201 - Lecture 1 - Electrostatics

Chapter 1: Review of Forces and Introduction to Electrostatic Force [00:00:00]

Professor Ramamurti Shankar: So, I’ve got to start by telling you the syllabus for this term — not the detailed one, just the big game plan. The game plan is: we will do electromagnetic theory. Electromagnetism is a new force that I will introduce to you and go through all the details. And I will do optics, and optics is part of electromagnetism. And then near the end we will do quantum mechanics. Now, quantum mechanics is not like a new force. It’s a whole different ball game. It’s not about what forces are acting on this or that object that make it move, or change its path. The question there is: should we be even thinking about trajectories? Should we be even thinking about particles going on any trajectory? Forget about what the right trajectory is. And you will find out that most of the cherished ideas get destroyed. But the good news is that you need quantum mechanics only to study very tiny things like atoms or molecules. Of course the big question is, you know, where do you draw the line? How small is small? Some people even ask me, “Do you need quantum mechanics to describe the human brain?” And the answer is, “Yes, if it is small enough.” So, I’ve gone to parties where after a few minutes of talking to a person I’m thinking, “Okay, this person’s brain needs a fully quantum mechanical treatment.” But most of the time everything macroscopic you can describe the way you do with Newtonian mechanics, electrodynamics. You don’t need quantum theory.

All right, so now we’ll start with the brand new force of electromagnetism. But before doing the force, I’ve got to remind you people of certain things I expect you all to understand about the dynamics between force, and mass, and acceleration that you must have learned last term. I don’t want to take any chances. I’m going to start by reminding you how we use this famous equation of Newton. So you’ve seen this equation, probably, in high school, but it’s a lot more subtle than you think, certainly a lot more subtle than I thought when I first learned it. So I will tell you what I figured out over these years on different ways to look at F = ma. In other words, if you have the equation what’s it good for? The only thing anybody knows right away is a stands for acceleration, and we all know how to measure it. By the way, anytime I write any symbol on the board you should be able to tell me how you’d measure it, otherwise you don’t know what you’re talking about as a physicist.

Acceleration, I think I won’t spend too much time on how you measure it. You should know what instruments you will need. So I will remind you that if you have a meter stick, or many meter sticks and clocks you can follow the body as it moves. You can find its position now, its position later, take the difference, divide by the time, you get velocity. Then find the velocity now, find the velocity later, take the difference, divide by time, you’ve got acceleration. So acceleration really requires three measurements, two for each velocity, but we talk of acceleration right now because you can make those three measurements arbitrarily near each other, and in the limit in which the time difference between them goes to zero you can talk about the velocity right now and acceleration right now. But in your car, the needle points at 60 that’s your velocity right now. It’s an instantaneous quantity. And if you step on the gas you feel this push. That’s your acceleration right now. That’s a property of that instant.

So we know acceleration, but the question is can I use the equation to find the mass of anything. Now, very often when I pose the question the answer given is, you know, go to a scale, a weighing machine, and find the mass. And as you know, that’s not the correct answer because the weight of an object is related to being near the earth due to gravity, but the mass of an object is defined anywhere. So here’s one way you can do it. Now you might say, “Well, take a known force and find the acceleration it produces,” but we haven’t talked about how to measure the force either. All you have is this equation. The correct thing to do is to buy yourself a spring and go to the Bureau of Standards and tell them to loan you a block of some material, I forgot what it is. That’s called a kilogram. That is a kilogram by definition. There is no God-given way to define mass. You pick a random entity and say that’s a kilogram. So that’s not right and that’s not wrong. That’s what a kilogram is.

So you bring that kilogram, you hook it up on the spring, and you pull it by some amount, maybe to that position, and you release it. You notice the acceleration of the 1 kilogram, and the mass of the thing is just one. Then you detach that mass. Then you ask — Then the person says, “What’s the mass of something else?” I don’t know what the something else is. Let’s say a potato. And you take the potato or anything, elephant. Here’s a potato. You pull that guy by the same distance, and you release that, and you find its acceleration. Since you pulled it by the same amount, the force is the same, whatever it is. We don’t know what it is, but it’s the same. Therefore we know the acceleration of 1 kilogram times 1 kilogram is equal to the unknown mass times the acceleration of the unknown mass. That’s how by measuring this you can find what the mass is. In principle you can find the mass of everything.

So imagine masses of all objects have been determined by this process. Then you can also use F = ma to find out what forces are acting on bodies in different situations, because if you don’t know what force is acting on a body you cannot predict anything. So you can go back to the spring and say, “I want to know what force the spring exerts when it’s pulled by various amounts. Well, you pull it by some amount x. You attach it to a non-mass and you find the acceleration, and that’s the force. And if you plot it, you’ll find F as a function of x will be roughly a straight line and it will take the form F = -kx, and that k is called a force constant. So this is an example of your finding out the left hand side of Newton’s law. You’ve got to understand the distinction between F = -kx and F = ma. What’s the difference? This says if you know the force I can tell you the acceleration, but it’s your job to go find out every time what forces might be acting on a body. If it’s connected to a spring, and you pull the spring and it exerts a force, someone’s got to make this measurement to find out what the force will be.

All right, so that’s one kind of force. Another force that you can find is if you’re near the surface of the earth, if you drop something, it seems to accelerate towards the ground, and everything accelerates by the same amount g. Well, according to Newton’s laws if anything is going to accelerate, it’s because there’s a force on it. The force on any mass m must be mg, because if I divide by m I’ve got to get g. So the force on masses near the earth is mg. That’s another force. Something interesting about that force is that unlike the spring force where the spring is touching the mass, you can see it’s pulling it, or when I push this chair you can see I’m doing it, the pull of gravity is a bit strange, because there is no real contact between the earth and the object that’s falling. It was a great abstraction to believe that things can reach out and pull things which are not touching them, and gravity was the first formally described force where that was true. And another excursion in the same theme is if this object gets very far, say like the moon over there, then the force is not given by mg, but the force is given by this law of gravitation. For every r near the surface of the earth, if you put r equal to the surface of the earth you will get a constant force that is just mg, but if you move far from the center of the earth you’ve got to take that into account, and that’s what Newton did and realized the force goes like 1 over r².

So every time things accelerate you’ve got to find the reason, and that reason is the force. Many times many forces can be acting on a body, and if you put all the forces that are acting on a body and that explains the acceleration, you’re done, but sometimes it won’t. That’s when you have a new force. And the final application of F = ma is this one. If you knew the force, for example, on a planet, and here’s a planet going around the sun and it is here. This is the sun, and you know the force acting on it given by Newton’s Law of Gravity you can find the acceleration that will help you find out where it will be one second later, and you repeat the calculation, you will get the trajectory. So F = ma is good for three things, that’s what I want you to understand: to define mass, to calculate forces acting on bodies by seeing how they accelerate, and finally to find the acceleration of bodies given the forces. This is the cycle of Newtonian dynamics.

And what I’m going to do now is to add one more new force, because I’m going to find out that there is another force not listed here. I’m going to demonstrate to you that new force, okay? Here’s my demonstration. The only demonstration you will see in my class, because everything else I’ve tried generally failed, but this one always works. So, I have here a piece of paper, okay? Then I take this trusty comb and I comb the part of my head that’s suited for this experiment, then I bring it next to this, and you see I’m able to lift that. Now, that’s not the force of gravity because gravity doesn’t care if you comb your hair or not, okay? And also when I shake it, it falls down.

So you’re thinking, “Okay, maybe there is a new force but it doesn’t look awfully strong because it’s not able to even overcome gravity, because it eventually yielded to gravity and fell down,” but it’s actually a mistake to think so. In fact this new force that I’m talking about is 10 to the power of 40 stronger than gravitational force. I will tell you by what metric I came up with that number, but it’s an enormously strong force. You’ve got to understand why I say it is such a strong force when, when I shook it the thing fell down. So the reason is that if you look at this experiment, here’s the comb and here’s the paper, the comb is trying to pull the paper, but what is trying to pull it down? What is trying to pull it down? So here is me, here is that comb, here’s the paper. The entire planet is pulling it down: Himalayas pulling it down, Pacific Ocean, pulling it down, Bin Laden sitting in his cave pulling it down. Everything is pulling it down, okay? I am one of these people generally convinced the world is acting against me, but this time I’m right. Everything is acting against me, and I’m able to triumph against all of that with this tiny comb. And that is how you compare the electric force with the gravitational force. It takes the entire planet to compensate whatever tiny force I create between the comb and the piece of paper. To really get a number out of this I’ll have to do a little more, but I just want to point out to you this is a new force much stronger than gravitation.

So I want to tell you a few other experiments people did without going into what the explanation is right now, but let me just tell you if you go through history what all did people do. So one experiment you can do: You take a piece of glass and you rub it on some animal that’s passing by, water buffalo. That’s why I cannot do all the experiments in class. You rub it on that guy, then you do it to a second piece of glass, and you find out that they repel each other, meaning if you put them next to each other they tend to fly apart. Then you take a piece of hard rubber and you rub that on something else. I forgot what, silk, Yeti, some other thing. Then you put that here. So I’ll give a different shape to that thing. That’s the rubber stick. And you find when you do that to this, these two attract each other. Sometimes they repel, sometimes they attract. Here’s another thing you can do: Buy some nylon thread. You hang a small metallic sphere, and you bring one of these rods next to it. It doesn’t matter which one. Initially they’re attracted and suddenly when you touch it and you remove it, they start repelling each other. What’s going on? That’s another thing you could do. Last thing I want to mention is if you took two of these things which are repelling each other, let’s say. Let’s say they’re attracting each other like this. Then you connect them with a piece of nylon and you take it away, nothing happens. If you connect them with a piece of wire and take away the wire, they no longer attract each other.

Chapter 2: Coulomb’s Law [00:15:21]

So these are examples of different things. I’m just going to say, you do this, you do this, you do that, then finally you need a theory that explains everything. So that’s the theory that I’m going to give you now. That’s the theory of electrostatics. And I don’t have time to go into the entire history of how people arrived at this final formula, so I’m just going to tell you one formula that really will explain everything that I’ve described so far, and that formula is called Coulomb’s Law. Even though Mr. Coulomb’s name is on it, he was not the first one to formulate parts of the law, but he gave the final and direct verification of Coulomb’s Law that other people who had contributed. So Coulomb’s Law says that certain entities have a property called charge. You have charge or you don’t have charge, but if you have charge the charge that you have, you meaning any of these objects, is measured in coulombs. Remember, that was not Coulomb’s idea to call it coulomb. Whenever you make a discovery, you’re breathlessly waiting that somebody will name it after you, but it’s not in good taste to name to after yourself, but it carries Coulomb’s name.

So he didn’t say call it coulomb, okay, but he certainly wrote down this law. The law says that if you’ve got one entity which has some amount of charge called q₁, and there’s another entity that has some amount of charge q₂ they will exert a force on each other which is given by q₁q₂ times this constant which is somehow written as 1 over 4Πε₀. That’s 1 over r². But r is the distance between them, and you can ask in this picture, what do you mean by distance? I mean, is it from here to there, or is it from center to center? We’re assuming here that the distance between them is much bigger than the individual sizes. For example, you say, how far am I from Los Angeles, well, 3,225 miles, but you can say are you taking about your right hand or your left hand? Well, I’m a point particle for this purpose so it doesn’t matter. So here we’re assuming that either they’re mathematically point charges or they’re real charges with a finite size but separated by a distance much bigger than the size, so r could stand, if you like, for center to center. It doesn’t matter too much. So this is what Coulomb said.

Now, if you look at this number here, 1 over 4Πε₀, its value is 9 times 10 to the 9^th. What that means is the following: If you take one body with 1 coulomb of charge, another body with 1 coulomb of charge and they’re separated by 1 meter, then the force between them will be this number, because everything else is a 1. It’ll be 9 times 10 to the 9 newtons. That’s an enormous force, and normally you don’t run into 1 coulomb of charge, but the reason why a coulomb was picked is sort of historical and it has to do with currents and so on. But anyway, this is the definition. But if you want to be more precise, I should write a formula more carefully because force is a vector. Also I should say force on whom and due to what. So let’s say there are two charges, and say q₁ is sitting at the origin and q₂ is sitting at a point whose position is the vector r. Then the force on 2 due to 1 is given by q₂q₁ over 4Πε₀ times 1 over r². That’s the magnitude of the force, but I want to suggest that the force is such that q₁ pushes q₂ away.

So I want to make this into a vector, but I’ve got the magnitude of the vector. As you know, to make a real vector you take its magnitude and multiply it by a vector of unit length in that same direction. The unit vector we can write in many ways. One is just to say e_r, e_ris a standard name for a vector of length 1 in the direction of r. But I’ll give you another choice. You can also write it as r divided by the length of r. That also would be a vector of unit length parallel to r. So there are many ways to write the thing that makes it a vector. And F₂₁ is minus of F₁₂. Now, how do we get attraction and how do we get repulsion? We get it because q₁ and q₂, if they’re both positive and you if you use the formula, you’ll find they repel each other, but if they’re of opposite signs, you’ll do the same calculation, but you’ll put a minus sign in front of the whole thing. That’ll turn repulsion into an attraction. So you must allow for the possibility that q can be of either sign; q can also be 0. There are certain entities which don’t have any electric charge, so if you put them next to a million coulombs nothing happens. So some things have plus charge. Some things have minus charge. Some things have no charge, but they’re all contained in this Coulomb’s Law.

Chapter 3: Conservation and Quantization of Charge [00:21:11]

Now, again, skipping all the intermediate discoveries, I want to tell you a couple of things we know about charge. First thing is: q is conserved. Conserved is a physics terms for saying — does not change with time. For example, when you say energy is conserved, it means particles can come and collide and do all kinds of things, but if you add that energy before, you’ll get the same answer afterwards, and whenever that happens, the quantity is conserved. The claim is electrical charge is conserved. So electrical charge may migrate from A to B or B to A, but if you add up the total charge, say the chemical reaction of any process, including in big particle accelerators where things collide and all kinds of stuff comes flying out, the charge of the final products always equal to the charge of the incoming products.

But charge conservation needs to be amended with one extra term, extra qualification. It’s called local. Suppose I say the number of students in the class is conserved? That means you count them any time, you’ve got to get the same number. Well, here’s one possibility. Suddenly one of you guys disappears and appears here at the same instant. That’s also consistent with conservation of student number because the number didn’t change. What disappeared there, appeared here. But that is not a local conservation of charge because it disappears in one part of the world and appears in another one. And it’s not even a meaningful law to have in the presence of relativity. Can any of you guys think of why that might be true, why a charge disappearing somewhere and appearing somewhere else cannot be a very profound principle? Yes?

Student: [inaudible]

Professor Ramamurti Shankar: Yep?

Student: Well, if it’s in the same instant disappearing from one place and appearing another place, it’s traveling faster than light?

Professor Ramamurti Shankar: Well, we don’t know that it was the same thing that even traveled. It may not have traveled. It may even be — Here’s another thing. Suppose an electron, suppose a proton disappears there and a positron appears here. That still conserves charge, but we don’t think that the proton traveled and became the positron, right? So it is not that it has traveled. You are right. I hadn’t thought about that. It’s a good point that it implies it traveled infinitely fast, but that’s not the reason you object to it. Yep?

Student: It’s not necessarily simultaneous.

Professor Ramamurti Shankar: That is the correct answer. The answer is it is not simultaneous in every frame of reference. You must know from the special theory that if two events are simultaneous in one frame of reference, if you see those same two events in a moving train, or plane, or anything they will not be simultaneous. Therefore, in any other frame of reference, either the charge would have been created first and then after a period of time reappeared somewhere, I mean, destroyed somewhere and appeared after a delay, or the appearance could take place before the destruction, so suddenly you’ve got two charges. So conservation of charge, which is conserved non-locally, cannot have a significance except in one frame of reference, but if you believe that all observers are equivalent and you want to write down laws that make sense for everybody it can only be local. So electrical charge is conserved and it is local, locally conserved. In other words, stuff doesn’t just disappear. Stuff just moves around. You can keep track of it, and if you add it up you get the same number.

The second part of q, which is not necessary for any of these older phenomena, is that q is quantized. That means the electrical charge that we run into does not take a continuum of possible values. For example, the length of any object, you might think at least in classical mechanics, is any number you like. It’s a continuous variable, but electric charge is not continuous. As far as we can tell, all the charges we have ever seen are all multiples of a certain basic unit of charge, which turns out to be 1.6 times 10 to the -19 coulombs. Every charge is either that or some multiple of it. Multiple could be plus or minus multiple. So charge is granular, not continuous.

Chapter 4: Microscopic Understanding of Electrostatics [00:26:16]

Okay, so I’m going to give you a little more knowledge we have had since the time of Coulomb that sort or explains these things. I mean, what’s really going on microscopically? We don’t have to pretend we don’t know. We do, so we might as well use that information from now on. What we do know is that everything is made up of atoms, and that if you look into the atom it’s got a nucleus, a lot of guys sitting here. Some are called protons and some are called neutrons, and then there are some guys running around called electrons. Of course we will see at the end of the semester that this picture is wrong, but it is good enough for this purpose. It’s certainly true that there are charges in an atom which are near the center and other light charges which are near the periphery, are outside. All things carrying electric charge in our world in daily life are either protons or electrons. You can produce strange particles in an accelerator. They would also carry some charge which would in fact be a multiple of this charge, but they don’t live very long. So the stable things that you and I are made of and just about everything in this room is made of, is made up of protons, neutrons and electrons. The charge of the neutron, as you can guess, is 0. The charge of the electron, by some strange convention, was given this minus sign by Franklin. And the charge of the proton is plus 1.6 times into -19 coulombs.

There are a lot of amazing things I find here. I don’t know if you’ve thought about it. The first interesting thing is that every electron anywhere in the universe has exactly the same charge. It also has exactly the same mass. Now, you might say, “Look, that’s a tautology,” because if it wasn’t the same charge and if it wasn’t the same mass you would call it something else. But what makes it a non-empty statement is that there are many, many, many, many electrons which are absolutely identical. Look, you try to manufacture two cars. The chance that they’re identical is 0, right? I got one of those cars so I know that. It doesn’t work. It’s supposed to. So despite all the best efforts people make, things are not identical.

But at the microscopic level of electrons and protons, every proton anywhere in the universe is identical. And they can be manufactured in a collision in another part of the universe. This can be manufactured in a collision in Geneva, the stuff that comes out identical. That is a mystery, at least in classical mechanics it’s a mystery. Quantum Field Theory gives you an answer to at least why all electrons are identical, and why all protons are identical. The fact that they’re absolutely identical particles is very, very important. It also makes your life easy, because if every particle was different from every other particle, you cannot make any predictions. We know that the hydrogen atom on a receding galaxy is identical to the hydrogen atom on the Earth. That’s why when the radiation coming from the atom has a shifted wavelength of frequency, we attributed to the motion of the galaxy. From the Doppler Shift we find out its speed. But another explanation could be, well, that’s a different hydrogen atom. Maybe that’s why the answer’s different. But we all believe it’s the same hydrogen atom, but it’s moving away from us.

Therefore, one of the remarkable things is that all electrons and all protons are equal, but a really big mystery is why is the charge of the electron exactly equal and opposite the charge of the proton. They are not the same particle. Their masses are different. Their other interactions are different. But in terms of electrical charge these two numbers are absolutely equal as far as anybody knows. That’s another mystery. Two different particles, not related by any manifest family relationship, have the same charge, except in sign. And there are theories called Grand Unified Theories which try to explain this, but certainly not part of any standard established theory, but it’s key to everything we see in daily life because that’s what makes the atom electrically neutral.

Okay, now we can understand the quantization of charge, because charge is carried by these guys and these guys are either there or not there, so you can only have so many electrons. We cannot have a part of an electron, or part of a proton. Now, let’s try to understand all these experiments in terms of what we know. First of all, when you take this piece of glass, and you rub it, the atoms in glass are neutral. They’ve got equal number of protons and electrons, but when you rub it, the glass atom loses some electrons to whatever you rubbed it on. Therefore, it becomes positively charged, because some negative has been taken out. In the case of the rubber stick, it gains the electrons and whatever animal you rubbed it on, it loses the electrons. So actually real charge transfer takes place only through electrons. Protons carry charge, but you are never going to rip a proton out unless you use an accelerator. It’s really deeply bound to the nucleus. Electrons are the ones who do all the business of electricity in daily life. The current flowing in the wire, in the circuit, it’s all the motion of electrons. So from this and Coulomb’s Law, can you understand the attraction between these two? How many people think you can, from Coulomb’s Law, understand the attraction between these two rods? Nobody thinks you can? Well, why do you think you cannot? You know why?

Student: Because they’re not point charges?

Professor Ramamurti Shankar: Okay, any other reason why Coulomb’s Law is not enough? Well, how will we apply Coulomb’s Law to understand the attraction between these two rods? What will you have to do?

Student: You’d have to apply it to F = ma.

Professor Ramamurti Shankar: No. Once you got the F, the a will follow, but can you compute the force between two rods? One of them has got a lot of positive charge. One of them has a lot of negative charge given Coulomb’s Law. Yes?

Student: You don’t know the exact quantities of the charges.

Professor Ramamurti Shankar: Pardon me?

Student: You don’t know the exact quantities of the charges.

Professor Ramamurti Shankar: Suppose I tell you. I tell you how many charges there are. Yes?

Student: You don’t which direction the attraction is.

Professor Ramamurti Shankar: No, we do know, because the plus and minus will be drawn towards each other. Okay, I’ll tell you what it is. It’s an assumption we all make, but you’re not really supposed to make it. It’s not a consequence of any logic. Coulomb’s Law talks about two charges, two point charges. What if there are three charges in the universe? What is the force this one will experience due to these two? This is q₁. This is q₂. This is q₃. Coulomb’s Law doesn’t tell you that. It tells you only two at a time, but we make an extra assumption called superposition which says that if you want the force on 3 (should read 1), when there is q₁ and q₂, you find the force due to q₂ and you find the force due to q₃ and you add them up. The fact that you can add these two vectors is not a logical requirement. In fact, it’s not even true at an extremely accurate level that the force between two charges is not affected by the presence of a third one. But it’s an excellent approximation, but you must realize it is something you’ve got to find to be true experimentally. It’s not something you can say is logical consequence. Logically there is no reason why the interaction between two entities should not be affected by the presence of a third one. But it seems to be a very good approximation for what we do, and that’s the reason why eventually we can find the force between an extended object, another extended object by looking at the force on everyone of these due to everyone of those and adding all the vectors.

Okay, so superposition plus Coulomb’s Law is what you need. Then you can certainly understand the attraction. How about the comb and the piece of paper? That’s a very interesting example and it’s connected to this one. See, the piece of paper is electrically neutral. So let me do paper and comb instead of this one. It’s got the same model. Here’s the piece of paper. Here’s the comb. The comb is positively charged. The paper is neutral. So anyway, there’s nothing here to be attracted to this one, but if you bring it close enough, there are equal amount of positive and negative charges, but what will happen is the negative charges will migrate near these positive charges from the other end, leaving positive charges in the back, so that the system will separate into a little bit of negative closer to the positive, and the leftover positive will be further away. Therefore, even though it’s neutral the attraction of plus for this minus is stronger than the repulsion of this plus with this plus. That’s called polarization.

So polarization is when charge separates. Some materials cannot be polarized, in which case no matter how much you do this with a comb it won’t work. Some materials can be polarized. The piece of paper is an example of what can be polarized. We can understand that too. And in this example, if you bring a lot of plus charges here, and you look at what’s going on here, the minus guys here will sit here and the plus will be left over in the back, and then this attraction between plus and minus is bigger than this repulsion, so it will be attracted to it. But once it touches it, this rod touches that, then what you have is a lot of plus charges here. They repel each other. They want to get out. Previously they couldn’t get out. They were stuck on the rod, but now that you’ve made contact, some of them will jump to that one. Then when you separate them, you will have a ball with some plus charges, and you will have a rod with more plus charges, and they will repel each other. And finally I said if you take two of these spheres, suppose one was positively charged, one was negatively charged, they’re attracting each other. If you connect them with a nylon wire or a wooden stick nothing happens, but if you connect them with an electrical wire, what happens is that the extra negative charges here will go to that side, and then when you are done they will both become electrically neutral.

Okay, so that’s why. So the point of this one is: electric charges can flow through some materials, but not other materials. If it can flow through some materials, it’s called a conductor. If it cannot flow through them, it’s called an insulator. So real life you’ve got both. So when you’re changing the light bulb, if you don’t want to get an electric shock you’re supposed to stand on a piece of wood before you stick your finger in, unless you’ve got other intentions. Then, you will find that you don’t get the shock because the wood doesn’t conduct electricity. But if you stand on a metallic stool, on a metallic floor and put your hand in the socket, you’ll be part of an electrical circuit. The human body is a good conductor of electricity, but what saves you is that it cannot go from your feet to the floor. Now, there are also semiconductors, which are somewhere in between, but in our course either we’ll talk about insulators, which don’t conduct electricity, and perfect conductors, which conduct electricity.

Okay, so a summary of what I’ve said so far is that there’s a new force in nature. To be part of that game you have to have charge. If you have no charge, you cannot play that game. Like neutrons cannot play this game. Nothing’s attracted or repelled by neutrons and neutrons cannot attract or repel anything. So you’ve got to have electric charge. It happens to be measured in coulombs. So let me ask you another question. Suppose I tell you, here is Coulombs Law. Let me just write the number 1 over 4Πε₀. How are we going to test that this law is correct? Okay, I’m giving you a bonus. You don’t have to discover the law. I’m giving you the law. All you have to do is to verify it, and don’t use any other definitions other than this law itself. How will you know it depends on q₁ and q₂ in this fashion? How will you know it depends on r in that fashion? That’s what I’m asking you. Can anybody think of some setup, some experiment you will do? Let me ask an easier question. How will you know it goes like 1 over r²? Yep?

Student: Vary the distance between them, and show that the force falls off.

Professor Ramamurti Shankar: Well, you’re right that if you vary the distance between them and show the force falls like that, but how do you know what the force is? Yes?

Student: Could you use a spring here?

Professor Ramamurti Shankar: What was your plan?

Student: Observe acceleration.

Professor Ramamurti Shankar: You are right. Both of you are right. You can maybe hold this guy fixed, and let this go, and see how it accelerates. And if you knew the mass of this guy then you know the force. Then you can vary the distance to another distance, maybe half the distance. At half the distance if you get four times the force you verified 1 over r² law. The other one is with the spring. You can take a spring. Say maybe there are two metals, uncharged objects, then you dump some charge on this and some charge on that, and then the spring will expand, and you can see what force the spring expands, exerts, and see if it is proportional to 1 over r². That’s how Newton deduced the 1 over r² force law. He found the acceleration of the apple is 3,600 times the acceleration of the moon towards the earth, and the moon was 60 times further than the apple, and 60 squared is 3,600. That’s how he found 1 over r². Now, he was very lucky. It could have been 1 over r to the 2.110 or 1.96, but it happens to be exactly 1 over r². Anyway, that’s how we can find even if it’s not 1 over r². If it’s 1 over r³, or 1 over r⁴, whatever it is you can find by taking two charges.

See, we don’t have to know what q₁ and q₂ are. That’s what I’m trying to emphasize here. If all you’re trying to see is does it vary like 1 over r², keep everything the same except r. Double the r and see what happens. And best way is what you said. Watch the acceleration, and if it falls to one fourth of the value for doubling the distance, it is 1 over r². All right, suppose I got 1 over r². I want to know it depends on the charges as the first power of q₁ and the first power of q₂. So how should we do that? And don’t say put 10 electrons once and then 20 electrons because you cannot see electrons that well. In the old days people did not even know about electrons, and yet they managed to test this. So how will you vary the charge in a known way? Yep?

Student: You could have many identical spheres, and maybe keep touching them to each other.

Professor Ramamurti Shankar: Ah! Okay, many identical spheres.

Student: And then put charge on one and then touch it to the second one and you’ll get half as much.

Professor Ramamurti Shankar: Very good. Let me repeat what she said. First you take many identical spheres. Well, I not going to even try to draw identical spheres because I haven’t learned how to draw spheres, but let’s imagine you’ve got a whole bunch of these guys. You put some charge on this. You don’t know what it is, okay? We don’t know what q is. We’re trying to find out. You don’t have to know what q is. So let this be one of the objects. That’s my q. For the other object, keep a fixed-object containing some other q. This has got charge q. Don’t vary the r. Question is, can you change q to q/2, and her answer was: if it’s got some charge, maybe a plus, bring it in contact with the second identical sphere. If it really is identical, you have to agree that when you separate them they must exactly have half each. That’s a symmetry argument. Because for any reason you give me for why one of them should have more, I will tell you why the other one should have more. You cannot, so they will split it evenly and therefore charge will split evenly to q/2 here and q/2 here. Then you can take this and put it there — you’ve got q/2. Then you can do other combinations. For example, you can take this q/2 and connect it to the ground so it becomes neutral. So this has got 0 again. You can touch that with the q/2 and separate them. Then each will have q/4. So in this way you can vary the charge in a known way, maybe half of it, double it. I give you some homework problem where you want to get 5/16 of a coulomb. By enough spheres you can do that.

Again, what I want you to notice is that you did not know what q was, but all you knew is that q went to q/2 when you brought two identical spheres and separated them. That’s how we can find that it depends linearly on q₁. Of course, it also depends linearly on q₂ because it’s up to you to decide who you want to call q₁, and who you want to call q₂. Okay, so I want you people to understand all the time that you should be able to tell me how you measure anything, okay? That’s very, very important. That’s why you should think about it. If you think in those terms you’ll also find you’re doing all the problems very well. If you’re thinking of pushing symbols and canceling factors of Π you won’t get the feeling for what’s happening. So everything you write down you should be able to measure. If you say, “Oh, I want to measure the force,” you’ve got to be sure how you’ll measure it, and one way is like you said, find m times a. If you knew the m you can measure the force. For everything make sure you can measure it. If I give you a sphere charged with something, then of course we’ve got to decide. Suppose I give you a sphere. It’s got some charge, and I want you to find how much charge is on that sphere. This time I want you to tell me how many coulombs there are. What will you do? What process will you use? Well, then you have a problem because you are not able to figure out, but if I tell you here’s an object, it is 3 meters long, you can test it because you’ll go and bring the meter stick from the Bureau of Standards and measure it three times. I’m asking you, if I give you a certain charge and say how much charge is there, by what process can we calibrate the charges? Yep?

Student: Put it in the vicinity of a reference charge and then measure the acceleration.

Professor Ramamurti Shankar: That’s correct. If you knew one standard charge, somehow or other we knew its value, then bring the unknown one next to it, put it at a known distance, right? You know the r. You know the 4Π. You know the ε₀. You find the force, you can find this charge. So all we need to know is how to get a reference charge, right? So how do I know something has a coulomb? How do I get 1 coulomb of charge just to be sure? You know what you could do, because you haven’t defined yet the reference, so you should think about how will I get a coulomb charge, or any other charge? So I could take these two spheres that she talked about, each with the same charge q. We don’t know what it is. I put them at 1 meter distance and I measure the force, namely how hard should I hold one from running away to the other one. Once I got the force, the only thing unknown in the equation is q times q. I know r. I know 1 over 4Πε₀. I can get q. So every time you write something think about how you’ll measure it, because in that process you’re learning how the physics is done. If you try to avoid that you’ll be just juggling equations, and that doesn’t work for you and that doesn’t work for me. Anybody who wants to do good physics should be constantly paying attention to physical phenomena, and not to the symbols that stand for physical objects.

All right, so the final thing I want to do in this connection is to give this number I mentioned, F_gravity over F_electric. I said gravity is 10 to the -40 times weaker. Well, you have to precise on how you got the number. See, it’s not like selling toothpaste where you can say it is 7.2 times whiter. I don’t know how those guys measure whiteness in a unit with two decimal places, but that’s a different game. It’s not subject to any rules, but here you have to say how you got the number. In what context did you make the comparison? It turns out the answer does depend on what you choose. There’ll be some variations, but those tiny variations are swamped by this enormous ratio I would get. So what you could do is take any two bodies, and find the ratio of gravity to electric force. One option is to take two elementary particles, whichever two you like. So I will take an electron and a proton, but you can take an electron and a positron, or a proton and a proton. It doesn’t matter.

These two guys attract each other gravitationally and electrically. So I will write the force of gravitation, which is G, mass of the proton, mass of the electron, over r² divided by q_electron, q_proton over 4Πε₀ times 1 over r². Notice in this experiment, in this calculation, r² does not matter, so you don’t have to decide how far you want to keep them, because they both go like 1 over r² ,so you can pick any r. So whatever you pick is going to cancel and you will be left with this number. A q₁, q₂ and the 1 over 4Πε₀ is 9 times 10 to the 9^th. So now we put in some numbers. So G is 10 to the -11 with some pre-factors, maybe 6 in this case. I’m not going to worry about pre-factors. But the mass of the proton is 10 to the -27 kilograms, the mass of the electron 10 to -30 kilograms. So don’t say how come they all have these nice round numbers. They are not. There are factors like 1 and 2. I’m not putting them because I’m just counting powers of 10. q₁ is 1.6 times 10 to the -19, so two of those q’s is 10 to the -38. Then 9 times 10 to the 9^th is roughly 10 to the 10^th. If you do all of that you will find this is 10 to the -40, if it is some typical situation that you took, and you found this ratio of forces. If there are two elementary particles, which are like the building blocks of matter, and you brought them to any distance you like you compare the electric attraction to the gravitational attraction.

So one question is: if gravity is so weak, how did anyone discover the force of gravity? If all you had was electrons and protons, you’d have to measure the force between them. Suppose you knew only about electricity, didn’t know about gravitation. One way to find there is an extra force is to measure the force to an accuracy good to 40 decimal places, and in the 40th decimal place you find something is wrong. You fiddle around and figure out the correction comes from m₁m₂ over r², but that’s not how it was done, right? You guys know that. So how did anyone discover the force of gravity when it’s overwhelmed? Yes?

Student: Most things are neutral?

Professor Ramamurti Shankar: Yes. Most things are electrically neutral. In other words, electric force, even though it’s very strong, comes with opposite charges. It can occur with a plus sign or with a minus sign. Therefore, if you take the planet Earth, it’s got lots and lots of charges in every atom, but every atom is neutral. You’ve got the moon, ditto, lots and lots of atoms, but they’re all neutral. But the mass of the electron does not cancel the mass of the proton. So mass can never be hidden, whereas charge can be hidden. Mass never cancels. That’s the reason why, in spite of the incredible amount of electrical forces they’re potentially capable of exerting, they present to each other neutral entities. Therefore, this remaining force which is not shielded is what you see, and has a dramatic role in the structure of the universe, force of gravity. But in most cosmological calculations you can forget mainly the electric force. It’s all gravitational force. That’s because electricity can be neutralized. So you cannot hide gravity. Everything has mass. Even photons which have no mass have energy. They’re also attracted by gravitation.

So gravity cannot be hidden, and that’s the origin of something called dark matter. So how many of you guys heard about dark matter? Okay? Anyone want to volunteer? Someone whose name begins with T, anybody’s name begins with T and also knows the answer to this? The trouble is, you people are plagued with one quality which is not good for being in physics, namely you’re modest. So you don’t want to tell me the answer. So I have to give an excuse for whoever gives the answer. If your seat has a number 142, anybody in seat 142? Maybe they’re not even numbered. Look, anybody with a red piece of clothing knows the answer to this — go ahead. Yes?

Student: [inaudible]

Professor Ramamurti Shankar: Pardon me?

Student: [inaudible]

Professor Ramamurti Shankar: Right. Basically there’s no way you can see it, and there’s dark matter right in this room, okay? And there’s dark matter everywhere, but the reason, the way people found out there is dark matter, do you know how that was determined? Yep?

Student: The rotation of galaxies didn’t line up with the matter that was visible, so…

Professor Ramamurti Shankar: So yes. Maybe one example I can talk is about our own galaxy. So here’s our visible galaxy, okay, the old spiral. Now, if something is orbiting this galaxy just by using Newtonian gravity, by knowing the velocity of the object as it goes around, you can calculate how much mass is enclosed by the orbit. That’s a property of gravitation — from the orbit, you can find out how much mass is enclosed. So what you will find is, if you found something orbiting the center of the galaxy at that radius, you’ll enclose some mass. If you take objects at bigger and bigger radius, you’ll enclose more and more mass, until you find orbits as big as the galaxy. Then the mass enclosed as a function of radius should come and stop, because after that the orbit’s getting bigger, but not enclosing any more mass. But what people found, that even after you cross the nominal size of the galaxy, you still keep picking up mass, and that is the dark matter halo of our galaxy. So it’s dark to everything, but you cannot escape gravity. That’s what I meant to say. You cannot avoid gravitational force.

So people are trying to find dark matter. People at Yale are trying to find dark matter. The thing is, you don’t know exactly what it is. It’s not any of the usual suspects, because then they would have interacted very strongly. So you’re trying to find something not knowing exactly what it is. And you’ve got to build detectors that will detect something. And you go through it everyday in your lab, and you’re hoping that one of these dark matter particles will collide with the stuff in your detector, and trigger a reaction. Of course there will be lots of reactions everyday, but most of them are due to other things. That’s called background. You’ve got to throw the background out, and whatever is left has got to be due to dark matter. And again, how do you know it’s dark matter? How do you know it’s not something else? Well you can see that if you’re drifting through dark matter in a moving Earth, you will be running into more of them in the direction of motion and less in the other direction, because you’re running into the wind. So by looking at the direction dependence, you can try to see if it’s dark matter. Anyway, dark matter was discovered by simple Newtonian gravitation. The particles that form dark matter are very interesting to particle physicists. There are many candidates in particle theory, but the origin of the discrepancy came from just doing Newtonian gravity.

All right, the final thing today before we break is that there’s one variation of Coulomb’s Law. By the way, I do not know your mathematical training and how much math you know, so you have to be on the lookout, say, if I write something that looks very alien to you, you’ve got to go take care of that, in particular, how to do integrals in maybe more than one dimension. Anyway, what I wanted to discuss today is the following: we know how to do Coulomb’s Law due to any number of point charges. So if you put another charge q here you want the force on this guy due to all these. You draw those lines, you take the 1 over r² due to that, 1 over r² due to that, add all the vectors. That’s very simple. But we will also take problems where the charges are continuous. So here’s an example. Here’s a ring of charge. The ring has some radius. You pick your radius r, and the charge on it is continuous. It’s not discrete, or it could be in real life everything is discrete, but to a coarse observer it will look like it’s continuous. So we can draw some pictures here, charges all over the ring, and λ is the number of coulombs per meter. Let me see, if you snipped one meter of the wire it’ll have λ coulombs in it. And you want to find the electric force on some other charge q due to this wire.

So you cannot do a sum. And you have to do an integral. That’s what I’m driving at, and I’m going to do one integral, then we’ll do more complicated ones later. So I want to find the force on a charge q here. So what I will do is, I will divide this into segments each of length, say dl. Then I will find the force of the charge here, dF. I will add the forces due to all the segments. The force of this segment will be the charge — this segment is so small, you can treat it as a point charge, and the amount of charge here is λ times dl. That’s the q₁. The q₂ is the q I put there. Then there’s the 4Πε₀, r², r² will be this distance z times this radius r will be — maybe I shouldn’t call it r. Let me call it capital R, and it’s R² plus z². That’s the distance. But now that force is a vector that’s pointing in that direction, but I know that the total force is going to point in this direction because for every guy I find in this side I can find one in the opposite direction pointing that way. So they will always cancel horizontally.

The only remaining force will be in the z direction. So I’m going to keep only the component of the force in the z direction. I denote it by dF in the z direction. For that, you have to take this force and multiply by cosine of that θ. I hope you know how to find the component of a force in a direction. It’s the cosine of the angle between them. That angle is equal to this angle, and cosine of this is z divided by R² plus z² on the root. That is the dF due to this segment, and the total force in the z direction is integral of this, and what that integrate. λ, q, all these are constant, R, z, everything is a constant. You have to add all the dl’s, if you add all the dl’s you will get the circumference. In other words, this is going to be λqz divided by 4Πε₀R² plus z² to the 3/2 integral of dl. Integral of dl is just 2ΠR. In other words, every one of them is making an equal contribution, so the integrand doesn’t depend on where you are in the circle, so you’re just measuring the length of the circle. That’s the answer. The force looks like λ times 2ΠR, what is that? λ is the charge per unit length. That, times the length of the loop, is the charge on the loop. It’s the charge you’re putting there divided by 4Πε₀ divided by R² plus z² to the 3/2. That’s an example of calculating the force which will be in this direction. Now, once you’ve done this calculation you may think maybe I missed a factor of Π or factor of e, something. Can you think of a way to test this? What test would you like to apply to this result? Yep?

Student: Put the z equal to 0 and have it in the middle. There should be no forces on it.

Professor Ramamurti Shankar: Very good. What he said is, if you pick z equal to 0 you’re sitting in the middle of the circle, and you’re getting pushed equally from all sides, and you better not have a force, and that’s certainly correct. This vanishes when z goes to 0. Anything else? Any other test? Yep?

Student: You could put it underneath by negative z. The force should be negative.

Professor Ramamurti Shankar: Yes, it will point down and be negative. That’s correct, but how about the magnitude of the force itself, rather than just the direction? Yep?

Student: If you go infinitely far away it should look like a point charge.

Professor Ramamurti Shankar: Yes. If you go very, very far, someone’s holding a loop, you cannot see that it’s even a loop. It’s some tiny spec, and it should produce the field. So what field should it produce? It should produce the coulomb force q₁q₂, or 4Πε₀ times distance squared. And when z is much, much, much bigger than R, this is one kilometer, this is two inches. You forget this. You get z² to the 3/2 is then z cubed. That means the whole thing here reduces to 1 over z² and it looks like the force between two point charges. So I would ask you whenever you do a calculation to test your result.

Okay, before going I’ve got to tell you something about those who come late. I realize that you guys come from near and far, so when you come late let me give you my preference for doors, okay? Door number one is that one. That’s the least problematic. Door number two is this one, because in the beginning of the lecture I’m usually on that side of the board, so you guys can come in. Door number three is that one where Jude is taking the picture, but do not stand in front of the camera and contemplate your future. If you do I will make sure you don’t have a future, okay? So don’t do that. If you come fashionably late, never come through that door, maybe this one. In fact if you come through that door because I have reached this side of the board, you are very, very late, so I think you should take the day off and start fresh next time, all right? Okay, thank you.

[end of transcript]

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