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# PHYS 201: Fundamentals of Physics II

## Lecture 1

## - Electrostatics

### Overview

The course begins with a discussion of electricity. The concept of charge is introduced, and the properties of electrical forces are compared with those of other familiar forces, such as gravitation. Coulomb’s Law, along with the principle of superposition, allows for the calculation of electrostatic forces from a given charge distribution.

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html## Fundamentals of Physics II## PHYS 201 - Lecture 1 - Electrostatics## Chapter 1: Review of Forces and Introduction to Electrostatic Force [00:00:00]
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 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 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 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 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 q they will exert a force on each other which is given by _{2}q times this constant which is somehow written as 1 over 4_{1}q_{2}Πε. That’s 1 over _{0}r. But ^{2}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 4 ^{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 q is sitting at the origin and _{1}q is sitting at a point whose position is the vector _{2}r. Then the force on 2 due to 1 is given by q over 4_{2}q_{1}Πεtimes 1 over _{0 }r. That’s the magnitude of the force, but I want to suggest that the force is such that ^{2}q pushes _{1}q away._{2}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 eis a standard name for a vector of length 1 in the direction of _{r}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 F is minus of _{21}F. Now, how do we get attraction and how do we get repulsion? We get it because _{12}q and _{1}q, 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 _{2}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: 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?
The second part of ## 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?
q. This is _{2}q. 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 _{3}q and _{1}q, you find the force due to _{2}q and you find the force due to _{2}q 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._{3}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 4 q and _{1}q in this fashion? How will you know it depends on _{2}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 r? Yep?^{2}
r. That’s how Newton deduced the 1 over ^{2}r 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 ^{2}r. Now, he was very lucky. It could have been 1 over ^{2}r to the 2.110 or 1.96, but it happens to be exactly 1 over r. Anyway, that’s how we can find even if it’s not 1 over ^{2}r. If it’s 1 over ^{2}r, or 1 over ^{3}r, whatever it is you can find by taking two charges.^{4}See, we don’t have to know what q are. That’s what I’m trying to emphasize here. If all you’re trying to see is does it vary like 1 over _{2}r, keep everything the same except ^{2}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 r. All right, suppose I got 1 over ^{2}r. I want to know it depends on the charges as the first power of ^{2}q and the first power of _{1}q. 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?_{2}
Again, what I want you to notice is that you did not know what q because it’s up to you to decide who you want to call _{2}q, and who you want to call _{1}q. 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 _{2}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?
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 4Πε. I can get _{0}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. 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._{electric}These two guys attract each other gravitationally and electrically. So I will write the force of gravitation, which is q, _{electron}q over 4_{proton}Πε times 1 over _{0}r. Notice in this experiment, in this calculation, ^{2}rdoes not matter, so you don’t have to decide how far you want to keep them, because they both go like 1 over ^{2 }rso you can pick any ^{2 },r. So whatever you pick is going to cancel and you will be left with this number. A q, _{1}q and the 1 over 4_{2}Πεis 9 times 10 to the 9_{0 }^{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. q is 1.6 times 10 to the -19, so two of those _{1}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 r, 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?^{2}
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?
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 rdue 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 ^{2 }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 is the _{2}q I put there. Then there’s the 4Πε, _{0}r, ^{2}rwill be this distance ^{2 }z times this radius r will be — maybe I shouldn’t call it r. Let me call it capital R, and it’s Rplus ^{2 }z. 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.^{2}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 zon the root. That is the ^{2 }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 4Πε_{0}Rplus ^{2 }zto the 3/2 integral of ^{2 }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 4Πε divided by _{0}R plus ^{2}z 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 ^{2}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?
Πε times distance squared. And when _{0}z is much, much, much bigger than R, this is one kilometer, this is two inches. You forget this. You get z to the 3/2 is then ^{2}z cubed. That means the whole thing here reduces to 1 over z and it looks like the force between two point charges. So I would ask you whenever you do a calculation to test your result.^{2}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] Back to Top |
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