Prof: 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 r^(2) 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 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_1, and there’s another entity that has some amount of charge q_2 they will exert a force on each other which is given by q_1q _2 times this constant which is somehow written as 1 over 4Πε _0 That’s 1 over r^(2) 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 4Πε _0, 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 q_1 is sitting at the origin and q_2 is sitting at a point whose position is the vector r Then the force on 2 due to 1 is given by q_2q _1 over 4Πε_0 times 1 over r^(2) That’s the magnitude of the force, but I want to suggest that the force is such that q_1 pushes q_2 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_r_ is 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 F_21 is minus of F_12 Now, how do we get attraction and how do we get repulsion? We get it because q_1 and q_2, 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 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: > Prof: Yep? Student: Well, if it’s in the same instant disappearing from one place and appearing another place, it’s traveling faster than light? Prof: 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 Prof: 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 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? Prof: 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 Prof: 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 Prof: Pardon me? Student: You don’t know the exact quantities of the charges Prof: Suppose I tell you I tell you how many charges there are Yes? Student: You don’t which direction the attraction is Prof: 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 q_1 This is q_2 This is q_3 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 q_1 and q_2, you find the force due to q_2 and you find the force due to q_3 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 4Πε _0 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 q_1 and

q_2 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 r^(2)? Yep? Student: Vary the distance between them, and show that the force falls off Prof: 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? Prof: What was your plan? Student: Observe acceleration Prof: 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 r^(2 )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 r^(2) That’s how Newton deduced the 1 over r^(2) 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 r^(2) 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 r^(2) Anyway, that’s how we can find even if it’s not 1 over r^(2) If it’s 1 over r^(3), or 1 over r^(4), whatever it is you can find by taking two charges See, we don’t have to know what q_1 and q_2 are That’s what I’m trying to emphasize here If all you’re trying to see is does it vary like 1 over r^(2), 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 r^(2) All right, suppose I got 1 over r^(2) I want to know it depends on the charges as the first power of q_1 and the first power of q_2 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 Prof: 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 Prof: 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 q_1 Of course, it also depends linearly on q_2 because it’s up to you to decide who you want to call q_1, and who you want to call q_2 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 Prof: 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 ε _0 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 4Πε _0 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 r^(2 )divided by q_electron, q_proton over 4Πε _0 times 1 over r^(2) Notice in this experiment, in this calculation, r^(2 )does not matter, so you don’t have to decide how

far you want to keep them, because they both go like 1 over r^(2 ),so you can pick any r So whatever you pick is going to cancel and you will be left with this number A q_1, q_2 and the 1 over 4Πε_0 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 q_1 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 m_1m_2 over r^(2), 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? Prof: 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: > Prof: Pardon me? Student: > Prof: 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 Prof: 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 r^(2 )due to that, 1 over r^(2 )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 q_1 The q_2 is the q I put there Then there’s the 4Πε _0, r^(2), r^(2 )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 R^(2 )plus z^(2) 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 R^(2 )plus z^(2 )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 4Πε _0R^(2 )plus z^(2 )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 4Πε _0 divided by R^(2) plus z^(2) 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 Prof: 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 Prof: 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 Prof: 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 q_1q _2, or 4Πε _0 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 z^(2) to the 3/2 is then z cubed That means the whole thing here reduces to 1 over z^(2) 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