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# PHYS 201: Fundamentals of Physics II

## Lecture 15

## - Maxwell's Equations and Electromagnetic Waves II

### Overview

The physical meaning of the components of the wave equation and their applications are discussed. The power carried by the wave is derived. The fact that, unlike Newton’s laws, Maxwell’s equations are already consistent with relativity is discussed. The existence of magnetism is deduced from a thought experiment using relativity.

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html## Fundamentals of Physics II## PHYS 201 - Lecture 15 - Maxwell's Equations and Electromagnetic Waves II## Chapter 1: RecapSolving Maxwell’s Equations [00:00:00]
So last time I told you that if you want to solve Maxwell’s equation in free space you can write down any function you want for (ky - ωt), and B was iBsin_{0}(ky - ωt).When you take this assumed form with this First of all you know it’s a periodic function in This whole graph describes the situation at one given time only at points on the Then you can ask yourself what happens later. You can do it in two ways. First of all let me tell you the relation between (2Πy/λ - 2Πt/T). This is completely equivalent to what I wrote because these are the definitions of ω and k in terms of λ and T. Maybe writing it this way is more transparent: that if you add a λ to y on the top, if you say y goes to y + λ you’re adding a 2Π. If you add capital T to small t upstairs you’re subtracting the 2Π. That doesn’t matter. This shows you the periodicity in space of λ and in time of capital T. You can also write ω as 2Π/T, but 1/T is the frequency. This is what we normally think of as frequency, so many hertz, megahertz or whatever. That’s the frequency. The angular frequency omega we use is 2Π times that, because every time the thing completes a cycle it goes by 2Π, and if it does f revolutions per second the angular frequency is 2Πf radians per second. That’s the difference between revolutions per second and radians per second. Okay, so this is another way to write it, and B will be the same thing. So you should understand what this wave does. So imagine the plane wave is coming from the blackboard towards you. If it’s polarized this way — now I should tell you what polarization is. Polarization is the direction of E. B is understood to be perpendicular to it, and the direction of propagation is perpendicular to both of them. You can see E x B is the direction of propagation of the wave. You do the cross product from E to B and that’s the way it advances. So what you will see is that if these rows in this classroom were not curved, but straight, then all the people in the first row will see exactly the same electric field. So maybe it’s pointing up. Then people in the second row may see no electric field because they happen to be sitting at that distance. People in the third row may see a negative electric field.So every half-wave length it’ll go from 0 to maximum to 0 to minimum back to 0, and so it oscillates at one time. If you wait a little bit the whole pattern will shift towards the back of the class. So what these guys saw now you will saw a little later. How much later? It’s the time light takes to go from this row to your row. That’s the delay, So one example of transverse waves is if you hold a string; I give it to the people in the last row and I hold it and then I shake it. These ripples will go from me towards the end of the room. The ripples are going like this but the wave is going that way. That’s a transverse wave. Or if I shake it this way, same thing, it’s going from side to side. The wave is going from me to you. A longitudinal wave is a sound wave. For example, when I talk now some diaphragm in the throat pushes the air out and compresses it, and maybe when it goes the other way decompress it, and the motion of the air is back and forth, namely in the same direction as the propagation. So sound waves are longitudinal, light waves are transverse. Yes?
You’ve got to be careful when you say rotate. For example, if you have a grandfather clock and you rotate only the clock it won’t work the same way. That’s because the clock is very sensitive to the earth, but if you took the earth and the clock, you rotate both of them it doesn’t matter. In fact, that’s happening all the time. So all the things relevant should be rotated, but for electromagnetic field in vacuum there’s nothing else to rotate that matters so you rotate it. Now another thing you can do instead of rotating around that axis you can rotate it around this axis, around the axis of propagation. You can turn All right, so now we’ve got a few properties of waves, and one thing maybe worth mentioning is the light from the light bulbs in this room is a chaotic set of waves being sent to you. Each atom emits light at a different polarization not in sync with the other atom, so it doesn’t have a definite frequency. It’s a big jumble, a big mess, but it’s a superposition of elementary plane waves. Actually, I should amend myself. Plane waves are an idealization. Every wave you have, like if you turn on the light bulb the waves go out spherically from the center, but far from the center when this sphere is 10,000 miles in radius if you’re a little creature at the end it will look like a plane wave, just like a sphere would look like a sheet to somebody who is very near like the earth does to us. Yep?
To roughly double that is all you can see. Stuff on the other side is ultra violet, then you’ve x-rays, on the other side you’ve got infrared, you’ve got radio waves, but they’re all electromagnetic waves. All you’re doing is varying ## Chapter 2: Deriving the Energy and Intensity of an Electromagnetic Wave [00:18:19]All right, so let me tell you now about the energy contained in this wave. You’ve got to agree that when you have an electromagnetic wave you have energy that you did not have before. Let me ask you a first question. If I’m sending an electromagnetic wave towards you, and I ask you, which way is it polarized, what will you need to check that, any idea? Look, I’ve told you. If you cannot measure something or tell me in principle how we’ll measure it you don’t know what you’re saying. Yes? You have an idea? Anybody? Yeah, go ahead.
And the remarkable thing about the field is that any expression you derive for it is a local expression. It only cares about what the field is at this point. It doesn’t care what the origin is. It doesn’t matter if this is produced by static charges or maybe it’s an electric field produce by changing magnetic field. It does not matter. The answer does not depend on the context. This is the energy density, energy per unit volume. For a magnetic field it looks like _{0}, and you all know by now that whatever ε_{0} does μ_{0} does in the other place. If this guy’s up that guy’s down, or in the field laws μ_{0}/2Π’s up and 1/4Πε_{0} epsilon is down. Therefore, if you have a region where there is, let’s say, no electromagnetic field and suddenly a wave goes by that region has now got energy, and how much energy do we have? The electrical energy, you can see, is ε_{0} over 2Esin_{0}^{2}^{2}(ky - ωt). And the magnetic energy is 1 over 2μ_{0}Bsin_{0}^{2}^{2}(ky - ωt). I will now show you that these energy densities are actually equal.They’re equal because ½μ E/c, so let’s write it as E. From the definition of the velocity of light and μ_{0}^{2}/c^{2}_{0} and ε_{0} this is simply ε_{0} over 2E. Sorry, _{0}^{2}E plus all the sine squares, which I forgot. Therefore, the total energy is equal to _{0}^{2}Eε_{0}^{2}_{0}sin^{2}(ky - ωt). You got a half from this and you got a half from that. Even though the formula looks different by the time you put in the relation between μ_{0} and ε_{0} and B and _{0}E it turns out to be equal. So even though the magnetic field is weaker than the electric field by a factor of 1/_{0}c you might think it’s negligible in terms of energy, but it’s got same energy density, and you add them up you get that. Now you can see this energy density is time dependent and space dependent because it’s oscillating with time at a given point, and oscillating with space at a given time. So you can ask yourself, let me sit at one place and ask myself, “What does the average energy density average over a full cycle?” If something is up sometime and down sometime what’s the average? The average will be ε_{0}E/2, because the average value of sin_{0}^{2}^{2}θ over a full cycle is 0 to 2Π divided by 2Π which is ½. This is something we have done before in circuits. You should know that. Average of sine squared is half, average of cosine squared is half, and the check is that sine squared plus cosine squared average is 1 because that’s 1 identically. So try to remember this. So this is the average energy density.Next I want to ask the following question. What is the rate at which energy is coming at me? And that is called intensity, I, and it is equal to the watts per meter squared, just like a flow of liquid except this is the flow of energy. So what I want to do is take a square meter, stand in the way of the beam and ask how many joules cross per second, or if you want, how many watts per square meter. That’s easily calculated from the energy density and I’ll give you the reason. It’s identical to the reasoning for currents traveling in a wire, or fluid flowing in a tube. Suppose you take a region, a cylindrical cross section through which electromagnetic waves are traveling? This is cross section A and I wait 1 second it’ll go a distance So let us now calculate that and you’ll get a very interesting result. The intensity is equal to the energy density times ^{2}(ky - ωt). This is not the average energy. This is instantaneous. I’ll average it in a moment. Are you guys with me now somewhere here, here? Now I’m going to write it as follows, ε_{0 }and E then at _{0}B — I forgot a _{0}c here. You realize that? That’s another c. Then E_{0} is B_{0} times another c times sin^{2}(ky - ωt). And what is that? cε^{2}_{0} is 1/μ_{0}. So E over μ_{0} B_{0}_{0} sin^{2}(ky - ωt). But if you now define a vector S, it’s called a Poynting vector, which is spelled with a p, to be E x B over μ_{0}. The magnitude of that vector is precisely the intensity. The magnitude of S, which you can either call S or you can call it I, is exactly this.So So here’s one example. At the surface of the earth if you take a square meter, and here’s the sun, emitting light. You can ask, “What’s the intensity of sunlight?” Anybody have an idea how many watts per square meter from the sun.
_{0}. All these averages are simply half, anything involving in sine squared. I’m not going to worry about the half, but if you took this to be the average intensity you can ask, “How big is the electric field that comes with it,” because that light is going to be this electric and magnetic field. That’s all light is. So the sunlight produces electromagnetic field, is electromagnetic waves and I’m asking, “How big is the E vector?”And all you have to do is stick that into this number. If you want you can get rid of ## Chapter 3: The Origin of Electromagnetic Waves [00:30:44]So let’s talk about one thing which I’ve not discussed at all, which is, where are these electromagnetic fields coming from? I said you don’t need ρ, you don’t need I, you don’t need the current, you don’t need the charge, these can exist in free space. But what’s the origin of the electromagnetic waves? Anybody know? When can you get them? It didn’t happen in anything we studied. Yes?
When you have a time dependent electric field you’ll have a magnetic field going around it because the line integral of B will involve the rate of change of electric flux. And that will also be time dependent, but if that gets time dependent there’ll be an electric field going around that. So basically these will curl around each other whenever they’re dependent on time, and they can then free themselves loose from the capacitor and take off. All you need is two plates, and an AC source, and you connect them, you will make electromagnetic waves. You’ll make them at the frequency of the source, so you won’t be able to see it. Your dog won’t be able to see, but some gadget will be able to pick it up. That’s all you need, oscillating charges. So what happens in a radio station is you could imagine a simple radio station with an LC circuit, the current is oscillating at some rate, and part of the circuit is in the antenna, and the charges are going up and down as the current goes back and forth that sends out the waves, and the waves come to your house. So here’s the picture. Here’s the radio station, and the waves are emitted in big circles, and this is your house, and here’s your little antenna. It’s a piece of wire, and the electric field, if it’s polarized this way, will move the charges up and down, and charges can be part of an LC circuit. So here are the antennas, if you like; part of the circuit. As the charge goes up and down the AC current will try to flow here. And if you tune this capacitor so that it resonates with the frequency you’ll get a hefty signal from the radio station. So in the end it’s all charges. Charges produce the field. Charges respond to the field. That was true in the static case. That’s true in the time dependent case. Yes?
r you will get a negligible number compared to the actual electric field. So it’s really like you and your parents. I mean, at some point you are free from your parents. You are able to manage on your own, but you had parents somewhere sometime, right? That’s what it is. The electromagnetic waves can go on their own, but they are not produced on their own. They’re produced by charges. It is just that unlike the static fields, which are very near the currents and charges that produce them, the time dependent fields propagate on their own. ^{2}E keeps B alive and B keeps E alive. If E tries to die there’s a dE/dT that produces a B. If the B tries to go down it produces at dB/dT that produces an E, so they go back and forth. It’s really like oscillations in which you have kinetic to potential transfer. You can have energy transfer. So the fields cannot die. They are self-sustaining, but to get all of that physics you had to put the term that Mr. Maxwell put in. Without that term you don’t have this phenomenon. You don’t get it from statics.So one typical problem you have is this radio station is, let’s say, 100 kilowatts and you’re sitting here at some distance ## Chapter 4: Relativity and Maxwell’s Equations [00:37:07]Okay, so I want to switch now to my favorite theme. The remarkable thing about electromagnetism is that you can ask what happened when physics went through the Einstein’s revolution with special relativity. We know everything changed after Einstein, and all the Newtonian mechanics had to be modified. And so far I never mentioned the word relativity, so you can ask yourself, “How are these modified by Einstein’s work?” So first let me tell you, remind you, now you guys did relativity last term, right? Is there anybody who’s never seen it before? Okay, well you don’t have to know a whole lot, but let me just say the following. In the Newtonian world there was a principle of relativity according to which the equations like That means an accelerating plane is a reference frame in which people cannot use x is the rest length of the spring and _{0}x - x is the deviation from that. That’s the equation. Now you and I differ by what?_{0}You and I differ by a constant velocity. The constant velocity just means x prime is equal to F = ma doesn’t have to be modified by going to a moving frame. And that is the relativity of Newtonian mechanics. If these are the laws of motion we can understand why if you’re inside a train, which is completely closed, you cannot look outside and it’s going at uniform velocity with respect to the ground, you cannot tell. You cannot tell because nothing you do will be different. Because everything you observe in the world is controlled by Newton’s Laws, and Newton’s Laws are unaffected by adding a constant velocity to everything. So when you wake up and I say, “Is the train moving,” you cannot tell.Okay, so that’s by doing physics experiments. You cannot say, “It says Amtrak so I know it’s not moving.” That kind of argument is based on sociological axioms. I’m just saying can you — after all it’s possible, theoretically, Amtrak trains can move, so you must admit the possibility. Okay, now you come to Maxwell’s equations and electromagnetic theory. Let me write down what we have. The first thing we have is c/^{2})d^{2}Edt^{2} is 0. These are some of the results we got from electromagnetic theory. Now comes the important question. This is the velocity of a particle according to whom? That question came up earlier in the class. Who is supposed to use it? I may assume it worked for me, but how do I know then when you see it it’s got a different velocity will you get the same physical world that I get. Now let’s look at this equation, this also came from Maxwell’s equations, and compare the equation for a string. Let me call ψ as the displacement of the string rather than y, d is ^{2}ψ/dx^{2}(1/v. I just wrote the equation shifting to the other side. They look very similar. Here v is the velocity of the waves according to a person for whom the string is at rest. Okay, according to whom the string is at rest. You understand that? This velocity, because the waves are traveling in the string, a at speed ^{2})d^{2}ψ/dt^{2}v. So this equation is to be used in its present form only by a person for whom the string is at rest. If you want to see the string from a moving frame then x’ is x - ut, and in classical mechanics t’ is equal to t. You can do the change of variables with these partial derivatives. I don’t want to do that, but you can say d/dx of ψ is equal to d/dx’ of ψ times dx’ over dx, then dψ/dt’, then dt’ over dx. Now t’ is the same as t, but formally we can change variables, and we can take this equation and rewrite it in a frame moving to the right at speed u, and I promise you it won’t look like this. It will look very different. More importantly at least understand conceptually that the velocity u of the moving observer will appear in the final equations, because these derivatives, x’ over x and so on contain the velocity u. Can you see that, dx’/dt has the velocity u? So when you make all these changes and put them in you’ll get a new equation involving x’ and t’ in which the velocity u will appear. So if you are that person you have to decide what’s the velocity to use for you, and the answer is unique. It’s your velocity relative to the string. The string is anchored in the lab. If you happen to have a speed u relative to that that’s the speed you should put in. Therefore, the equation is not the same for everybody. There is a special observer for whom the string is at rest, namely in the laboratory frame for whom this equation works, and v is the velocity for that person. Now we come to this equation. It has no reference to the velocity of the observer. It’s got a velocity of light, and you can ask who is supposed to use it. Whereas in the string we know the privileged frame of reference is where the string is nailed down, but the light is traveling in vacuum.There is no frame of reference. People thought maybe even the vacuum contains a medium called ether. Because everything needs a medium to travel they said there is an ether. Then, of course, this is to be used only by people who live in that ether at rest, but we are moving relative to the ether because we are on the earth which is going around the sun. You may say, “Well, maybe today I just happen to be at rest relative to the ether. That’s possible, but then tomorrow I cannot be because I’m going around the sun. Six months from now I’m going the opposite way around the sun at a huge speed, but what I find is every single day of the year I’m able to use these equations. That means they apply to me no matter what my velocity is. So these equations, it turns out, are valid for any observer who is inertial, namely one for whom at low velocities Newton’s Laws apply. Now what people were worried about in the old days they said, “Look, let’s take That was what they were worried about until Einstein came and said, “There is no either and this is the wrong set of transformations.” If you use t’ is t - ux over c divided by the same square root, if you change coordinates this way you’ll find amazingly, if you change all the ^{2}d/dx’s to d/dx’ and did the whole partial derivatives and chain rule and so on you will find that in the new frame of reference you’ll find the equation will look like this. It will look the same for anybody moving relative to me at any speed. That’s why it’s not clear that I’m the privileged user. All people at uniform relative motion can use the very same equation with the very same number c entering. So this was the great triumph of the Maxwell theory. It was that it was already consist with the relativity. In fact it is what led to Einstein’s revolution because this equation said no matter who you are a light pulse is going to travel at a speed c for you, no matter who you are. That’s very strange because every signal we know has a property that if you move along the signal its speed is reduced, right?If you’ve got a bullet going at 700 miles per second if you travel at 400 you will think it is going at 300, but if it’s a beam of light it’s supposed to have the same velocity for everybody, even those moving in the same direction and the opposite direction. It doesn’t matter. Therefore, something had to change with our definition of space, and time, and velocities, and Einstein replaced it with these new equations. And one of the consequences of that equation is that if I have an object that is going at a speed v and they’re moving the same direction at the speed u in the old days you will subtract your speed in the same direction, but the correct answer is — this is 1 - v is the speed of an object according to me. You are moving in the same direction at speed u. You will measure the speed w given by this. If u and v are much smaller than c you can forget this and it looks like the good old Newtonian days, but if u and v are comparable to c then the denominator is one less than, one minus something, so the whole thing will be a little bigger than what you thought because you’re dividing by 1 minus something.And finally, if what I was looking at was a light pulse, that means c. So this law of transformation of velocity has the amazing property that if I’m observing a light pulse that’s got a speed c you will also get a speed c. So it’s a very beautiful way in which the mystery was resolved. So Maxwell gave these equations, did not give a preferred frame of reference. Then you’ve got to ask yourself, “What coordinate transformation should exist so that the equation has the same form for everybody,” because it doesn’t tell you who supposed to use it. Then you get this equation. You can get to this equation simply by demanding that two observers looking at a light pulse somehow get the same speed. If you fiddle with that and the symmetry between the two observers you will get this. Anyway, I just wanted to tell you that there are many things you have to change, but you don’t have to change any of electromagnetic theory. The equations I wrote down are correct.## Chapter 5: Deducing the Presence of Magnetism [00:51:46]Okay, the last thing I want to do is something I promised long ago, which is the following thing. If you believe in relativity, namely if you believe that the laws of physics should have the same form for people in uniform relative motion, you can deduce the presence of magnetism given just the presence of electrostatics. In other words, suppose you never heard of magnetism. I can show you that it must exist. magnetic forces must exist, and I show that as follows. Before I show that you need a couple of results that you guys may not remember all the time. First result is if you’ve got a wire and it’s got Second thing to know, if I took a rod of length v you know it will shrink, therefore these plus signs will be compressed, and to a person seeing the moving rod the density n will be n (the density at rest) divided by 1 − _{0}v^{2} over c. This is a relativistic effect: that the number of charges get squeezed because the rod itself gets squeezed. So a moving rod which is charged will appear to have a higher charge density. And the final result I’m going to invoke is the following. What is the charge per unit length? If the charge per unit volume is ^{2}n then I claim the answer is n times a. That’s also easy to understand. If you took a unit length of this wire, unit length, the volume of that is just A times 1 and that times the density is the amount of charge that’s there. So if you have a very thin wire you may like to think about charge per unit length rather than charge per unit volume, and this is the way to go from one to the other. Now we are ready.Now I’m ready to show you how just by thinking you can deduce the presence of magnetism. And I like this argument because quite often this is how people make, theorists make discoveries. They will take something that’s known. They’ll appeal to a principle like symmetry, or relativity or whatever. Then they will say, “This implies that there is a new force, and I’m going to tell you what the new force is.” So here’s what I want you to imagine. There’s a very long infinite line of charge and right on top of it there’s another infinite line of negative charge. If they’re just sitting there there’s nothing interesting. Now what I want to do is I want to have the upper thing going at a speed So what is the current? The current is going to be the density of plus charges, times the velocity, times area, times the value of each charge. But for the wire to be neutral — this is where I want you to follow me closely. The two of them cannot have the same densities at rest because if the plus charge had the same density at rest as the minus charge when it starts moving and it starts compressing the plus density will exceed the minus density and the wire will be neutral. So I cook it up so that Okay, now imagine a particle at rest here, and you know all about electrostatics. What do you think it will do? You’ve never heard of magnetism. You’ve heard of electrostatics. What will this charge do? Yep?
So if previously they’re balanced now they won’t balance. In fact, ^{- }over this, which is n divided by 1 - _{0}^{+}v. Therefore the wire will appear to be negatively charged because the negative charges are that value, the positive charges are that value, so net charge on the wire, net charge density will be equal to -^{2}/c^{2}nover 1 - _{0}^{+}v + ^{2}/c^{2}n. If you like, this is the plus charge and that’s the minus charge times _{0}^{+}e, if you like. And I’m going to make an approximation. I’m going to approximate this is ntimes 1 + _{0}^{+}v. There are more terms, but in the binomial expansion I’m going to stop with the first term. So this is the result that’s very good in the limit of small ^{2}/c^{2}v/c, but v/c is not set equal to 0. It is set equal to a finite but small amount. So if you compare those two you will find the net charge is - n. That’s going to be the charge on the wire. So to the person moving with this charge the wire looks charged. In fact it looks negatively charged. So you know, knowing just electrostatics, what the charge will do. The charge will be drawn to the wire, start moving towards the wire. That’s a fact._{0}^{+} v^{2}/c^{2}Well, if it’s moving towards the wire for me it better be moving towards the wire for the original person also because even if you and I move horizontally the fact that your moving transverse to a velocity is going to be a true statement in both frames. From that you will conclude that even in the original laboratory frame this moving charge would have been attracted to that wire, and it’s attracted by virtue of its velocity. So that means there is a new force in which a charge is attracted to the current in the same direction, right? So let’s ask, “What is the force of attraction?” In Newtonian mechanics — I mean, at this point once you’ve got this length contraction you can just think in terms of Newtonian formulas because the errors you make involve higher powers of n. You have a charge per unit length which is the area of the wire times that. I sorry, that’s charge _{+}^{0 }v^{2} /c^{2}e. That’s the density. That’s the actual charge. That’s the λ. So what’s the electric field of a wire with charge per unit λ? I remind you it’s λ/2Πε_{0 }r.So for this particular lambda it’s ctimes ½Πε^{2} eA _{0}r. In other words in its particle rest frame there’ll be an electrical field because there’s a net charge, negative charge, of that strength on the wire. Once you’ve got a charge per unit length λ/2Πε_{0}r is the field attracting it toward the center, so that’s the force. So I’m going to write it as follows. I’m going to write it as nover 2Πε_{+}^{0}veA_{1} _{0 }c times another ^{2}e and another v. This is the force, this e times E. In Newtonian approximation the force perpendicular to the motion is the same for all observers, therefore I ask you — and this force must be present even in the lab, but look at this force. This guy is the current, nevA is the current. This guy 1/ε_{0}c^{2} is μ_{0}/2Π. This is ev. You see, this is the exactly the magnetic force we got, μ_{0}I/2Πr. I forgot a 1/r here. That is the azimuthal magnetic field around the wire that times ev is the force. Because you can tell that no matter where the charge was, whether it was here, or whether it was there or anywhere around the wire it’ll have the same attraction.So you can deduce in the laboratory frame there is an attractive force towards the wire, and that agrees with the exact formula you got for the magnetism. If you don’t ignore the higher powers of v you can see this. There’s only one thing I did which is a little fudge, which you guys may not have noticed, is that the density here is ^{2}/c^{2}nwhereas the actual current in the lab is _{+}^{0}nand the two of them differ by this factor. However, if you have a _{+}v in front of your expression and you’ve ignored ^{2}/c^{2}v or ^{4}c there is no point in keeping this guy here because that makes an error of order ^{4}v. If I’m going to keep that I should go back right here and keep such terms. So the leading order you don’t have to worry about the difference in density. This is a very subtle calculation because sometimes you worry about the difference and sometimes you don’t. So the rule always is if you’re trying to do calculations to a certain order, namely ^{4}/c^{4}v/c the whole thing squared then things that make corrections of order v/c to the fourth can be dropped.Anyway, I’m not that worried about the details, but I want you to understand at least what the logic is. That’s more important. I know about electrostatics. I cleared the neutral current carrying wire. It’s neutral because the negative charges are at rest at some density. The positive charges in the moving rod are also at the same density, but the rest density is somewhat lower. Then I predicted the particle won’t move because it’s got no electric attraction. Then I go to the rest frame of the particle, then I find that the positive charges have come to rest, and therefore to a lower density. Negative charges are moving the other way, therefore at the higher density. They no longer cancel. The wire is charged. I expect this charge to be attracted to the wire. That means going back to the lab I expect the moving charge also to be attracted to the wire because when something goes towards the wire that goes towards the wire according to all people. But you can go beyond and actually compute the force and equate the force and deduce that that is a new force now. Whenever you have a current carrying wire with current Okay, so I’ll see you guys after the break, and my suggestion to you on what to do with the break is to try to carry your textbook with you, and the textbook has got lots of problems, some of which are a lot simpler than the one I gave in the early days, but roughly the same level of difficulty as the midterm. And do as many problems as you can. Look at the worked examples and try to solve them. [end of transcript] Back to Top |
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