B1 中級 4652

 Prof: All right, class, I thought I'd start as usual by telling you what happened last time. Not the whole thing, but just the highlights so you can follow what's happening today. The main things I did last time were the notion of an electric field, which is going to be with you from now till the end of the course. The idea of the electric field is that if you've got lots of charges, q_1, q_2, q_3, instead of worrying about the force they exert on each other, you ask yourself, at a generic point where there's nothing, if I put 1 coulomb here, what will it experience? What force will it experience? You compute that. So in your mind, imagine a coulomb, and you find the force due to q_1, it looks like that. q_2 may exert a force that way. q_3 could be of opposite sign, so maybe it will exert a force that way. You add all these vectors, they add up to something. That something is called the electric field at that point. There is nothing there except the electric field, but it's very real, because if you put something, something happens to it. So the electric field is everywhere. The charges are in a few places. Electric field is defined everywhere, except right on top of the charges, which is where it probably is infinite. Once you know the electric field anywhere, if you put another charge q, a real charge, it will experience a force equal to qE, because electric field was the force you would have had on the 1 coulomb, and if you put q coulombs, it will be qE. That's the electric field. So you can imagine computing it for any given distribution of charges, because you know what each one of them does. Then I said there's one nice way to visualize the electric field, which is to draw the field lines. You go to each point and you ask, if I put a charge here, a positive test charge, which way will it move? Then you follow that thing as it moves and you get that line and you get that line and you get that line. You can draw these lines. The lines give you one piece of information which is very obvious, namely, if you are here, the force is along that line. But I also pointed out to you, you get more than just the direction of the force. You also can understand the strength of the force. The strength of the force is contained in the density of these lines. Now density has to be defined carefully. It's not like mass per unit volume. It's an area density. That means if you get yourself 1 square meter, you know, meter by meter piece of wood, like a frame, and you hold it there, perpendicular to the lines and see how many lines go through, that's called the area density of lines. That you can see will fall like 1/r^(2), because if you draw a sphere of radius r, that area is 4Πr^(2), and the same number of lines are going through that as any other sphere, so it will be proportional to 1/r^(2). But so is the electrical field proportional to 1/r^(2), and I said, let us agree that we will draw 1/ε_0 lines per coulomb. This is a necessity; it's just a convenience. It's like saying I want to measure distances in inches and centimeters for daily life. You can measure them in parsecs and angstroms, but you'll be dealing with nasty numbers. So it's a convenience, and the convenience here is, let's pick 1/ε_0 lines per coulomb. You'll see the advantage of that, because if you ask, what's the density of lines here on a sphere of radius r, if there's charge q at the center, I got that many lines, and the area of the sphere is 4Πr^(2), so the density of lines per unit area, you see, is precisely equal in magnitude to the electric field at that point. If you had not drawn 1/ε_0, but maybe 5/ε_0 lines, then the line density will be 5 times the electric field. It will still represent the electric field, but we don't want to simply represent it. We want it to be the electric field. It makes it easier. Then I said, let's take a slightly more complicated situation, two charges. This is called a dipole, one pole and another pole, and you can draw the field lines here. You can see if you put a test charge, it'll run away from the plus to the minus. If you leave it somewhere here, it'll go like that and loop around and come back, and you can calculate them. This is no longer guesswork. If you had enough time, I hope you all agree, you can go to any point you like and find the force of attraction due to this one, the force of repulsion due to that one, add them up and you will get an arrow that direction. So one can draw these lines, and the lines tell you a story. Then I said, let us find the field in an analytic expression due to the dipole. Yes? Student: I have a question about the dipole. If you put a test particle going in the positive direction of the x axis, would it also _________? Prof: No. She's got a good point. If you put somebody here, it will never loop around. Can you see why? Because as it goes further away, this is trying to pull it back. It's always going to be closer to this guy, so it's never going to come back. This line will go like that and this line will go like this. But anything else at any other angle will loop around and come back. All right, now the field strength, if you want to calculate it, you can use the formula for E due to this one and E due to that one and add them, and I did that for you. I don't want to go into the details, but I remind you in all cases, the electric field fell like 1/r^(3), because the field of each charges goes like 1/r^(2). And if these were on top of each other, they will completely cancel each other. So the reason you have a non-zero field is thanks to a. Therefore the answer has to contain an a in front of it, at least n the first approximation. But that a, from dimensional analysis, has to come with a 1/r so that the whole thing has the same dimension as before. That's where you get a/r^(3). That a times q times 2 and so on, that became the dipole moment. That was true here. We verified that's true here. Later on, we'll verify it everywhere, because there are easier ways to calculate than what I'm using. Then I said, forget about the field due to charges. Let's look at what charges do when you put them in a field. So I took two examples. One was a very simple example. These are two parallel plates. They are not two lines; they are plates coming out of the blackboard. They're filled with charge and this has got charge on it. This has got - charge on it. And therefore, the electric field will look like this, right? Because if you leave a test charge, it will go away from the positive, towards the negative. Then I said, suppose there really is a particle here with some velocity, v_0, what will it do? You can see that the force on this guy is going to be q times E. E is pointing down. If you divide by the mass, that's the acceleration, also pointing down, and it's constant. So that's like a particle in a gravitational field, except g is replaced by this number. So it will just curve like that, and you can calculate the trajectory. The final thing I did was, what happens when you put a dipole in a uniform field. Here as well, I think I was rushing near the end, and even I couldn't read my stuff in the corner. So I will go over that. If there's something that you didn't follow, then I will be happy to repeat that part for you. But you should understand what the question is. There is an electric field which is pointing like this, as if you have two plates here, charge is here, - charge is there. They're producing a constant electric field in the horizontal direction. In that environment, I take an electric dipole whose - charge and charge, q and -q, happen to be oriented like that. Question is, what will happen to this guy? If you want, you can imagine that it's a little massless stick, and one end you glue q coulombs, other end you glue -q coulombs, and your let it sit there. What will it do? First of all, it won't feel any net force, because the force in this direction is q times E and the force that direction is also q times E, if you want, it's -q times E and they cancel. But that doesn't mean it won't react. It will react, because you can all see intuitively, it's trying to straighten this guy out and applying a torque like that. You follow that? That's what it will do. And the way to find the torque, the torque is the product of the force and the distance between the point of rotation and the force, and the sine of the angle between them, that is to say, sine of this angle. What the sine of the angle does is to take the component of the force perpendicular to this axis, because if you resolve the force into that part and that part, this part is no good for rotation. That's trying to stretch the dipole along its own length. It's the perpendicular part that's going to rotate something, so you get that times sine theta. That you can write now as the vector equation p x E. Because p is equal to--I'm sorry. I need a 2 here. I forgot the 2, because this charge will have a torque and that charge will have a torque and the two torques are additive. They are both going the same way. Then 2q times a is p, and this is E and the sinθ comes in the cross product. I'm assuming all of you know about the cross product. Okay, final thing I did, which is, if you have a force, you can associate with that force a potential energy. Again, this is something you must have seen last time, but I will remind you. As long as it's not a frictional force, you can say the force is connected to potential energy in this following fashion. Or the potential energy at x minus potential energy at some starting point x_0 -- I'm sorry, x_0 - x is the integral of the force from x_0 to x. This is the relation between the force, as delivered to the potential, and the potential is the integral of the force. For example, for a spring, U is ½kx^(2), and F = -kx. If you go to this one, we tell you U(x_0) - U(x) is equal to the integral of -kx dx from x_0 to x. So that gives you kx_0^(2)/2 − kx^(2)/2. And by comparison, you can see U(x) = ½kx^(2). Actually, this is not the unique answer. Do you know why? Given this formula, can I immediately say this guy corresponds to that, this one corresponds to that one? Is there some latitude here? Yes? Student: You can always add in a constant. Prof: You can add a constant to both, because if I said that, that certainly works. If I add 92 to both, it still works, because the 92 extra doesn't matter when you take the difference. So it's conventional to simply pick the constant so that the formula looks simple. Coming to other expression, if you had a torque, which is pEsinθ, you can ask--that's the torque--it's minus, because it's trying to reduce the angle θ-- you can ask what U leads to that. And you can see it's -pEcosθ. See that, take the -U and take the derivative, you'll get the torque. But if you got two vectors, p and E and you see the cosθ, I hope you guys know that you can write it as a dot product. So that's the end of what I did last time, okay? The potential energy is proportional to the dot product of p with E. The torque is equal to the cross product of p with E. And what does pEcosθ mean if you plot it as a function of angle? It will look like this. This is Π and this is 0. That means it likes to sit here and if you deviate a little bit and let it go there, it'll rattle back and forth, just like a mass spring system. In fact, you can very easily show, near the bottom of the well, the potential energy is proportional to θ^(2). That's because cosθ can be written as 1 − θ^(2)/2 θ^(4)/4!, etc. for small angles. If you just keep that term, you will find it looks like this. Not very different from U = kx^(2). So what x does with forces, θ does with rotations. All right, so this is what we did last time. Now I'm going to do the new stuff. So new stuff is going to give you--I think it's useful, because it tells you the level at which you should be able to do calculations. So here's a typical problem. You have an infinite line of charge, of which I will show that part, and somebody has sprinkled on it λ coulombs per meter. So it is not a discrete set of charges; it's assumed to be continuous and it's everywhere. I'm just showing a few of them, and if you cut out one meter, you'll find there's λ coulombs there. And you want to compute the electric field you will get due to this distribution, everywhere. So you want to go somewhere here and ask what's the electric field. That's what we're going to do. Let's go here. You will see why. Now first of all, you've got to have an intuition on which way the electric field will point. You have a feeling? Yes. It will point here, this way. Why not like that? Yeah? Student: > Prof: Okay. She said the horizontal parts will cancel. That's correct. Another argument from symmetry is that if anybody can give you a reason why it should tilt to the left, I can say, "Why don't you use the same argument to say it will tilt to the right?" Because this is an infinitely long wire and things look the same if you look to the left and if you look to the right. And the field you get should have the same property. If this was a finite wire, I wouldn't say that, because in a finite wire, that can be a tilt, somewhere here. But infinite wire, it cannot tilt to the left or to the right, because each point has the same symmetric situation to its right and to its left. In a finite wire, it's not true. Life to the left is different from the life to the right, but for infinite wire, you know it cannot be biased one way or the other. It's got to go straight up. Secondly, you can find the field here, here, here or at anywhere at the same distance, you've got to get the same answer. Again, because if you move two inches to the right, it doesn't make any difference with an infinite wire. You've still got infinite wire on either side. So we'll pick a typical point and calculate the field and we know that answer is going to be good throughout that line. So now I take this point here. I want the field here. I'm going to make that my origin. Then I take a piece of wire of length dx. dx is so small that I can treat it as a point. Now the dx I've drawn is not a point, but in the end, we're going to make dx arbitrarily small so that it's good enough. It's like a point and it is at a distance x from the origin, and let's say that distance is a. So let's find the field due to just this guy, the shaded region. Think of a charge. How much charge is sitting here? I hope you all agree, the charge sitting in this region is just λ times dx. That's just the definition. If you've got that many coulombs, imagine a test charge of 1 here. Well, it will push it this way, and the field due to that, I'm going to draw as infinitesimal, so I'm going to call it dE. You can call it E, but dE is to remind you, it's a tiny field, due to a tiny section dx. Now that electric field is biased to the left, but for every such section you find here, I'll find a section on the other side that's precisely biased to the right by the same amount. Therefore the only part that's going to survive due to this guy, combined with this, will be the portion here, which is the vertical part of that force. So let me find the contribution first, only from this one, then we will add the contribution for this one. For that one, you just find the vertical projection. So how much is that? Remember, the Coulomb's law for the electric field is q/4Πε _0r^(2). So this is the q. That's the 4Πε _0. r^(2) is that distance squared, which is x^(2) a^(2). That's really like the field of a point charge at that distance. But now this is the magnitude of the electric field vector at this angle, but I want the part along the y direction. So I've got to take cosine of that θ. You guys follow that? If you took this vector, this part is dEcosθ. That part is dEsinθ. But that angle and that angle are equal, and cosθ for this triangle, you can see, is a divided by (a^(2) x^(2))^(½). So this is the electric field in the y direction. I'm going to call it dE, in the y direction, due to the segment dx. The total electric field is obtained by adding all the dEs or adding all the contributions from all the segments on this line. And that goes from - infinity to infinity. All right, so now it's a matter of just doing this integral. So this gives me (λa/4 Πε _0) dx/(x^(2) a^(2))^(3/2), integrated from - to infinity. Now I can make life a little easier by saying that this function is an even function of x. That means when you change x to -x, it doesn't care. Therefore the contribution from a positive x region is the same as the contribution from a negative x region. That also makes sense in this picture here, because if you look at the field I'm computing, this section and this section give equal contributions in the y direction. But even if you did not know any of that background, as a mathematician, if you see this integral, you would say, "Hey, put a 0 there and put a 2 here" namely,integrate over half the region, because the second half is giving you the same answer. So you double the integral, but cut the region of integration in half. So at this point, you are free to look up a book, if it was an exam, but maybe not even if it was an exam at your level. You should be able to do this integral. So integrals have been around from the time of Newton and the question of an integral is, find the area of some graph with this particular functional form. And the answer to any integral is that function whose derivative is the integrand. So what you have to do is guess many answers until you get the right one. But people have been guessing for hundreds of years, and there's big tables of integrals with all the integrals you want. But you should still be able to do some integrals from scratch and I'm going to tell you how to do this one. But before you do the integral, you've got to have some idea what the answer is going to look like. I want you to get some feeling about this. Answer depends on what, is the first question. What's the answer going to depend on? Student: a. Prof: a, you understand? Whatever this is depends only on a because 0 and infinity are not going to be present as part of the answer. If the lower limit was 5, it can depend on 5, but it doesn't depend on any other thing, other than a. Then from dimension analysis, I got a length squared to the 3/2, that's length cubed, and a length on the top, so whole thing should look like something over length squared. The only length I have is a, so it's going to look like 1 over a^(2) times a number. Once you got the number, you're done. So I'm going to do all the work now to show you that the number is actually just 1. This number will turn out to be 1, in which case, you will find it's λ/2Πε _0a. Well, let's see how we get the number to be 1. So does anybody know what trick you use to do this integral? This is whatever, math 120 or-- yes? Student: Use substitution? Prof: Yes. What substitution? Student: x^(2) a^(2) = U. Prof: That won't help you. Yes? Student: Can you use trigonometric substitution? Prof: Yes. Trigonometric substitution. Which one? Okay, look--no, no, I don't blame you. I know the answer because I've seen it, but if I have to work on it, I'll try for a while before I got it right. The whole idea is, we don't like all these 3 _______ here. We want to turn that into something nice. So I'll tell you what the answer is. You can all marvel at how wonderfully it works. So what we are going to do is to introduce an angle theta--nothing to do with the angle in the problem--so that x = tanθ. That means instead of going over all values of x, I'll go with the suitable values of θ-- I'm sorry, this would not even be correct dimensionally. x = atanθ. You can see that every x that I want, I can get by some choice of theta, because tanθ goes from 0 to infinity when θ goes from 0 to Π/2. You cannot say x = acosθ, for example. You are doomed. If x is acosθ, the biggest x you can get is a, whereas I want this x to go from 0 to infinity. So when you make the change of variables, you've got to make sure that for every x you want, there is some θ that will do it. Then the next thing you do, you say dx/dθ = a times derivative of tanθ which is sec^(2)θ. Then you write that as dx = that. What that means is an integral dx is related to an integral dθ by this factor. Therefore going to the integration here, I'm just doing that part, which is going to be a sec^(2)θ dθ. θ goes from 0 to Π/2. Now let's look downstairs. Downstairs I've got x^(2) a^(2). x itself is atanθ, therefore a^(2) times 1 tan^(2)θ. 1 tan^(2)θ happens to be sec^(2)θ. That to the power 3/2, which is what I want, will give me an a^(3)sec^ (3)θ. So what do we get? You can see as promised I get a 1/a^(2) and I get integral of 1/secθ, which is cosθdθ from 0 to Π/2. Yes. And integral of cosθ is sinθ from Π/2 to 0. That just happens to be 1. So the final answer is what I gave you here, E. Well, E is the vector. I've just shown you the magnitude, but we've all agreed what the direction is. The direction is away from the wire. So if you like, if you look at this wire from the end, the lines will look like this. If the infinite wire is coming out of the blackboard towards you and you look at it this way, if you go too close, you'll poke your eyes out. Look from here, you'll see the lines are going out radial everywhere. The question is, how do the fields get weak? How does it weaken with distance? It weakens like 1/a. That's a big of a surprise, right? The field away from the wire doesn't fall like 1 over distance squared, but falls like 1 over distance. The reason is that every individual portion of the wire has a contribution that does fall like 1 over distance squared, but it is an infinite wire. When you add it all up, the net answer goes like 1 over the distance. The field away from a wire falls like the distance from the wire, on the perpendicular from the wire and there's pointing away from the wire. That's it. Okay, so that's one calculation. Then I'm going to do one more and that's going to be the end of the tough calculations. Second calculation is going to be an infinite sheet. On the infinite sheet, the appropriate quantity is called the charge density, which is coulombs per meter squared. That means if you cut out a tiny piece, the charge on it will be sigma times the area of that piece. So there is positive charge everywhere here, and the number of coulombs per unit area is called sigma. These are standard. λ is coulombs per meter, σ is used for coulombs per unit area. The question is, what's going to be the electric field at some point away from that plane? Once again, I think we can all agree that the electric field at some point from the plane will not depend on where in front of the plane you are standing. Are you standing here or are you standing there? It doesn't matter, because it's an infinite plane. If I moved 1 inch--I'll tell you why it won't matter. If I moved 1 inch and the answer changed, I should get the same change if I didn't move and somebody moved the sheet 1 inch the other way. But when I move an infinite sheet the other way by 1 inch, it looks exactly the same. It's got to produce exactly the same field. So you can always ask, what will happen if I move to the left, the same as what will happen if the sheet moves to the right? The sheet moving to the right looks exactly like the sheet before. The answer won't change, therefore the answer won't change for you if you move to the left. I've got infinite plane below you. As long as you don't change the distance from the plane, you navigate perpendicular to it, no matter where you are, you will get the same answer. Same answer, meaning same direction of the field, same magnitude. And that direction has to be perpendicular to the plane, again for symmetry reasons. If you tilt it in any one direction, you have no reason to do it. For example, if you tilt it this way, I can take the infinite plane and rotate it, then the tilt will be in some other direction, maybe like that, but the rotated infinite plane looks the same. In other words, if the cause does not change, the effect should not change. If I can do certain things to the infinite plane that leave it invariant, then I can do the same transformation to the location of the point, and that shouldn't have a different answer. So the plane has the property that when you slide it up and down parallel to itself, or twist it and turn it, it looks the same, therefore the field pattern should have that property. Therefore the field has to be the same at all distances from the plane anywhere on top of the plane, and it's going to point this way. But you can also find out in a minute--by the way, you don't need any of the symmetry arguments. You just do the calculation by brute force, it will have these properties. But it's good to know what to anticipate, because maybe you made a mistake somewhere. It's good to know some broad features. So none of this is needed to calculate, even in that problem. Go ahead and find the electric field not where I found it, but 2 inches to the right. You'll find the answer looks the same. So those symmetry properties will come out of the wash, but it's good for you to anticipate that, and that's where you should look at the symmetry of the source. For example, the source was a ball of charge. You know if you rotate the ball, when I'm not looking to rotate the ball, it's going to look the same. That means the field pattern should have the property, when you rotate it, it looks the same, because the same cause should produce the same effect. Anyway, going to this problem now, let's find the electric field here. Okay, now this is going to be a stretch for me to draw, so I'm going to try, but you'll have to go look at some textbook if you want a really nice looking picture, but this is the best I can do. I take a ring of radius r and thickness dr. I take an annulus, and I ask, what will that ring do to this point? So let's take a tiny part of that ring, this guy. Well, for that, you just did what you did, you draw the line here. You'll produce a dE that looks like this. What is its magnitude? Magnitude is just given by Coulomb's law. The q there is sigma times the tiny area, dA. Let's call this dA. dA is the name for a small area. σ times dA is the name for a small charge. That charge will produce a force, 1/4Πε_0, square of the distance, r^(2) a^(2). Finally, here is where the symmetry comes in, can you see that for every section here, I can find an opposite section that will cancel everything but the part perpendicular to the plane? So I should only keep this portion of it. Namely, I should take the cosθ. The cosine of that θ is the distance a, just like in the other problem, a/(r^(2) a^(2))^(½). This is now dE. If you want, you can put this following symbol, dE_perp, meaning perpendicular to the plane. Yes? Student: Do you need to multiply by 2 again, because you're __________? Prof: Let's be careful. Her question was, should I multiply by 2, because of this guy here, right? In fact, I should multiply by all kinds of other numbers, because so far, I've found the field only due to this segment here. I've got to add the field due to that and that and that and so on, right? What will that contribution be? For every one of them, this factor, (r^(2) a^(2))^(3/2) is the same. They all contribute to the same factor, so when I added the shaded region, I'll just get the area of the shaded region. All these dAs, if you add them up, what will I get? It will be sigma over 4Πε _0. Now you've got to ask yourself, what's the area of an annulus of radius r and thickness dr? So take that annulus, take a pair of scissors and you cut it, and you stretch it out like that, it's going to look like this. This is dr and this is 2Πr. So the area of an annulus is just 2Πr dr. So the sum of all these areas is 2Πr dr and then I've got here (r^(2) a^(2))^(3/2). But now this is the dE, due to annulus of thickness of dr. Then I've got to integrate over all values of r, but r goes from 0 to infinity. So I have here σ/2ε_0 times rdr, divided by (r^(2) a^(2))^(3/2), 0 to infinity. Student: > Prof: Did I miss a pi? Student: Shouldn't it be sigma over ____ pi? Prof: There is a 2 pi here. Student: Oh, okay, yeah. Prof: 2ε_0. So do you understand what I did? I broke the plane into concentric rings and I took one ring and looking head on at that ring, I took a portion of that ring and see what field is produces. And I know that even though the field due to that is at an angle, the only part that's going to remain is the part perpendicular to the plane, because the counterpart to this one on the other side will produce a similar field with the opposite angle here that will cancel, so only the part perpendicular will survive. Then I found out that the contribution from every dA had exactly these factors. They all had the same r and they all had the same a, so some of all the dAs, all I have to add is 2Πr dr. And that's the contribution from this annulus, then I still have to look at annulus of every radius, so that's the integral over dr. Yes? Student: What happened to the a _______? I thought it was a over-- Prof: Oh, I'm sorry. It's there. Thank you. There is an a still here. Yeah, I would have caught that guy in a while, but I'm always happy when you do that. That's correct. Okay, so now about how this integral. Do you have any idea what you might do now? Yes? Student: Use substitution. Prof: Right. What substitution? "Use substitution" is a pretty safe answer, but you've got to go a little beyond that. Student: Substitute r^(2) a^(2) for the ________. Prof: Yes. You can do that in this problem, because there's an r on the top. If you didn't have the r, you couldn't do that, but now you can. I'll tell you how it works. First of all, you can always do that tanθ substitution even here. It will always work. The tanθ substitution, if you put it here, it will still work. You can go home and verify that, but I will do it a different way now. I will say, let w = r^(2), then dw is equal to 2r dr. So if I come here, I can write it as aσ/ 4ε_0. I borrow a 2 top and bottom to make it dw. w also goes from 0 to infinity, but now I get (w a^(2))^(3/2). Now this is simple integral, dx/x some number to some power is x^(n 1)/(n 1), but n is now -3 over 2. So you get aσ/ 4ε_0, divided by (w a^(2))^(-½), divided by -½, which is -2 on the top, and that goes from infinity to 0. So I'm not going to do this much slower than this. This is the kind of integral that you can see right away, or you can go and work out the details. This is something you should do. If you have trouble with such integrals, then you should work harder than people who don't have trouble with such integrals, because you should be able to do x^(n 1)/(n 1), and know that n 1 is -½, and when it comes upstairs, it becomes -2. Now if you look at this integral, in the upper limit omega's infinity, you get 1 over infinity, which is 0. The lower limit when omega is 0, you get 1/a, and that will cancel the a here, and you will get σ/2ε_0. So that's the final answer. So the electric field of this infinite plane, if you look at it from the side, looks like this. The σ/2ε_0. So what do you notice about this one that's interesting? Student: It doesn't depend on the distance. Prof: It doesn't fall with distance. No matter how far you go from this infinite plane, the field is the same. Again, each part of it makes a contribution that falls like 1 over distance squared. As you go further and further out, you might think the field should get weaker, right? How could it not get weaker? They're moving away from everything. At least with the line charge, it didn't go weaker like 1/r^(2), but it did get weaker. How can you go further and further from a plane? You are going further away from everybody? How could it not matter? Yes, any ideas? For example, if you go very close to the plane, what happens? If you go really close to the plane what happens is, the field in each section here looks like this. Therefore the part that's useful is very small. If you go further away, you get things like that. Maybe it's a little weaker, but the part that's useful, this one, is getting bigger. So by magic, these tendencies cancel in the end. It doesn't depend on distance. Now unless you do the integral, you will not know it doesn't depend on the distance, because you can give arguments for why it'll get weaker, arguments for why it'll get stronger. The fact that it'll precisely be independent of distance, you have to do the calculation. Yes. Student: What's the negative sign? Prof: Negative sign where, here? Student: Yes. Prof: -2 is there, but the upper limit is infinity. Student: Oh, okay. Prof: All right. Now here's the third problem, and the good news is, I'm not going to solve it for you, but I'll tell you what it is. Here is a solid ball of charge. It's got some charge density ρ coulombs per meter cubed. So ρ is the standard name. You use density for mass over volume and you use the same symbol rho for charge per unit volume. So somebody's assembled a blob of electrical charge, and 1 cubic meter of that has ρ coulombs. You want to find the field due to this one.