WEBVTT 00:00:03.350 --> 00:00:10.350 What do rain, bees, and vectors have in common? They're all ways that MIT physicists visualize 00:00:10.600 --> 00:00:16.160 the concept of "flux." In this video, we'll explore flux as it relates to Gauss' law. 00:00:16.160 --> 00:00:20.480 If you've ever had trouble choosing the right Gaussian surface, getting a quantity out of 00:00:20.480 --> 00:00:25.970 an integral, or just wanted to know what flux is, get ready: this video is for you. 00:00:25.970 --> 00:00:29.330 This video is part of the Derivatives and Integrals video series. 00:00:29.330 --> 00:00:34.190 Derivatives and integrals are used to analyze the properties of a system. Derivatives describe 00:00:34.190 --> 00:00:38.999 local properties of systems, and integrals quantify their cumulative properties. 00:00:38.999 --> 00:00:44.249 "Hello. My name is Peter Fisher. I am a professor in the physics department at MIT, and today 00:00:44.249 --> 00:00:48.239 I'll be talking with you about electric flux and Gauss' Law. 00:00:48.239 --> 00:00:53.149 By this time in your course you've seen and used Gauss' Law quite a bit. You have a good 00:00:53.149 --> 00:00:57.899 handle on the electric field, what it does and how it works. We will also assume that 00:00:57.899 --> 00:01:02.559 you know how to take an area integral, a skill that is important to Gauss' Law but which 00:01:02.559 --> 00:01:04.510 is not taught in this video. 00:01:04.510 --> 00:01:09.180 Our end goal is to improve your ability to use Gauss' Law. To do this, we'll help you 00:01:09.180 --> 00:01:14.030 develop an organized view of electric flux and what goes into it. We'll also spend a 00:01:14.030 --> 00:01:19.549 lot of time talking about symmetry and how to use it to your advantage. 00:01:19.549 --> 00:01:24.020 We're going to start by thinking about what goes into flux and the interrelationships 00:01:24.020 --> 00:01:29.049 between these quantities, so that we can use flux more effectively. 00:01:29.049 --> 00:01:33.509 As a refresher, here are the two equations we'll be working with today: the definition 00:01:33.509 --> 00:01:38.729 of flux and Gauss' Law. You may have seen these written with a double integral, or with 00:01:38.729 --> 00:01:43.320 slightly different notation, so take a quick look to make sure you're familiar with all 00:01:43.320 --> 00:01:46.770 the pieces. 00:01:46.770 --> 00:01:51.609 To really use symmetry to our best advantage in Gauss' Law, we need to understand a few 00:01:51.609 --> 00:01:57.810 more basic things. We need to be able to use vector areas, to categorize charge distributions, 00:01:57.810 --> 00:02:02.109 and to determine when and how the two can work together. 00:02:02.109 --> 00:02:07.359 The underlying reason we want to use symmetry is that integrals are difficult, significantly 00:02:07.359 --> 00:02:12.050 more difficult than derivatives. There are many situations in which we cannot determine 00:02:12.050 --> 00:02:17.140 the flux integral. Therefore, we seek to simplify. 00:02:17.140 --> 00:02:21.450 The integrals we're dealing with are surface integrals, also called area integrals. There 00:02:21.450 --> 00:02:26.970 are two basic kinds, open and closed, as you can see on the screen. An open surface uses 00:02:26.970 --> 00:02:32.440 an open integral, and a closed surface uses a closed integral. Flux can be defined for 00:02:32.440 --> 00:02:37.000 any surface, but Gauss' Law uses only closed integrals. 00:02:37.000 --> 00:02:43.220 When we take the integral, we use dA, a tiny piece of the area. dA is a vector piece of 00:02:43.220 --> 00:02:49.500 the area, so it has a direction to it as well. dA always points perpendicular to the surface. 00:02:49.500 --> 00:02:54.740 With closed surfaces we usually choose dA to point outward from the surface. 00:02:54.740 --> 00:03:01.740 dA usually gets represented as two other differentials, such as dx times dy. Which two depends on 00:03:02.060 --> 00:03:07.840 the coordinate system you choose, with some being more complex than others. Sometimes 00:03:07.840 --> 00:03:14.120 we will need these, so it's good to have them. The good part is that if we use symmetry correctly, 00:03:14.120 --> 00:03:18.970 we can avoid having to do an integral at all. 00:03:18.970 --> 00:03:25.090 Let's pursue our goal of not doing an integral. We have this integral of E dot dA. Let's write 00:03:25.090 --> 00:03:30.470 the dot product as a cosine. Under certain circumstances, we can remove the E from the 00:03:30.470 --> 00:03:36.200 integral. We can only do that when our electric field has uniform magnitude over the entire 00:03:36.200 --> 00:03:43.000 surface. You can see some examples on the bottom of the screen. 00:03:43.000 --> 00:03:48.230 We can remove the cosine(theta) from our integral only when the angle between the area and the 00:03:48.230 --> 00:03:53.340 electric field doesn't change. The left-hand example shows a field that always points in 00:03:53.340 --> 00:03:58.829 the same direction, and an area that always points in the same direction. The middle example 00:03:58.829 --> 00:04:03.800 has a field whose direction changes, but it always points outward, just like the area 00:04:03.800 --> 00:04:10.390 vector from the cylinder points outward. It still works. In the right-hand example, the 00:04:10.390 --> 00:04:15.800 field has a constant direction, but the area vector does not. The angle between them changes, 00:04:15.800 --> 00:04:19.370 so we couldn't remove the cosine from the integral. 00:04:19.370 --> 00:04:26.350 If we want to move both the field and the cosine out of the integral, we need to fulfill 00:04:26.350 --> 00:04:32.470 both conditions. The left-hand and center examples only fill one condition each. The 00:04:32.470 --> 00:04:38.500 example on the right has both, and would be ideal to use. 00:04:38.500 --> 00:04:43.930 Before we move on, a reminder about flux and angles. Only electric field lines that actually 00:04:43.930 --> 00:04:49.010 pass through a surface provide flux. When the surface is parallel to the field, no flux 00:04:49.010 --> 00:04:54.080 is provided. This is particularly useful for some three-dimensional surfaces, such as the 00:04:54.080 --> 00:04:58.730 empty cylinder to the right. If we remember this, we can really simplify our integrals. 00:04:58.730 --> 00:04:59.690 Another thing to remember: Flux is a scalar. As long as the field strength is the same, 00:04:59.690 --> 00:05:00.660 and the angle between the field and surface stays the same, the flux will be the same. 00:05:00.660 --> 00:05:01.560 We don't have to worry about the flux pointing in a particular direction, because it's just 00:05:01.560 --> 00:05:01.840 a number with no direction. 00:05:01.840 --> 00:05:06.720 Now we know how we would like the field and the area to line up. Next we need to understand 00:05:06.720 --> 00:05:11.680 the circumstances that make that possible. We need to look at electric charge distributions 00:05:11.680 --> 00:05:13.400 and their symmetry. 00:05:13.400 --> 00:05:19.070 Electric charge can come as a single point, as in an electron or proton, but it can also 00:05:19.070 --> 00:05:24.040 take other shapes. It might be stretched into a line, spread throughout a volume, or spread 00:05:24.040 --> 00:05:30.480 over a surface. Not a Gaussian surface - remember, that's something we make up to solve a problem. 00:05:30.480 --> 00:05:35.490 Charge is spread on a physical object. You can stretch the charge out into lines, spread 00:05:35.490 --> 00:05:40.970 it on thick wires or spherical shells, build a solid ball out of it, or even just smear 00:05:40.970 --> 00:05:43.220 it around on an object. 00:05:43.220 --> 00:05:47.800 An object has symmetry when it has exactly similar parts that are either facing each 00:05:47.800 --> 00:05:53.290 other, or arranged around an axis. On the screen are some pictures of objects that have 00:05:53.290 --> 00:05:54.510 symmetry. 00:05:54.510 --> 00:06:01.310 There are three categories that we'll investigate, and three types of symmetry to consider. Our 00:06:01.310 --> 00:06:05.460 hope is to have a highly symmetric charge distribution, with any of the three types 00:06:05.460 --> 00:06:09.389 of symmetry. That's when Gauss' Law is easiest to apply. 00:06:09.389 --> 00:06:13.490 These examples have little or no symmetry to them, and are not the sort of thing for 00:06:13.490 --> 00:06:16.280 which we can use Gauss' Law. 00:06:16.280 --> 00:06:21.330 These three charge distributions have visible symmetry, but it's not enough. We need a charge 00:06:21.330 --> 00:06:26.580 distribution so symmetric that the field it generates is also symmetric. 00:06:26.580 --> 00:06:32.290 Let us show why. Let's say we have a charged cube, which is certainly a symmetric object. 00:06:32.290 --> 00:06:37.669 We can draw the field from it fairly easily. If we place a cubical Gaussian surface around 00:06:37.669 --> 00:06:43.310 the charge, we can see the problem: the field and the surface have different angles in different 00:06:43.310 --> 00:06:48.139 places. There's some symmetry here, but not enough to allow us to pull everything out 00:06:48.139 --> 00:06:50.870 of the integral. 00:06:50.870 --> 00:06:57.639 These cases are ideal: a point charge, a spherical shell, an infinite sheet of charge, or an 00:06:57.639 --> 00:07:03.419 infinite cylindrical shell of charge. No doubt you've used Gauss' Law with these charge distributions 00:07:03.419 --> 00:07:09.639 before. How can we describe such distributions and find others like them? 00:07:09.639 --> 00:07:14.630 All three of those have a special quality: if you move in two directions, they look exactly 00:07:14.630 --> 00:07:20.510 the same. No matter how far you move across the plane in the x or y directions, it looks 00:07:20.510 --> 00:07:26.699 identical. Move along the cylinder or around it, and it looks the same. Rotate around the 00:07:26.699 --> 00:07:31.990 sphere around the equator or the poles, and you can't tell the difference. 00:07:31.990 --> 00:07:37.580 We can also add similar objects together or scale them however we like. By integrating 00:07:37.580 --> 00:07:43.199 the fields from a lot of shells, we can create a solid object, just like these Russian nesting 00:07:43.199 --> 00:07:44.400 dolls. 00:07:44.400 --> 00:07:50.210 Of course, perfectly symmetric charge distributions are rare in real life. This is probably because 00:07:50.210 --> 00:07:56.510 infinitely long cylinders are so hard to find. Thus we often use Gauss' Law in cases where 00:07:56.510 --> 00:08:02.040 the charge distributions are close to one of our symmetric cases. We can usually obtain 00:08:02.040 --> 00:08:05.800 excellent estimates of the true electric field in this way. 00:08:05.800 --> 00:08:10.699 Two approximations are particularly common: when the object is so large that we can treat 00:08:10.699 --> 00:08:16.430 it as being infinite, or when we are so close to the object that its large-scale features 00:08:16.430 --> 00:08:23.150 become unimportant. In these cases, we often treat the object as being an infinite plane. 00:08:23.150 --> 00:08:28.370 Here's an example of the first approximation. Compare the distance from your point of interest 00:08:28.370 --> 00:08:34.250 to the object, to the size of the object itself. If the smallest dimension of the object is 00:08:34.250 --> 00:08:39.999 ten times your distance from it, you can rely on a very good estimation from Gauss' Law, 00:08:39.999 --> 00:08:42.938 even if your plane isn't really infinite. 00:08:42.938 --> 00:08:48.089 To use the other approximation, we want a situation where the curvature is very small 00:08:48.089 --> 00:08:52.709 in the place we care about. For example, it would be very difficult for us to find the 00:08:52.709 --> 00:08:58.699 field at this point. However, if we zoom in, we can reach a point where the surface looks 00:08:58.699 --> 00:09:05.699 like a flat plane. Finding an estimate for the field at this point will be much easier. 00:09:06.589 --> 00:09:11.290 Now we have the background we need. It's time to put all the pieces together and pick our 00:09:11.290 --> 00:09:12.699 Gaussian surface. 00:09:12.699 --> 00:09:18.249 We'll do a quick review in case you missed something, and give you an example or two. 00:09:18.249 --> 00:09:23.430 After that, it's your turn to pick the best surface for a given situation. 00:09:23.430 --> 00:09:28.350 In order to simplify our integrals as much as possible, we seek out situations with as 00:09:28.350 --> 00:09:33.110 much symmetry as we can get. The symmetry in the charge distribution isn't something 00:09:33.110 --> 00:09:38.439 we can control, but our choice of surface will determine whether there is symmetry in 00:09:38.439 --> 00:09:43.410 how the electric field meets the Gaussian surface. And that's always the key: you choose 00:09:43.410 --> 00:09:47.980 the surface to make the problem easier for you. Finally, it's important to remember that 00:09:47.980 --> 00:09:53.480 in some cases, Gauss' Law won't be helpful, and it's good to look for another method. 00:09:53.480 --> 00:10:00.480 Here are three charge distributions: a long cylinder, a sphere, and a flat plane. To maximize 00:10:00.699 --> 00:10:06.499 the effects of symmetry, we choose surfaces that match up well with the charge distributions. 00:10:06.499 --> 00:10:11.629 To find the field near a cylinder, we surround it with another cylinder. To find the field 00:10:11.629 --> 00:10:17.619 near a sphere, we encase it in a sphere. To find the field near a charged plane, we use 00:10:17.619 --> 00:10:22.949 a box-shaped surface. Sometimes people will use a cylindrical surface instead, and count 00:10:22.949 --> 00:10:29.100 the flux that goes through the top of the cylinder. Both approaches are valid. 00:10:29.100 --> 00:10:35.490 Here is your first challenge: choose the Gaussian surface that will best fit this charge distribution. 00:10:35.490 --> 00:10:41.879 We want to know the electric field at a certain distance from the center of this solid sphere. 00:10:41.879 --> 00:10:46.220 What surface would you choose? 00:10:46.220 --> 00:10:53.220 A sphere inside our existing sphere will be easiest. This example helps us remember that 00:10:54.920 --> 00:10:59.170 the Gaussian surface can be inside an object. The area of the surface will be the usual 00:10:59.170 --> 00:11:03.379 four thirds pi r cubed. We'll need to remember to use only the charge inside our volume, 00:11:03.379 --> 00:11:07.980 not the whole charge of the sphere. 00:11:07.980 --> 00:11:13.240 Here's challenge number two. We would like to know the electric charge contained in a 00:11:13.240 --> 00:11:17.059 thundercloud. Let's say that the inside of the cloud has an electric field that looks 00:11:17.059 --> 00:11:18.860 like this. 00:11:18.860 --> 00:11:25.860 What surface would you choose? 00:11:27.689 --> 00:11:33.829 A box-shaped surface will take advantage of the Cartesian symmetry in this electric field. 00:11:33.829 --> 00:11:37.959 Because of the alignment between the sides of the box and the electric field, they will 00:11:37.959 --> 00:11:43.139 not contribute to the flux integral. We can use just the area of the top and bottom of 00:11:43.139 --> 00:11:50.139 the box, and the field at those locations, to find the flux. This is a common technique. 00:11:51.249 --> 00:11:55.540 Here's challenge number three: finding the electric field at a certain distance from 00:11:55.540 --> 00:11:58.869 the center of charged disc. 00:11:58.869 --> 00:12:05.869 What surface would you choose? 00:12:06.959 --> 00:12:11.600 No surface will work! The problem is that there's not enough symmetry in this situation 00:12:11.600 --> 00:12:13.959 to pick an appropriate surface. 00:12:13.959 --> 00:12:19.739 If we can't get an exact solution, we would like to approximate the field. Unfortunately, 00:12:19.739 --> 00:12:24.199 if we look at the distances involved, we are not close enough to the disc to make a good 00:12:24.199 --> 00:12:25.730 approximation. 00:12:25.730 --> 00:12:30.899 In the end, this is not a good Gauss' Law problem. It is solved much more easily through 00:12:30.899 --> 00:12:35.929 other techniques, such as integration and Coulomb's Law. An example can be found in 00:12:35.929 --> 00:12:39.319 the supplemental materials for this video. 00:12:39.319 --> 00:12:44.300 Here is the final challenge. A top and side view of the field are shown. 00:12:44.300 --> 00:12:51.300 What surface would align well with this field? 00:12:52.589 --> 00:12:57.769 A cylindrical surface would work very well with this circular field. You can see that 00:12:57.769 --> 00:13:03.350 the field is perpendicular to the area vector at all points. Our dot product is now giving 00:13:03.350 --> 00:13:09.569 us a cosine of ninety degrees at all times. This means a zero value for the flux at all 00:13:09.569 --> 00:13:14.589 points. We don't even need to determine the area of the cylinder. 00:13:14.589 --> 00:13:20.559 If our flux is zero, the total charge inside must also be zero. We might guess that positive 00:13:20.559 --> 00:13:26.149 and negative charges together could create this field, but as it turns out, they cannot. 00:13:26.149 --> 00:13:30.529 Fields that look like this are created by a changing magnetic field, using Faraday's 00:13:30.529 --> 00:13:34.559 Law, which you will learn more about later in your course. 00:13:34.559 --> 00:13:38.749 There are some times when you'll just have to integrate; there's no way around it. Not 00:13:38.749 --> 00:13:44.139 every situation is perfectly symmetric. When you come to these situations, look for the 00:13:44.139 --> 00:13:50.649 following things to help you out. First, an area vector that you can define easily. Second, 00:13:50.649 --> 00:13:56.959 an electric field that you can easily find or that is defined for you in a problem. Third, 00:13:56.959 --> 00:14:01.730 see if you can create a surface where either the magnitude of the field or its angle is 00:14:01.730 --> 00:14:07.809 uniform. The more of these you can take advantage of, the better. 00:14:07.809 --> 00:14:08.939 Let's review. 00:14:08.939 --> 00:14:13.519 The more symmetry we can take advantage of in our problem, the easier time we'll have 00:14:13.519 --> 00:14:20.209 with Gauss' Law and flux. Ideally, we want a situation where the charge, the field it 00:14:20.209 --> 00:14:25.049 creates, and the Gaussian surface we choose all have the same symmetry, so that we can 00:14:25.049 --> 00:14:29.600 simplify the integral as much as possible. 00:14:29.600 --> 00:14:35.660 For the final segment of this video, we wanted to answer a common question about electromagnetism. 00:14:35.660 --> 00:14:40.549 What is flux? What does it mean? How can we understand it? 00:14:40.549 --> 00:14:45.499 We took a video camera into the physics department at MIT to find out how physics professors 00:14:45.499 --> 00:14:52.499 and students think about flux. Here's what we found out. 00:14:59.110 --> 00:15:06.110 Thank you 00:15:19.879 --> 00:15:26.879 for watching this video. We hope that it has 00:17:32.190 --> 00:17:36.620 improved your understanding of flux and Gauss' Law. Good luck in your future exploration 00:17:36.620 --> 00:17:42.490 of electromagnetism.