1 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 2 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. 3 00:00:16,160 --> 00:00:20,480 If you've ever had trouble choosing the right Gaussian surface, getting a quantity out of 4 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. 5 00:00:25,970 --> 00:00:29,330 This video is part of the Derivatives and Integrals video series. 6 00:00:29,330 --> 00:00:34,190 Derivatives and integrals are used to analyze the properties of a system. Derivatives describe 7 00:00:34,190 --> 00:00:38,999 local properties of systems, and integrals quantify their cumulative properties. 8 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 9 00:00:44,249 --> 00:00:48,239 I'll be talking with you about electric flux and Gauss' Law. 10 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 11 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 12 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 13 00:01:02,559 --> 00:01:04,510 is not taught in this video. 14 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 15 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 16 00:01:14,030 --> 00:01:19,549 lot of time talking about symmetry and how to use it to your advantage. 17 00:01:19,549 --> 00:01:24,020 We're going to start by thinking about what goes into flux and the interrelationships 18 00:01:24,020 --> 00:01:29,049 between these quantities, so that we can use flux more effectively. 19 00:01:29,049 --> 00:01:33,509 As a refresher, here are the two equations we'll be working with today: the definition 20 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 21 00:01:38,729 --> 00:01:43,320 slightly different notation, so take a quick look to make sure you're familiar with all 22 00:01:43,320 --> 00:01:46,770 the pieces. 23 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 24 00:01:51,609 --> 00:01:57,810 more basic things. We need to be able to use vector areas, to categorize charge distributions, 25 00:01:57,810 --> 00:02:02,109 and to determine when and how the two can work together. 26 00:02:02,109 --> 00:02:07,359 The underlying reason we want to use symmetry is that integrals are difficult, significantly 27 00:02:07,359 --> 00:02:12,050 more difficult than derivatives. There are many situations in which we cannot determine 28 00:02:12,050 --> 00:02:17,140 the flux integral. Therefore, we seek to simplify. 29 00:02:17,140 --> 00:02:21,450 The integrals we're dealing with are surface integrals, also called area integrals. There 30 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 31 00:02:26,970 --> 00:02:32,440 an open integral, and a closed surface uses a closed integral. Flux can be defined for 32 00:02:32,440 --> 00:02:37,000 any surface, but Gauss' Law uses only closed integrals. 33 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 34 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. 35 00:02:49,500 --> 00:02:54,740 With closed surfaces we usually choose dA to point outward from the surface. 36 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 37 00:03:02,060 --> 00:03:07,840 the coordinate system you choose, with some being more complex than others. Sometimes 38 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, 39 00:03:14,120 --> 00:03:18,970 we can avoid having to do an integral at all. 40 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 41 00:03:25,090 --> 00:03:30,470 the dot product as a cosine. Under certain circumstances, we can remove the E from the 42 00:03:30,470 --> 00:03:36,200 integral. We can only do that when our electric field has uniform magnitude over the entire 43 00:03:36,200 --> 00:03:43,000 surface. You can see some examples on the bottom of the screen. 44 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 45 00:03:48,230 --> 00:03:53,340 electric field doesn't change. The left-hand example shows a field that always points in 46 00:03:53,340 --> 00:03:58,829 the same direction, and an area that always points in the same direction. The middle example 47 00:03:58,829 --> 00:04:03,800 has a field whose direction changes, but it always points outward, just like the area 48 00:04:03,800 --> 00:04:10,390 vector from the cylinder points outward. It still works. In the right-hand example, the 49 00:04:10,390 --> 00:04:15,800 field has a constant direction, but the area vector does not. The angle between them changes, 50 00:04:15,800 --> 00:04:19,370 so we couldn't remove the cosine from the integral. 51 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 52 00:04:26,350 --> 00:04:32,470 both conditions. The left-hand and center examples only fill one condition each. The 53 00:04:32,470 --> 00:04:38,500 example on the right has both, and would be ideal to use. 54 00:04:38,500 --> 00:04:43,930 Before we move on, a reminder about flux and angles. Only electric field lines that actually 55 00:04:43,930 --> 00:04:49,010 pass through a surface provide flux. When the surface is parallel to the field, no flux 56 00:04:49,010 --> 00:04:54,080 is provided. This is particularly useful for some three-dimensional surfaces, such as the 57 00:04:54,080 --> 00:04:58,730 empty cylinder to the right. If we remember this, we can really simplify our integrals. 58 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, 59 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. 60 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 61 00:05:01,560 --> 00:05:01,840 a number with no direction. 62 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 63 00:05:06,720 --> 00:05:11,680 the circumstances that make that possible. We need to look at electric charge distributions 64 00:05:11,680 --> 00:05:13,400 and their symmetry. 65 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 66 00:05:19,070 --> 00:05:24,040 take other shapes. It might be stretched into a line, spread throughout a volume, or spread 67 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. 68 00:05:30,480 --> 00:05:35,490 Charge is spread on a physical object. You can stretch the charge out into lines, spread 69 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 70 00:05:40,970 --> 00:05:43,220 it around on an object. 71 00:05:43,220 --> 00:05:47,800 An object has symmetry when it has exactly similar parts that are either facing each 72 00:05:47,800 --> 00:05:53,290 other, or arranged around an axis. On the screen are some pictures of objects that have 73 00:05:53,290 --> 00:05:54,510 symmetry. 74 00:05:54,510 --> 00:06:01,310 There are three categories that we'll investigate, and three types of symmetry to consider. Our 75 00:06:01,310 --> 00:06:05,460 hope is to have a highly symmetric charge distribution, with any of the three types 76 00:06:05,460 --> 00:06:09,389 of symmetry. That's when Gauss' Law is easiest to apply. 77 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 78 00:06:13,490 --> 00:06:16,280 which we can use Gauss' Law. 79 00:06:16,280 --> 00:06:21,330 These three charge distributions have visible symmetry, but it's not enough. We need a charge 80 00:06:21,330 --> 00:06:26,580 distribution so symmetric that the field it generates is also symmetric. 81 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. 82 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 83 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 84 00:06:43,310 --> 00:06:48,139 places. There's some symmetry here, but not enough to allow us to pull everything out 85 00:06:48,139 --> 00:06:50,870 of the integral. 86 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 87 00:06:57,639 --> 00:07:03,419 infinite cylindrical shell of charge. No doubt you've used Gauss' Law with these charge distributions 88 00:07:03,419 --> 00:07:09,639 before. How can we describe such distributions and find others like them? 89 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 90 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 91 00:07:20,510 --> 00:07:26,699 identical. Move along the cylinder or around it, and it looks the same. Rotate around the 92 00:07:26,699 --> 00:07:31,990 sphere around the equator or the poles, and you can't tell the difference. 93 00:07:31,990 --> 00:07:37,580 We can also add similar objects together or scale them however we like. By integrating 94 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 95 00:07:43,199 --> 00:07:44,400 dolls. 96 00:07:44,400 --> 00:07:50,210 Of course, perfectly symmetric charge distributions are rare in real life. This is probably because 97 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 98 00:07:56,510 --> 00:08:02,040 the charge distributions are close to one of our symmetric cases. We can usually obtain 99 00:08:02,040 --> 00:08:05,800 excellent estimates of the true electric field in this way. 100 00:08:05,800 --> 00:08:10,699 Two approximations are particularly common: when the object is so large that we can treat 101 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 102 00:08:16,430 --> 00:08:23,150 become unimportant. In these cases, we often treat the object as being an infinite plane. 103 00:08:23,150 --> 00:08:28,370 Here's an example of the first approximation. Compare the distance from your point of interest 104 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 105 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, 106 00:08:39,999 --> 00:08:42,938 even if your plane isn't really infinite. 107 00:08:42,938 --> 00:08:48,089 To use the other approximation, we want a situation where the curvature is very small 108 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 109 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 110 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. 111 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 112 00:09:11,290 --> 00:09:12,699 Gaussian surface. 113 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. 114 00:09:18,249 --> 00:09:23,430 After that, it's your turn to pick the best surface for a given situation. 115 00:09:23,430 --> 00:09:28,350 In order to simplify our integrals as much as possible, we seek out situations with as 116 00:09:28,350 --> 00:09:33,110 much symmetry as we can get. The symmetry in the charge distribution isn't something 117 00:09:33,110 --> 00:09:38,439 we can control, but our choice of surface will determine whether there is symmetry in 118 00:09:38,439 --> 00:09:43,410 how the electric field meets the Gaussian surface. And that's always the key: you choose 119 00:09:43,410 --> 00:09:47,980 the surface to make the problem easier for you. Finally, it's important to remember that 120 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. 121 00:09:53,480 --> 00:10:00,480 Here are three charge distributions: a long cylinder, a sphere, and a flat plane. To maximize 122 00:10:00,699 --> 00:10:06,499 the effects of symmetry, we choose surfaces that match up well with the charge distributions. 123 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 124 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 125 00:10:17,619 --> 00:10:22,949 a box-shaped surface. Sometimes people will use a cylindrical surface instead, and count 126 00:10:22,949 --> 00:10:29,100 the flux that goes through the top of the cylinder. Both approaches are valid. 127 00:10:29,100 --> 00:10:35,490 Here is your first challenge: choose the Gaussian surface that will best fit this charge distribution. 128 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. 129 00:10:41,879 --> 00:10:46,220 What surface would you choose? 130 00:10:46,220 --> 00:10:53,220 A sphere inside our existing sphere will be easiest. This example helps us remember that 131 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 132 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, 133 00:11:03,379 --> 00:11:07,980 not the whole charge of the sphere. 134 00:11:07,980 --> 00:11:13,240 Here's challenge number two. We would like to know the electric charge contained in a 135 00:11:13,240 --> 00:11:17,059 thundercloud. Let's say that the inside of the cloud has an electric field that looks 136 00:11:17,059 --> 00:11:18,860 like this. 137 00:11:18,860 --> 00:11:25,860 What surface would you choose? 138 00:11:27,689 --> 00:11:33,829 A box-shaped surface will take advantage of the Cartesian symmetry in this electric field. 139 00:11:33,829 --> 00:11:37,959 Because of the alignment between the sides of the box and the electric field, they will 140 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 141 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. 142 00:11:51,249 --> 00:11:55,540 Here's challenge number three: finding the electric field at a certain distance from 143 00:11:55,540 --> 00:11:58,869 the center of charged disc. 144 00:11:58,869 --> 00:12:05,869 What surface would you choose? 145 00:12:06,959 --> 00:12:11,600 No surface will work! The problem is that there's not enough symmetry in this situation 146 00:12:11,600 --> 00:12:13,959 to pick an appropriate surface. 147 00:12:13,959 --> 00:12:19,739 If we can't get an exact solution, we would like to approximate the field. Unfortunately, 148 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 149 00:12:24,199 --> 00:12:25,730 approximation. 150 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 151 00:12:30,899 --> 00:12:35,929 other techniques, such as integration and Coulomb's Law. An example can be found in 152 00:12:35,929 --> 00:12:39,319 the supplemental materials for this video. 153 00:12:39,319 --> 00:12:44,300 Here is the final challenge. A top and side view of the field are shown. 154 00:12:44,300 --> 00:12:51,300 What surface would align well with this field? 155 00:12:52,589 --> 00:12:57,769 A cylindrical surface would work very well with this circular field. You can see that 156 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 157 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 158 00:13:09,569 --> 00:13:14,589 points. We don't even need to determine the area of the cylinder. 159 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 160 00:13:20,559 --> 00:13:26,149 and negative charges together could create this field, but as it turns out, they cannot. 161 00:13:26,149 --> 00:13:30,529 Fields that look like this are created by a changing magnetic field, using Faraday's 162 00:13:30,529 --> 00:13:34,559 Law, which you will learn more about later in your course. 163 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 164 00:13:38,749 --> 00:13:44,139 every situation is perfectly symmetric. When you come to these situations, look for the 165 00:13:44,139 --> 00:13:50,649 following things to help you out. First, an area vector that you can define easily. Second, 166 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, 167 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 168 00:14:01,730 --> 00:14:07,809 uniform. The more of these you can take advantage of, the better. 169 00:14:07,809 --> 00:14:08,939 Let's review. 170 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 171 00:14:13,519 --> 00:14:20,209 with Gauss' Law and flux. Ideally, we want a situation where the charge, the field it 172 00:14:20,209 --> 00:14:25,049 creates, and the Gaussian surface we choose all have the same symmetry, so that we can 173 00:14:25,049 --> 00:14:29,600 simplify the integral as much as possible. 174 00:14:29,600 --> 00:14:35,660 For the final segment of this video, we wanted to answer a common question about electromagnetism. 175 00:14:35,660 --> 00:14:40,549 What is flux? What does it mean? How can we understand it? 176 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 177 00:14:45,499 --> 00:14:52,499 and students think about flux. Here's what we found out. 178 00:14:59,110 --> 00:15:06,110 Thank you 179 00:15:19,879 --> 00:15:26,879 for watching this video. We hope that it has 180 00:17:32,190 --> 00:17:36,620 improved your understanding of flux and Gauss' Law. Good luck in your future exploration 181 00:17:36,620 --> 00:17:42,490 of electromagnetism.