1 00:00:00,050 --> 00:00:01,770 The following content is provided 2 00:00:01,770 --> 00:00:04,010 under a Creative Commons license. 3 00:00:04,010 --> 00:00:06,860 Your support will help MIT OpenCourseWare continue 4 00:00:06,860 --> 00:00:10,720 to offer high-quality educational resources for free. 5 00:00:10,720 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,209 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,209 --> 00:00:17,834 at ocw.mit.edu. 8 00:00:22,900 --> 00:00:24,210 PROFESSOR: Welcome back. 9 00:00:24,210 --> 00:00:32,836 And today we are going to look at a harder situation. 10 00:00:32,836 --> 00:00:37,930 At oscillations waves in the electromagnetic field. 11 00:00:42,890 --> 00:00:46,200 Why I say it's harder, for many reasons. 12 00:00:46,200 --> 00:00:49,150 First of all, so far we've always 13 00:00:49,150 --> 00:00:53,000 considered situations which we could either visualize or had 14 00:00:53,000 --> 00:00:59,100 some sensual way of getting a feel for what 15 00:00:59,100 --> 00:01:01,790 the physical situation is. 16 00:01:01,790 --> 00:01:05,069 When it comes to the electromagnetic field, 17 00:01:05,069 --> 00:01:09,570 as you well know, we can't see it, sense it, at all. 18 00:01:09,570 --> 00:01:14,520 And the only way to describe it is, in fact, 19 00:01:14,520 --> 00:01:16,230 in terms of mathematics. 20 00:01:16,230 --> 00:01:19,470 So there isn't, first, a word-- a description 21 00:01:19,470 --> 00:01:21,395 by analogy with what we see around. 22 00:01:24,470 --> 00:01:28,840 Secondly, it's more complicated. 23 00:01:28,840 --> 00:01:34,090 These are oscillations in three dimensions. 24 00:01:34,090 --> 00:01:37,570 And, as you well know, there both electric and magnetic 25 00:01:37,570 --> 00:01:38,630 fields. 26 00:01:38,630 --> 00:01:43,140 Overall, it is just much more difficult situation. 27 00:01:43,140 --> 00:01:51,760 So first of all, I start by a mathematical description 28 00:01:51,760 --> 00:01:53,190 of this system. 29 00:01:53,190 --> 00:01:55,760 Because, as I say, there is no other way 30 00:01:55,760 --> 00:01:58,770 we know of discussing it. 31 00:01:58,770 --> 00:02:05,710 And the mathematical description of the electromagnetic field, 32 00:02:05,710 --> 00:02:10,130 as you all know, are the so-called Maxwell's equations. 33 00:02:10,130 --> 00:02:17,750 I've written here the four Maxwell's equations for vacuum. 34 00:02:17,750 --> 00:02:23,230 So this is what the electric and magnetic fields 35 00:02:23,230 --> 00:02:26,240 have to satisfy. 36 00:02:26,240 --> 00:02:32,250 And I'm just reminding you that the definition 37 00:02:32,250 --> 00:02:34,440 of what electric and magnetic field-- 38 00:02:34,440 --> 00:02:38,510 the operational definition comes from the Lorentz force. 39 00:02:38,510 --> 00:02:41,700 Basically, this is just quickly to remind you, 40 00:02:41,700 --> 00:02:46,980 if I have a charge in vacuum and if it experiences a force, 41 00:02:46,980 --> 00:02:50,110 I know there is an electric field there. 42 00:02:50,110 --> 00:02:52,620 On the other hand, if it experiences 43 00:02:52,620 --> 00:02:55,640 a force when it's moving, then I know 44 00:02:55,640 --> 00:02:57,650 that there is a magnetic field. 45 00:02:57,650 --> 00:03:00,990 So this tells us that here, although we can't see it, 46 00:03:00,990 --> 00:03:05,140 there is an electromagnetic field. 47 00:03:05,140 --> 00:03:09,980 If one looks at these equations and plays around with them, 48 00:03:09,980 --> 00:03:15,500 one find that the electromagnetic field actually 49 00:03:15,500 --> 00:03:19,530 satisfy wave equations. 50 00:03:19,530 --> 00:03:22,840 This is the wave equation for the-- three-dimensional wave 51 00:03:22,840 --> 00:03:25,140 equation for the electric field. 52 00:03:25,140 --> 00:03:28,590 And this is for the magnetic field, where c is the phase 53 00:03:28,590 --> 00:03:32,630 velocity as always, and in the case of electromagnetism 54 00:03:32,630 --> 00:03:34,220 c is given by that. 55 00:03:34,220 --> 00:03:36,140 That's the speed of light, or the speed 56 00:03:36,140 --> 00:03:40,110 of electromagnetic waves. 57 00:03:40,110 --> 00:03:47,500 Now, so what this tells us, is that in vacuum, you 58 00:03:47,500 --> 00:03:51,170 can have excitations, oscillations 59 00:03:51,170 --> 00:03:54,000 of the electromagnetic and magnetic fields, 60 00:03:54,000 --> 00:03:55,920 which propagate. 61 00:03:55,920 --> 00:03:59,020 And we have all of the wave phenomena 62 00:03:59,020 --> 00:04:02,950 we've learned for other systems. 63 00:04:02,950 --> 00:04:08,780 The thing to keep in mind is that whatever 64 00:04:08,780 --> 00:04:11,960 the solution of the system is, whatever is propagating, 65 00:04:11,960 --> 00:04:17,950 it must satisfy all of these equations. 66 00:04:17,950 --> 00:04:21,329 Not every situation has to satisfy this. 67 00:04:21,329 --> 00:04:26,760 This is a subset of the infinite possibilities that 68 00:04:26,760 --> 00:04:29,970 are allowed by Maxwell's equations. 69 00:04:29,970 --> 00:04:31,300 OK. 70 00:04:31,300 --> 00:04:38,260 So now, instead of doing solutions 71 00:04:38,260 --> 00:04:42,870 to some specific situations with a specific boundary condition, 72 00:04:42,870 --> 00:04:46,660 et cetera, since it's already much more difficult, 73 00:04:46,660 --> 00:04:51,850 all I will do today is see how we 74 00:04:51,850 --> 00:04:56,280 can identify solutions of these equations. 75 00:04:56,280 --> 00:04:58,720 What kind of waves they correspond to. 76 00:04:58,720 --> 00:05:02,330 Or vice versa, if you want to describe 77 00:05:02,330 --> 00:05:05,310 in terms of mathematics some particular wave, 78 00:05:05,310 --> 00:05:08,070 how do we do that? 79 00:05:08,070 --> 00:05:12,820 That is the kind of problems I will be discussing today. 80 00:05:12,820 --> 00:05:15,920 So, let me come to the first problem. 81 00:05:22,830 --> 00:05:25,800 And probably using the word problem is a misnomer. 82 00:05:25,800 --> 00:05:28,380 The description. 83 00:05:28,380 --> 00:05:33,700 I'll consider first progressive wave 84 00:05:33,700 --> 00:05:35,165 solutions of these equations. 85 00:05:38,680 --> 00:05:47,060 Suppose we know that there is an electric field, which 86 00:05:47,060 --> 00:05:51,410 is a propagating electric field, sinusoidal. 87 00:05:51,410 --> 00:05:52,760 All right? 88 00:05:52,760 --> 00:05:59,820 I assure you, this does not contradict Maxwell's equations. 89 00:05:59,820 --> 00:06:01,440 You can try it. 90 00:06:01,440 --> 00:06:03,750 All right? 91 00:06:03,750 --> 00:06:06,195 It's not complete, as you'll see in a moment. 92 00:06:09,100 --> 00:06:12,360 The question here is, if you have an electric field 93 00:06:12,360 --> 00:06:17,460 like that, can we describe as well as 94 00:06:17,460 --> 00:06:21,860 possible in words, what kind of a wave this corresponds to? 95 00:06:21,860 --> 00:06:25,920 And secondly, answer the question 96 00:06:25,920 --> 00:06:29,980 if this is a real electromagnetic wave 97 00:06:29,980 --> 00:06:35,320 in the vacuum, what must be the corresponding magnetic field? 98 00:06:35,320 --> 00:06:38,610 By itself, this equation does not 99 00:06:38,610 --> 00:06:41,580 satisfy all the Maxwell's equations. 100 00:06:41,580 --> 00:06:46,930 You need a corresponding magnetic field. 101 00:06:46,930 --> 00:06:49,810 So, let's look at that. 102 00:06:49,810 --> 00:06:55,820 First of all, we know that any function, which 103 00:06:55,820 --> 00:07:03,080 is a function of x plus or minus vt describes 104 00:07:03,080 --> 00:07:06,300 a progressive wave. 105 00:07:06,300 --> 00:07:08,880 It satisfies the classical wave equation, 106 00:07:08,880 --> 00:07:12,090 you can try it and see. 107 00:07:12,090 --> 00:07:17,040 If these two terms-- the x and the t terms-- 108 00:07:17,040 --> 00:07:23,220 are of opposite sign, then this describes a progressive wave, 109 00:07:23,220 --> 00:07:27,030 which goes in the plus-x direction. 110 00:07:27,030 --> 00:07:29,580 If they are the same sign, then it 111 00:07:29,580 --> 00:07:32,720 goes in the opposite direction. 112 00:07:32,720 --> 00:07:37,100 And again I say, plot any function like this, 113 00:07:37,100 --> 00:07:41,270 and see what happens as you change t. 114 00:07:41,270 --> 00:07:44,660 The shape of the function will not change. 115 00:07:44,660 --> 00:07:48,180 But it will move either to the left or to the right 116 00:07:48,180 --> 00:07:49,300 as you change the time. 117 00:07:52,110 --> 00:07:54,610 So we immediately see, since this 118 00:07:54,610 --> 00:08:00,800 is a cosine of this, which is of this form, if I divide by a, 119 00:08:00,800 --> 00:08:03,240 this is x minus b over a. 120 00:08:03,240 --> 00:08:05,120 And I could take the a outside that. 121 00:08:05,120 --> 00:08:08,180 So this is a progressive wave. 122 00:08:08,180 --> 00:08:11,650 These two have opposite signs. 123 00:08:11,650 --> 00:08:13,310 It's a function of x and t. 124 00:08:13,310 --> 00:08:16,190 So this is a progressive wave, which is moving or progressing 125 00:08:16,190 --> 00:08:17,710 in the x direction. 126 00:08:17,710 --> 00:08:21,250 They're opposite signs, so in the plus-x direction. 127 00:08:21,250 --> 00:08:25,610 So immediately I know that this is a progressive wave. 128 00:08:25,610 --> 00:08:27,320 It is a sinusoidal one. 129 00:08:27,320 --> 00:08:29,690 Well, this is a cosine function, right? 130 00:08:29,690 --> 00:08:35,970 It's a sinusoidal wave. 131 00:08:35,970 --> 00:08:43,510 Now we know if I divide by a, I get minus-B over a t. 132 00:08:43,510 --> 00:08:45,670 So it becomes of this form. 133 00:08:45,670 --> 00:08:53,090 So the phase velocity of this wave will be B over a. 134 00:08:53,090 --> 00:08:57,102 And this we call normally, by the letter c, 135 00:08:57,102 --> 00:09:00,170 that's the phase velocity of electromagnetic waves 136 00:09:00,170 --> 00:09:00,840 in vacuum. 137 00:09:00,840 --> 00:09:03,540 Or commonly known as speed of light. 138 00:09:07,800 --> 00:09:13,340 That identifies it so far, as best as we can in words. 139 00:09:13,340 --> 00:09:20,930 This is, as I say, a progressive sinusoidal electric field 140 00:09:20,930 --> 00:09:22,410 moving in the plus-x direction. 141 00:09:27,850 --> 00:09:29,990 What are the a's and b's? 142 00:09:33,750 --> 00:09:40,450 By how much must you change x so that the wave gets 143 00:09:40,450 --> 00:09:43,130 the same amplitude as where you started? 144 00:09:43,130 --> 00:09:47,510 And the answer of that is that, of course, 145 00:09:47,510 --> 00:09:50,280 a must be 2 pi over lambda. 146 00:09:53,130 --> 00:09:57,060 because then if x changes by lambda, 147 00:09:57,060 --> 00:10:00,740 your cosine changes by 2 pi. 148 00:10:00,740 --> 00:10:05,290 So a in that equation must be 2 pi over lambda. 149 00:10:05,290 --> 00:10:08,840 That quantity is normally given the symbol k, 150 00:10:08,840 --> 00:10:12,810 it's called the wave number. 151 00:10:12,810 --> 00:10:19,720 Similarly, if I look at the time turn, 152 00:10:19,720 --> 00:10:23,860 b must be equal to 2 pi divided by the period. 153 00:10:26,810 --> 00:10:29,680 Because if t changes by the period, 154 00:10:29,680 --> 00:10:33,280 then that cosine-- the angle of that cosine, 155 00:10:33,280 --> 00:10:36,720 the phase of that function-- changes by 2 pi. 156 00:10:36,720 --> 00:10:39,350 And you're back where you started. 157 00:10:39,350 --> 00:10:41,410 So b must be 2 pi over t. 158 00:10:41,410 --> 00:10:45,980 So this tells you for that particular wave 159 00:10:45,980 --> 00:10:49,210 what the a must be, what the b is. 160 00:10:49,210 --> 00:10:51,980 2 pi over t is, of course, the same as 2 pi times 161 00:10:51,980 --> 00:10:54,250 the frequency, which we normally call 162 00:10:54,250 --> 00:10:57,400 the angular frequency, omega. 163 00:10:57,400 --> 00:11:01,000 So a is k, and b is omega. 164 00:11:05,590 --> 00:11:07,950 Next. 165 00:11:07,950 --> 00:11:16,310 I said that any solution that is real 166 00:11:16,310 --> 00:11:19,640 of the electromagnetic field must 167 00:11:19,640 --> 00:11:23,220 satisfy Maxwell's equations. 168 00:11:23,220 --> 00:11:26,000 So the same must be true of this. 169 00:11:26,000 --> 00:11:32,810 If this is the wave of the electron, electric field, 170 00:11:32,810 --> 00:11:42,140 there must be associated with it a magnetic field such that all 171 00:11:42,140 --> 00:11:46,450 of Maxwell's equations are satisfied. 172 00:11:46,450 --> 00:11:51,460 In particular, if we take this one-- Faraday's law-- 173 00:11:51,460 --> 00:11:55,850 we know that the rate of change of the magnetic field 174 00:11:55,850 --> 00:11:59,940 must be equal to minus the curl of the electric field. 175 00:12:03,350 --> 00:12:06,580 This you can look up in books of mathematics. 176 00:12:06,580 --> 00:12:09,940 If you look at all the components, the way I always 177 00:12:09,940 --> 00:12:13,570 remember it, it is the determinant 178 00:12:13,570 --> 00:12:16,760 where here you have the unit x direction yz. 179 00:12:16,760 --> 00:12:18,570 This is dx, dy, dz. 180 00:12:18,570 --> 00:12:21,634 And here is the x component of electric field, y component, 181 00:12:21,634 --> 00:12:22,300 and z component. 182 00:12:24,840 --> 00:12:29,600 For our particular electric field, 183 00:12:29,600 --> 00:12:33,120 I only have the z component. 184 00:12:33,120 --> 00:12:36,030 And it's only a function of x. 185 00:12:36,030 --> 00:12:39,090 So most of the terms of this expansion 186 00:12:39,090 --> 00:12:47,360 are 0, except the one-- the rate of change of with x of Ez. 187 00:12:47,360 --> 00:12:50,230 And that will be in the y direction. 188 00:12:50,230 --> 00:12:57,260 So this db dt must be equal to that if that is a solution 189 00:12:57,260 --> 00:12:59,450 the Maxwell's equations. 190 00:12:59,450 --> 00:13:07,940 If I take the x derivative of E up there. 191 00:13:07,940 --> 00:13:10,050 I end up-- and you could almost do it 192 00:13:10,050 --> 00:13:14,100 in your head-- db dt is minus this quantity. 193 00:13:17,960 --> 00:13:20,250 But if this is the rate of change of t, 194 00:13:20,250 --> 00:13:21,890 I can integrate this. 195 00:13:21,890 --> 00:13:28,360 And if I integrated it, B must be equal to-- the a 196 00:13:28,360 --> 00:13:34,320 comes from here, a minus a, a over-- sorry. 197 00:13:34,320 --> 00:13:37,820 The b comes from here, I misspoke. 198 00:13:37,820 --> 00:13:41,430 That comes out, and the integral of sine 199 00:13:41,430 --> 00:13:46,050 gives you cosine, so that must be satisfied. 200 00:13:46,050 --> 00:13:48,000 But since we integrated B, there will 201 00:13:48,000 --> 00:13:50,820 be a constant of integration. 202 00:13:50,820 --> 00:13:54,190 So if I add to this any constant B, 203 00:13:54,190 --> 00:13:56,800 this will still satisfy this equation. 204 00:14:02,300 --> 00:14:06,240 All of this is telling me is that if I 205 00:14:06,240 --> 00:14:09,660 have that electric field-- propagating electric field-- 206 00:14:09,660 --> 00:14:14,750 I must simultaneously have this propagating magnetic field. 207 00:14:14,750 --> 00:14:18,895 And on top of that, I can have any constant magnetic field. 208 00:14:23,610 --> 00:14:27,410 It means that is a more general situation where 209 00:14:27,410 --> 00:14:29,960 this electric field and these magnetic fields 210 00:14:29,960 --> 00:14:36,020 can exist with any constant B. I'll just call it 0. 211 00:14:36,020 --> 00:14:38,400 It's not an interesting part of this, 212 00:14:38,400 --> 00:14:40,350 it's not a propagating field. 213 00:14:40,350 --> 00:14:45,480 And so we end up that if you have that electric field 214 00:14:45,480 --> 00:14:49,460 propagating, and in with this magnetic field, 215 00:14:49,460 --> 00:14:55,270 then that system satisfies all Maxwell's equations. 216 00:14:55,270 --> 00:14:59,310 Both the E and B will satisfy these wave equations. 217 00:14:59,310 --> 00:15:02,080 Try it for yourself, and you'll see. 218 00:15:02,080 --> 00:15:07,153 So the answer to this is, what this 219 00:15:07,153 --> 00:15:13,000 is, this is a polarized-- plane-polarized electromagnetic 220 00:15:13,000 --> 00:15:17,430 wave, where we identified the wavelength, the frequency, 221 00:15:17,430 --> 00:15:20,680 it's propagating in the x direction. 222 00:15:20,680 --> 00:15:27,220 And the electric field is polarized in the z direction. 223 00:15:27,220 --> 00:15:29,640 One of the things we will learn from this so we don't have 224 00:15:29,640 --> 00:15:33,360 to repeat over and over again when we're looking at different 225 00:15:33,360 --> 00:15:38,950 formulae, which describe ways to help us to identify , them is-- 226 00:15:38,950 --> 00:15:43,390 notice that what we have found was that the electric 227 00:15:43,390 --> 00:15:47,640 and the magnetic fields are perpendicular to each other. 228 00:15:47,640 --> 00:15:49,620 The electric field in the z direction, 229 00:15:49,620 --> 00:15:51,840 the magnetic in the y direction. 230 00:15:51,840 --> 00:15:57,730 But the sinusoidal part and the phase velocity and everything 231 00:15:57,730 --> 00:16:02,100 else-- wavelength, frequency-- are exactly the same and phase. 232 00:16:02,100 --> 00:16:04,560 This is completely in general. 233 00:16:04,560 --> 00:16:10,940 If you have a progressive electromagnetic wave in vacuum, 234 00:16:10,940 --> 00:16:15,050 you find that the only way it can 235 00:16:15,050 --> 00:16:19,050 exist if you have simultaneously an electric 236 00:16:19,050 --> 00:16:21,620 and a magnetic field propagating. 237 00:16:21,620 --> 00:16:24,390 They are always at right angles to each other. 238 00:16:24,390 --> 00:16:27,960 This is the electric field, this will be the magnetic field. 239 00:16:27,960 --> 00:16:31,380 If it's propagating in that direction. 240 00:16:31,380 --> 00:16:36,860 It's always from e to b in a clockwise rotation, 241 00:16:36,860 --> 00:16:42,140 if they're propagating in that direction. 242 00:16:42,140 --> 00:16:44,840 So I drew a general sketch here. 243 00:16:44,840 --> 00:16:50,300 This is true for any progressive wave, 244 00:16:50,300 --> 00:16:52,890 electromagnetic progressive wave. 245 00:16:52,890 --> 00:16:57,310 And you have the electric field, magnetic field perpendicular 246 00:16:57,310 --> 00:17:01,770 to it, and the two propagate in that direction, 247 00:17:01,770 --> 00:17:03,830 given by this vector equation. 248 00:17:03,830 --> 00:17:08,000 Furthermore, if they satisfy Maxwell's equation the ratio 249 00:17:08,000 --> 00:17:11,740 of E to B, the magnitude, is equal to c. 250 00:17:11,740 --> 00:17:13,400 This is completely general. 251 00:17:13,400 --> 00:17:16,069 It is worth remembering when we're 252 00:17:16,069 --> 00:17:19,829 analyzing different situations. 253 00:17:19,829 --> 00:17:21,950 So that I went slowly through this, 254 00:17:21,950 --> 00:17:25,780 but that is one example where we see 255 00:17:25,780 --> 00:17:28,770 this mathematical description of something which 256 00:17:28,770 --> 00:17:32,730 we can recognize what it is, and which 257 00:17:32,730 --> 00:17:38,090 is a solution to Maxwell's equations in vacuum. 258 00:17:38,090 --> 00:17:41,210 What actually happens in the physical situation 259 00:17:41,210 --> 00:17:44,250 depends, as always, on all the boundary conditions, 260 00:17:44,250 --> 00:17:46,900 the initial conditions, et cetera. 261 00:17:46,900 --> 00:17:49,960 This doesn't address all those questions. 262 00:17:49,960 --> 00:17:56,800 All this says is this is one of the infinite possible solutions 263 00:17:56,800 --> 00:17:59,820 of Maxwell's equation. 264 00:17:59,820 --> 00:18:03,940 In other words, for electromagnetic fields 265 00:18:03,940 --> 00:18:10,450 corresponding to the plane wave propagating in one direction. 266 00:18:10,450 --> 00:18:11,845 Let's take a harder example. 267 00:18:16,260 --> 00:18:19,140 The question is the following. 268 00:18:19,140 --> 00:18:21,370 Can we now do the opposite? 269 00:18:21,370 --> 00:18:24,270 Not someone tells us the equation. 270 00:18:24,270 --> 00:18:29,510 Can we actually describe in mathematical forms 271 00:18:29,510 --> 00:18:33,510 a electromagnetic wave whose properties 272 00:18:33,510 --> 00:18:36,150 we know what we want and would like 273 00:18:36,150 --> 00:18:38,730 to write it mathematically. 274 00:18:38,730 --> 00:18:40,370 And I took a slightly harder one, 275 00:18:40,370 --> 00:18:45,350 so I said we would like to describe both 276 00:18:45,350 --> 00:18:49,630 the electric and the magnetic fields, 277 00:18:49,630 --> 00:18:54,890 which describes a monochromatic electromagnetic wave-- 278 00:18:54,890 --> 00:19:00,290 monochromatic means a single frequency, single wavelength-- 279 00:19:00,290 --> 00:19:05,610 with wavelength lambda, which propagates now 280 00:19:05,610 --> 00:19:10,350 not along the x or y or z axes that makes life easy. 281 00:19:10,350 --> 00:19:11,900 Let's say it goes at some angle. 282 00:19:11,900 --> 00:19:18,930 It goes at 45 degrees to the x-axis and y-axis. 283 00:19:18,930 --> 00:19:20,750 And the z is out of the board. 284 00:19:20,750 --> 00:19:23,570 So the wave-- we want the wave, which 285 00:19:23,570 --> 00:19:35,910 is propagating like this, where the wave front is-- let me come 286 00:19:35,910 --> 00:19:39,590 to it in a second-- where the vector 287 00:19:39,590 --> 00:19:43,750 perpendicular to the wave front is at 45 degrees 288 00:19:43,750 --> 00:19:45,696 to both the x-axis and y-axis. 289 00:19:48,760 --> 00:19:52,430 We want it plane-polarized, meaning 290 00:19:52,430 --> 00:19:58,730 that the electric vector is always in a plane 291 00:19:58,730 --> 00:20:01,560 and it's linearly polarized so it's 292 00:20:01,560 --> 00:20:06,415 in the same direction in the x-y plane. 293 00:20:09,400 --> 00:20:13,860 So how can we translate that into mathematics? 294 00:20:13,860 --> 00:20:17,990 Well, we'll use some of the knowledge 295 00:20:17,990 --> 00:20:20,750 we've just gained before. 296 00:20:20,750 --> 00:20:25,130 First of all, we know from what I discussed 297 00:20:25,130 --> 00:20:27,130 about the electric and magnetic field being 298 00:20:27,130 --> 00:20:30,190 perpendicular to each other and perpendicular 299 00:20:30,190 --> 00:20:35,140 to the direction of propagation that if the propagation is 300 00:20:35,140 --> 00:20:42,390 in this direction, then we know that the plane in which 301 00:20:42,390 --> 00:20:45,670 the electric and magnetic fields find themselves 302 00:20:45,670 --> 00:20:47,455 are perpendicular to that. 303 00:20:50,360 --> 00:20:52,280 Since this is propagating like this, 304 00:20:52,280 --> 00:21:02,100 the distance between the planes of equal phase will be lambda. 305 00:21:02,100 --> 00:21:04,500 That's the meaning of the wavelength. 306 00:21:04,500 --> 00:21:07,920 Once you've gone the distance of 1 lambda, 307 00:21:07,920 --> 00:21:09,750 the magnitude and direction is back 308 00:21:09,750 --> 00:21:14,290 to what it was before for both the electric and magnetic 309 00:21:14,290 --> 00:21:15,300 fields. 310 00:21:15,300 --> 00:21:17,240 So that's what it will look like. 311 00:21:20,960 --> 00:21:26,090 So the electric vector will be in this plane, 312 00:21:26,090 --> 00:21:28,970 but we are told furthermore it's in the xy, 313 00:21:28,970 --> 00:21:31,730 so it will be in this direction. 314 00:21:31,730 --> 00:21:34,050 If it's like this, and in this plane, 315 00:21:34,050 --> 00:21:37,000 so this must be the direction of the electric vector. 316 00:21:37,000 --> 00:21:40,150 So let's give it a magnitude E-zero. 317 00:21:40,150 --> 00:21:45,090 And what is this unit vector? 318 00:21:45,090 --> 00:21:49,550 Well, clearly that is in the x direction. 319 00:21:49,550 --> 00:21:56,500 It has a component like this, and in the y direction, 320 00:21:56,500 --> 00:21:58,430 it has a component like that. 321 00:21:58,430 --> 00:22:00,340 The magnitude of the components is 322 00:22:00,340 --> 00:22:03,380 the same, because of the 45 degrees. 323 00:22:03,380 --> 00:22:07,010 But for the x, it'll be negative, and for the y, 324 00:22:07,010 --> 00:22:08,310 positive. 325 00:22:08,310 --> 00:22:11,740 So the unit vector in the direction 326 00:22:11,740 --> 00:22:16,430 of the electric vector will be minus x-hat 327 00:22:16,430 --> 00:22:19,200 over root-2 plus y-hat over root-2. 328 00:22:19,200 --> 00:22:22,540 This is a unit vector, you can check for yourself. 329 00:22:22,540 --> 00:22:25,890 If you take square this, square that, take the square root, 330 00:22:25,890 --> 00:22:26,920 you get 1. 331 00:22:26,920 --> 00:22:32,240 So this is a unit vector in this direction where we wanted it. 332 00:22:32,240 --> 00:22:36,720 So if I write this as the amplitude 333 00:22:36,720 --> 00:22:39,600 and the direction of the electric field, 334 00:22:39,600 --> 00:22:44,940 I do have a field which is linearly polarized always 335 00:22:44,940 --> 00:22:47,030 in the same direction. 336 00:22:47,030 --> 00:22:48,840 We'll put a sine or cosine there, 337 00:22:48,840 --> 00:22:52,386 because we're talking about a monochromatic electromagnetic 338 00:22:52,386 --> 00:22:54,760 wave with the wavelengths, so it's a sinusoidal function. 339 00:22:58,450 --> 00:23:01,270 Where I put the sine or cosine or any other phase 340 00:23:01,270 --> 00:23:04,345 just determines where time equals 0. 341 00:23:04,345 --> 00:23:05,095 So let's put sine. 342 00:23:07,740 --> 00:23:13,920 It's going to be propagating in this direction, plus k, 343 00:23:13,920 --> 00:23:18,370 so these two will have opposite sign. 344 00:23:18,370 --> 00:23:21,760 This will be the frequency-- angular frequency-- 345 00:23:21,760 --> 00:23:24,730 of oscillations of this. 346 00:23:24,730 --> 00:23:31,670 And here we must describe a plane. 347 00:23:31,670 --> 00:23:37,200 Because along this plane, the phase has to be the same. 348 00:23:37,200 --> 00:23:38,720 That's what we mean by wavefront. 349 00:23:41,710 --> 00:23:45,720 Vectorially, how do we describe a plane? 350 00:23:45,720 --> 00:23:49,260 Well, we will have the plane which is perpendicular to k 351 00:23:49,260 --> 00:23:54,800 if we take k dot product of the vector r. 352 00:23:54,800 --> 00:23:56,070 r is the vector. 353 00:23:59,110 --> 00:24:03,790 Here is the vector r, from the origin 354 00:24:03,790 --> 00:24:08,300 to a point on the plane which I want to describe. 355 00:24:08,300 --> 00:24:10,630 So this is k dot r. 356 00:24:10,630 --> 00:24:17,640 So this now, we'll have k, which is the wave number, 357 00:24:17,640 --> 00:24:23,430 and this whole thing is called the k vector, 358 00:24:23,430 --> 00:24:28,300 will have a magnitude which is 2 pi over lambda. 359 00:24:28,300 --> 00:24:30,300 Same as in the other problem. 360 00:24:30,300 --> 00:24:36,300 But now it's pointing in this direction, which again, 361 00:24:36,300 --> 00:24:39,730 by analogy, how we calculated that is the unit vector 362 00:24:39,730 --> 00:24:43,460 x over root-2 plus unit vector y over root-2. 363 00:24:43,460 --> 00:24:45,480 So this is k. 364 00:24:45,480 --> 00:24:49,910 r is nothing I want to describe this point. 365 00:24:49,910 --> 00:24:53,560 I have x in the x direction, y in the y direction, 366 00:24:53,560 --> 00:24:55,210 z in the z direction. 367 00:24:55,210 --> 00:24:59,000 So that describes any point on that plane. 368 00:24:59,000 --> 00:25:01,830 If I take the dot product between them, 369 00:25:01,830 --> 00:25:09,160 I will get then a wave which is moving the the k direction. 370 00:25:09,160 --> 00:25:16,100 And this describes the position on the wavefront. 371 00:25:16,100 --> 00:25:20,180 So putting it all together, this electric field 372 00:25:20,180 --> 00:25:26,560 at every point of x, y, and t will have a magnitude 373 00:25:26,560 --> 00:25:32,130 is E-zero times this direction, the direction 374 00:25:32,130 --> 00:25:35,360 of polarization of the electric field, times sine. 375 00:25:40,660 --> 00:25:49,820 This is now telling me it's propagating in this direction. 376 00:25:49,820 --> 00:25:52,630 And with angular frequency omega. 377 00:25:52,630 --> 00:26:02,130 So that describes the electric part of this wave. 378 00:26:02,130 --> 00:26:03,730 How about the magnetic one? 379 00:26:03,730 --> 00:26:06,710 Well, we could do the same as before. 380 00:26:06,710 --> 00:26:10,420 The magnetic part is determined by this, 381 00:26:10,420 --> 00:26:14,480 because all Maxwell's equations have to be satisfied, 382 00:26:14,480 --> 00:26:16,420 including Faraday's law. 383 00:26:20,020 --> 00:26:23,570 But I told you, so it saves me doing it over and over again, 384 00:26:23,570 --> 00:26:27,510 we've learned once and for all, for a progressive wave 385 00:26:27,510 --> 00:26:30,130 the e and b are perpendicular to each other, 386 00:26:30,130 --> 00:26:33,970 and the ratio between them is c. 387 00:26:33,970 --> 00:26:39,860 So since I know what E is, the magnitude of the magnetic field 388 00:26:39,860 --> 00:26:41,880 is E-zero over c. 389 00:26:41,880 --> 00:26:46,320 It'll be at right angles to this direction 390 00:26:46,320 --> 00:26:49,470 and to the propagation, and therefore it 391 00:26:49,470 --> 00:26:51,830 will be out of the board. 392 00:26:51,830 --> 00:26:57,580 So that from E-cross-B, the vectors are in the k direction. 393 00:26:57,580 --> 00:27:00,280 So the b will be out of the board, which 394 00:27:00,280 --> 00:27:01,740 is easier this time. 395 00:27:01,740 --> 00:27:04,880 That's in the z direction. 396 00:27:04,880 --> 00:27:11,080 And it will be, as I said, exactly in phase in time 397 00:27:11,080 --> 00:27:14,390 and space with the electric field. 398 00:27:14,390 --> 00:27:16,510 The two are coupled together. 399 00:27:16,510 --> 00:27:21,000 So that now describes it entirely. 400 00:27:21,000 --> 00:27:25,720 So this is, in fact, the answer to our question. 401 00:27:25,720 --> 00:27:28,600 It describes an electric or magnetic field 402 00:27:28,600 --> 00:27:30,510 which is monochromatic. 403 00:27:30,510 --> 00:27:32,520 It's an electromagnetic wave. 404 00:27:32,520 --> 00:27:33,850 It has wavelength lambda. 405 00:27:33,850 --> 00:27:38,490 It propagates at 45 degrees to x and y axes, 406 00:27:38,490 --> 00:27:40,010 and is plane-polarized. 407 00:27:40,010 --> 00:27:45,080 e is always in the same direction and in the xy plane. 408 00:27:45,080 --> 00:27:46,400 So this is the answer. 409 00:27:46,400 --> 00:27:47,430 See, notice. 410 00:27:47,430 --> 00:27:49,840 In the past when we were doing problems, 411 00:27:49,840 --> 00:27:53,620 we focus more on things like what 412 00:27:53,620 --> 00:27:56,230 is the wave equation for this string? 413 00:27:56,230 --> 00:28:01,140 Or for a pipe with a gas in it? 414 00:28:01,140 --> 00:28:05,590 Or a transmission line, et cetera. 415 00:28:05,590 --> 00:28:12,460 Here, even guessing what solutions we're interested in, 416 00:28:12,460 --> 00:28:15,240 what kind of solution, it's already hard or even 417 00:28:15,240 --> 00:28:19,540 to describe the wave we're interested in. 418 00:28:19,540 --> 00:28:23,550 So this, for the other situations, 419 00:28:23,550 --> 00:28:26,030 this would have taken a few minutes. 420 00:28:26,030 --> 00:28:29,730 Here it needs a fair amount of analysis. 421 00:28:29,730 --> 00:28:34,090 And it takes much longer. 422 00:28:34,090 --> 00:28:35,800 Let me take one more case. 423 00:28:46,100 --> 00:28:51,290 The last case I'm going to exhibit is the following. 424 00:28:51,290 --> 00:28:58,160 Again the issue will be, there's this particular wave 425 00:28:58,160 --> 00:29:00,380 we want to produce. 426 00:29:00,380 --> 00:29:02,500 We know what we want, and we want 427 00:29:02,500 --> 00:29:06,000 to know how to describe it mathematically. 428 00:29:06,000 --> 00:29:10,470 So once again, we want to find a solution 429 00:29:10,470 --> 00:29:14,150 of our Maxwell's equations, which 430 00:29:14,150 --> 00:29:19,640 have the following property that correspond to a circularly 431 00:29:19,640 --> 00:29:22,260 polarized electromagnetic wave which 432 00:29:22,260 --> 00:29:24,560 is propagating in y direction. 433 00:29:24,560 --> 00:29:30,940 And it just says "any." so any, any circularly polarized 434 00:29:30,940 --> 00:29:35,345 electromagnetic wave which is propagating in the minus y 435 00:29:35,345 --> 00:29:35,845 direction. 436 00:29:39,600 --> 00:29:45,260 First of all, what we mean by circularly polarized wave? 437 00:29:45,260 --> 00:29:49,790 A circularly polarized wave is that, if I 438 00:29:49,790 --> 00:29:56,940 took a snapshot, if I could, at a given instant of time, 439 00:29:56,940 --> 00:30:02,910 one would find that the electric vector along the propagation 440 00:30:02,910 --> 00:30:11,890 direction is rotating like this on the spiral. 441 00:30:11,890 --> 00:30:17,290 If that wave is moving towards you, 442 00:30:17,290 --> 00:30:24,360 what you would see in any plane, a rotating electric field. 443 00:30:24,360 --> 00:30:29,445 And associated with a magnetic field at right angles to it. 444 00:30:32,360 --> 00:30:34,030 It doesn't tell us whether we want 445 00:30:34,030 --> 00:30:37,440 a left-handed or a right-handed rotated field. 446 00:30:37,440 --> 00:30:39,280 So just arbitrarily take one. 447 00:30:39,280 --> 00:30:42,520 And by the way, if ever you're interested in the left- 448 00:30:42,520 --> 00:30:46,190 and right-handed and figuring out which is which? 449 00:30:46,190 --> 00:30:47,310 It's a mess. 450 00:30:47,310 --> 00:30:50,290 Different communities use different definitions, 451 00:30:50,290 --> 00:30:52,230 what they mean by right- and left-handed. 452 00:30:52,230 --> 00:30:56,360 So I won't try to confuse you more than that. 453 00:30:56,360 --> 00:30:59,640 So here we want any wave, which corresponds 454 00:30:59,640 --> 00:31:03,920 to circular polarization, and is moving in the minus y 455 00:31:03,920 --> 00:31:05,310 direction. 456 00:31:05,310 --> 00:31:10,380 So if it's moving in plus or minus y direction, 457 00:31:10,380 --> 00:31:15,630 we know that the electric field will be in the xz plane 458 00:31:15,630 --> 00:31:18,120 at every instant of time. 459 00:31:18,120 --> 00:31:21,110 If it's circularly polarized, we know 460 00:31:21,110 --> 00:31:25,720 that the magnitude of the electric field at all locations 461 00:31:25,720 --> 00:31:29,630 of x, y, and z at all times will be the same. 462 00:31:29,630 --> 00:31:31,520 It does not change. 463 00:31:31,520 --> 00:31:32,320 It's a constant. 464 00:31:35,990 --> 00:31:38,040 Now so how do we create such a thing? 465 00:31:38,040 --> 00:31:40,900 Well, if we stop and think for a second, 466 00:31:40,900 --> 00:31:45,820 if we superimpose two solutions-- suppose 467 00:31:45,820 --> 00:31:50,750 we have one solution, which is a plane-polarized electromagnetic 468 00:31:50,750 --> 00:31:55,820 wave going towards you, and I superimpose 469 00:31:55,820 --> 00:32:03,190 on that another one which is out of phase with it and at 90 470 00:32:03,190 --> 00:32:09,040 degrees, then at every location in space, 471 00:32:09,040 --> 00:32:12,040 I'll have two components. 472 00:32:12,040 --> 00:32:16,950 If I make those components change, but in such a way 473 00:32:16,950 --> 00:32:19,230 that the vector addition of the two 474 00:32:19,230 --> 00:32:23,270 gives me a unit vector, a constant vector, 475 00:32:23,270 --> 00:32:26,750 I will have achieved what I wanted to do. 476 00:32:26,750 --> 00:32:35,110 So here is a equation which satisfies everything I've said. 477 00:32:35,110 --> 00:32:39,460 Let's consider an electromagnetic wave 478 00:32:39,460 --> 00:32:45,755 which is the same in all x and all z positions. 479 00:32:48,840 --> 00:32:52,640 The only variable is in the y direction. 480 00:32:52,640 --> 00:32:57,140 If I write that as the superposition 481 00:32:57,140 --> 00:33:01,520 of an electric field which is in the x direction, 482 00:33:01,520 --> 00:33:09,880 and propagating as a sine-- it's a sinusoidal wave-- 483 00:33:09,880 --> 00:33:17,760 and I add to it a cosine, which is at right angles. 484 00:33:17,760 --> 00:33:19,940 Furthermore I'll use the other information. 485 00:33:19,940 --> 00:33:22,540 It's going in the minus y direction. 486 00:33:22,540 --> 00:33:25,860 So I'll make these two opposite sign-- sorry, 487 00:33:25,860 --> 00:33:29,500 I make them the same sign, it is in minus-y direction. 488 00:33:29,500 --> 00:33:32,640 If it was in the plus-y, they would have opposite signs. 489 00:33:32,640 --> 00:33:37,590 If it's minus-y, this would have to be the same. 490 00:33:37,590 --> 00:33:41,690 So this is a sinusoidal wave moving 491 00:33:41,690 --> 00:33:43,960 in the minus-y direction. 492 00:33:43,960 --> 00:33:46,390 It'll have the wave number k, this is 2 pi over lambda. 493 00:33:46,390 --> 00:33:51,310 And this is 2 pi, the frequency or 2 pi over the period. 494 00:33:51,310 --> 00:33:58,650 Omega over k has to be c, the speed of electromagnetic waves. 495 00:33:58,650 --> 00:34:03,470 If I add to this, the resultant electric vector 496 00:34:03,470 --> 00:34:07,230 everywhere in space has a magnitude E-zero. 497 00:34:07,230 --> 00:34:08,690 I can check it. 498 00:34:08,690 --> 00:34:12,139 The magnitude of E is the square root 499 00:34:12,139 --> 00:34:14,550 of the x component of this squared 500 00:34:14,550 --> 00:34:17,719 plus the z component of this squared. 501 00:34:17,719 --> 00:34:21,770 So it's E-zero, the x component squared-- 502 00:34:21,770 --> 00:34:23,449 the sine squared of this. 503 00:34:23,449 --> 00:34:28,550 The z component is the cosine, so the squared is that. 504 00:34:28,550 --> 00:34:32,760 For all values of x, y, and z at all times, 505 00:34:32,760 --> 00:34:35,889 if I add these and take the square root, I get 1. 506 00:34:35,889 --> 00:34:37,580 And so this is E-zero. 507 00:34:37,580 --> 00:34:42,909 So this propagating wave does satisfy my requirement 508 00:34:42,909 --> 00:34:46,230 that everywhere is magnitude E-zero. 509 00:34:46,230 --> 00:34:48,110 It is a propagating wave. 510 00:34:48,110 --> 00:34:52,070 Each one of these are propagating 511 00:34:52,070 --> 00:35:02,310 with the speed of light in the direction of y. 512 00:35:02,310 --> 00:35:08,360 I'm sorry, forgive me, can't copy from one line to the next. 513 00:35:08,360 --> 00:35:11,210 This is plus, this is plus. 514 00:35:11,210 --> 00:35:11,830 All right. 515 00:35:11,830 --> 00:35:14,800 It's moving in the minus-y direction. 516 00:35:14,800 --> 00:35:17,420 The way I had it, it was going in the plus-y direction. 517 00:35:17,420 --> 00:35:18,230 I corrected it. 518 00:35:18,230 --> 00:35:20,670 This is in the minus-y direction. 519 00:35:20,670 --> 00:35:21,410 All right? 520 00:35:21,410 --> 00:35:24,310 And this is what was required. 521 00:35:24,310 --> 00:35:25,070 OK. 522 00:35:25,070 --> 00:35:31,960 So this mathematical description of the electric vector, 523 00:35:31,960 --> 00:35:35,180 how it's propagating. 524 00:35:35,180 --> 00:35:38,280 And now we want to know what the magnetic one is doing. 525 00:35:38,280 --> 00:35:44,900 Well, again, we could go back and make sure 526 00:35:44,900 --> 00:35:48,250 that Maxwell's equations are completely satisfied. 527 00:35:48,250 --> 00:35:52,880 And you'll find that here, in order for Faraday's law 528 00:35:52,880 --> 00:35:58,500 to hold, I have to have also a changing magnetic field. 529 00:35:58,500 --> 00:36:00,830 But instead of doing that, I'll make 530 00:36:00,830 --> 00:36:03,960 use of what we learned by the previous examples. 531 00:36:03,960 --> 00:36:07,860 We know that this is a superposition 532 00:36:07,860 --> 00:36:11,370 of two progressive waves. 533 00:36:11,370 --> 00:36:14,650 Each one of these is a solution of Maxwell's wave. 534 00:36:14,650 --> 00:36:16,140 I don't need both of them. 535 00:36:16,140 --> 00:36:20,620 I only needed both to get a circularly polarized wave. 536 00:36:20,620 --> 00:36:24,360 Each one of these has to satisfy Maxwell's equation. 537 00:36:24,360 --> 00:36:27,350 So associated with each of these components, 538 00:36:27,350 --> 00:36:32,420 I must have a magnetic field which satisfies the requirement 539 00:36:32,420 --> 00:36:35,570 that there is an electric vector and magnetic vector 540 00:36:35,570 --> 00:36:39,620 at right angle to each other moving together 541 00:36:39,620 --> 00:36:43,550 in the direction of propagation in phase and in time. 542 00:36:43,550 --> 00:36:47,440 So for each one of these, I will find 543 00:36:47,440 --> 00:36:49,740 the corresponding magnetic field, 544 00:36:49,740 --> 00:36:52,460 the magnitude will be E-zero over c, 545 00:36:52,460 --> 00:36:55,050 because we know that the ratio of the electric field 546 00:36:55,050 --> 00:36:59,990 to the magnetic field is always equal to c in vacuum. 547 00:36:59,990 --> 00:37:01,100 It's at right angles. 548 00:37:01,100 --> 00:37:05,380 This was in the x direction, this is in the z direction. 549 00:37:05,380 --> 00:37:09,330 And in this case, then add this one. 550 00:37:09,330 --> 00:37:12,040 Here, this was plus-z and this is minus-x. 551 00:37:12,040 --> 00:37:14,590 And you can draw yourself a little picture 552 00:37:14,590 --> 00:37:16,870 to make sure you get everything right. 553 00:37:16,870 --> 00:37:19,990 Let me just talk about, say, this one. 554 00:37:19,990 --> 00:37:21,930 The second component. 555 00:37:21,930 --> 00:37:23,460 What I have in the [INAUDIBLE], this 556 00:37:23,460 --> 00:37:26,860 is moving there in minus-y. 557 00:37:26,860 --> 00:37:30,370 This component is in the z direction, 558 00:37:30,370 --> 00:37:34,010 so it's over here, coming out of the board. 559 00:37:34,010 --> 00:37:37,830 If it's in this direction, moving down here, 560 00:37:37,830 --> 00:37:41,280 then the b must be in that direction. 561 00:37:41,280 --> 00:37:43,530 So it must be in this direction, which 562 00:37:43,530 --> 00:37:45,670 is minus-x, which is correct. 563 00:37:45,670 --> 00:37:49,190 So this is how I get this right. 564 00:37:49,190 --> 00:37:54,440 If I add these, I get the total magnetic field. 565 00:37:54,440 --> 00:38:02,000 This, now, describes one possible wave 566 00:38:02,000 --> 00:38:04,830 which satisfies this requirement. 567 00:38:04,830 --> 00:38:07,910 It's a circularly polarized electromagnetic wave 568 00:38:07,910 --> 00:38:11,840 propagating in the minus-y direction. 569 00:38:11,840 --> 00:38:17,360 OK, so let me stop at these examples of progressive waves, 570 00:38:17,360 --> 00:38:20,275 and I'll move over to standing waves. 571 00:38:24,290 --> 00:38:26,750 So let's continue in a second, thank you. 572 00:38:31,420 --> 00:38:33,860 So I've now erased the board, and I 573 00:38:33,860 --> 00:38:37,810 can continue talking about wave solutions 574 00:38:37,810 --> 00:38:39,300 to Maxwell's equations. 575 00:38:39,300 --> 00:38:40,940 But let's recap for a second. 576 00:38:44,640 --> 00:38:49,630 What we find is the following, that basically in vacuum 577 00:38:49,630 --> 00:38:53,210 at every location in space it's as if there was an oscillator. 578 00:38:55,950 --> 00:38:58,000 It can be displaced from equilibrium. 579 00:38:58,000 --> 00:39:00,750 It can be made to oscillate. 580 00:39:00,750 --> 00:39:04,480 Displacement from equilibrium means 581 00:39:04,480 --> 00:39:06,730 there is an electric field there, 582 00:39:06,730 --> 00:39:10,590 or there is a magnetic field there. 583 00:39:10,590 --> 00:39:12,940 These can oscillate. 584 00:39:12,940 --> 00:39:15,090 They don't have to oscillate. 585 00:39:15,090 --> 00:39:20,800 So for example, you could have a static field, 586 00:39:20,800 --> 00:39:25,150 just an electric field constant in time everywhere in space. 587 00:39:25,150 --> 00:39:30,660 That means every location space is displaced from equilibrium. 588 00:39:30,660 --> 00:39:33,570 There could be a constant magnetic field instead, 589 00:39:33,570 --> 00:39:34,845 or both constant. 590 00:39:38,770 --> 00:39:40,230 Imagine how complicated this is. 591 00:39:40,230 --> 00:39:45,670 At every location the direction of this displacement 592 00:39:45,670 --> 00:39:49,210 from equilibrium for the electric and magnetic fields, 593 00:39:49,210 --> 00:39:50,910 they are vectors. 594 00:39:50,910 --> 00:39:53,550 There are possibility of the electric field facing 595 00:39:53,550 --> 00:39:57,270 a different directions of the magnetic field. 596 00:39:57,270 --> 00:40:00,740 What we find is that whatever that combination is 597 00:40:00,740 --> 00:40:04,490 in space and time, that combination 598 00:40:04,490 --> 00:40:08,060 has to satisfy Maxwell's equations. 599 00:40:08,060 --> 00:40:11,940 That completely describes what happens in vacuum 600 00:40:11,940 --> 00:40:15,830 at every point in space and time. 601 00:40:15,830 --> 00:40:21,540 Now there are in particular combinations 602 00:40:21,540 --> 00:40:25,720 of these displacements of oscillations in space and time, 603 00:40:25,720 --> 00:40:31,350 which satisfy the wave equation for the electric and magnetic 604 00:40:31,350 --> 00:40:31,850 fields. 605 00:40:35,390 --> 00:40:40,580 It's a tiny subset of total, but there are such. 606 00:40:40,580 --> 00:40:46,980 And we are considering now for that tiny subset what kind 607 00:40:46,980 --> 00:40:50,850 of solutions exist, how to describe them. 608 00:40:50,850 --> 00:40:53,440 And even there, we're limiting ourselves 609 00:40:53,440 --> 00:40:58,700 to a tiny subset of a tiny subset. 610 00:40:58,700 --> 00:41:03,270 So far, I took the subset where this displacement 611 00:41:03,270 --> 00:41:08,540 from equilibrium of the electric and magnetic fields 612 00:41:08,540 --> 00:41:12,160 is a progressive wave. 613 00:41:12,160 --> 00:41:17,140 And what we found, in order to make sure 614 00:41:17,140 --> 00:41:21,130 that the Maxwell's equations are satisfied, 615 00:41:21,130 --> 00:41:23,760 you can't have any old electric field 616 00:41:23,760 --> 00:41:28,480 wave, or any old magnetic field wave. 617 00:41:28,480 --> 00:41:30,040 There's an interplay. 618 00:41:30,040 --> 00:41:33,960 There is, in reality, just one electromagnetic field, 619 00:41:33,960 --> 00:41:36,750 and that propagates. 620 00:41:36,750 --> 00:41:41,950 We'll now go and look for other solutions of these equations. 621 00:41:41,950 --> 00:41:50,980 And very interesting solutions are standing waves. 622 00:41:50,980 --> 00:41:56,840 So let me take a concrete example and discuss it. 623 00:41:56,840 --> 00:42:01,020 So here is, you could call it a problem. 624 00:42:01,020 --> 00:42:07,130 Suppose that I have everywhere in space an electric field 625 00:42:07,130 --> 00:42:10,200 which consists of a standing wave. 626 00:42:10,200 --> 00:42:12,140 You can recognize this when we were talking 627 00:42:12,140 --> 00:42:19,020 about standing waves on strings, for example. 628 00:42:19,020 --> 00:42:22,660 Where you have the electric field always pointing 629 00:42:22,660 --> 00:42:24,730 in the x direction. 630 00:42:24,730 --> 00:42:28,420 It's oscillating at every point in space 631 00:42:28,420 --> 00:42:31,840 with the same frequency and phase, cosine omega t. 632 00:42:31,840 --> 00:42:35,570 It's oscillating with that angular frequency. 633 00:42:35,570 --> 00:42:43,130 And spatially, it not change in the x and y direction, 634 00:42:43,130 --> 00:42:45,940 but it does in the z direction. 635 00:42:45,940 --> 00:42:48,490 And that is a cosine like this. 636 00:42:48,490 --> 00:42:54,190 So this is a standing wave of electric field. 637 00:42:57,080 --> 00:43:01,840 This by itself cannot be a solution. 638 00:43:01,840 --> 00:43:06,230 Is not a situation you can have in vacuum. 639 00:43:06,230 --> 00:43:10,000 It violates, by itself, Maxwell's equation. 640 00:43:10,000 --> 00:43:17,330 If you look at them, you find that in order for this 641 00:43:17,330 --> 00:43:19,250 to satisfy Maxwell's equation, the 642 00:43:19,250 --> 00:43:23,260 must be associated with it a magnetic field that 643 00:43:23,260 --> 00:43:24,980 looks like that. 644 00:43:24,980 --> 00:43:27,280 And so the question, the first thing is, 645 00:43:27,280 --> 00:43:32,610 show that if you have this, you must also have this present. 646 00:43:32,610 --> 00:43:35,700 The second part is some more discussion 647 00:43:35,700 --> 00:43:39,790 about when you have these two present, when 648 00:43:39,790 --> 00:43:42,270 you have a standing wave in vacuum 649 00:43:42,270 --> 00:43:45,080 of electromagnetic waves, for example, 650 00:43:45,080 --> 00:43:51,770 then what is the energy density? 651 00:43:51,770 --> 00:43:55,030 You know, in an electric field or a magnetic field, 652 00:43:55,030 --> 00:43:57,190 if you have in space, if you take 653 00:43:57,190 --> 00:44:03,250 any value inside the volume, there will be energy. 654 00:44:03,250 --> 00:44:09,830 And the energy per unit volume per cubic meter 655 00:44:09,830 --> 00:44:11,270 is the energy density. 656 00:44:11,270 --> 00:44:14,180 So we're going to calculate how much energy density there 657 00:44:14,180 --> 00:44:16,980 is in this standing wave. 658 00:44:16,980 --> 00:44:21,700 And another quantity, which is for practical reasons 659 00:44:21,700 --> 00:44:26,150 very important, is when you have an electric and magnetic fields 660 00:44:26,150 --> 00:44:33,920 present, actually energy flows through that system. 661 00:44:33,920 --> 00:44:39,490 And the amount of energy per unit area 662 00:44:39,490 --> 00:44:43,320 that flows-- per unit area perpendicular 663 00:44:43,320 --> 00:44:46,690 to the direction of flow-- is called the Poynting vector. 664 00:44:46,690 --> 00:44:48,510 And by the way, the Poynting has nothing 665 00:44:48,510 --> 00:44:52,150 to do with a vector that points, it's to do with a gentleman 666 00:44:52,150 --> 00:44:54,430 by the name of Poynting, after which this was called. 667 00:44:57,390 --> 00:45:00,440 So the second part of the problem 668 00:45:00,440 --> 00:45:07,070 is, once we found a standing wave that satisfies everything 669 00:45:07,070 --> 00:45:11,150 possible [INAUDIBLE] in vacuum, for this particular case 670 00:45:11,150 --> 00:45:14,410 what is the energy density, the magnetic and electric fields, 671 00:45:14,410 --> 00:45:16,380 and what's the Poynting vector? 672 00:45:16,380 --> 00:45:17,895 OK, so how do we do this? 673 00:45:23,700 --> 00:45:26,260 We know what the electric field is doing, 674 00:45:26,260 --> 00:45:28,370 it's the standing wave. 675 00:45:28,370 --> 00:45:32,890 We know that it must satisfy all Maxwell's equations, 676 00:45:32,890 --> 00:45:36,320 in particular Faraday's law. 677 00:45:36,320 --> 00:45:45,330 As before, we can calculate the curl of the electric field. 678 00:45:45,330 --> 00:45:51,710 Now here, the electric field is only in the x direction. 679 00:45:51,710 --> 00:45:54,920 And it's a function of z. 680 00:45:54,920 --> 00:45:59,520 And so the curl of this, to be only just one component 681 00:45:59,520 --> 00:46:04,100 of that, and that is given by this quantity. 682 00:46:04,100 --> 00:46:08,600 So this is minus the curl of this E. 683 00:46:08,600 --> 00:46:11,510 And we know by Faraday's law that this 684 00:46:11,510 --> 00:46:15,200 must equal to the rate of change of the magnetic field 685 00:46:15,200 --> 00:46:20,110 at that place of x, y, and z. 686 00:46:20,110 --> 00:46:23,250 Now I can integrate this equation, 687 00:46:23,250 --> 00:46:28,250 and find what B is at every point in space and every time. 688 00:46:28,250 --> 00:46:29,760 And that's easy enough. 689 00:46:29,760 --> 00:46:34,390 We just have to integrate that, which gives you the sine here, 690 00:46:34,390 --> 00:46:38,280 and the omega comes down, and you get this. 691 00:46:38,280 --> 00:46:40,910 Whenever you integrate, there is a constant. 692 00:46:40,910 --> 00:46:46,160 All it's telling us is that I can satisfy Maxwell's equations 693 00:46:46,160 --> 00:46:50,390 not only with an oscillating electric field 694 00:46:50,390 --> 00:46:52,760 present with an oscillating magnetic field, 695 00:46:52,760 --> 00:46:55,950 but I can always add a constant magnetic field 696 00:46:55,950 --> 00:46:56,910 throughout space. 697 00:46:56,910 --> 00:46:59,440 I could have also added a constant electric field. 698 00:46:59,440 --> 00:47:03,200 So there's an infinite number of solutions I can superimpose. 699 00:47:03,200 --> 00:47:04,840 I'm not interested in them. 700 00:47:04,840 --> 00:47:11,260 I am interested in the standing wave, the time-dependent part. 701 00:47:11,260 --> 00:47:15,390 So might as well make that 0. 702 00:47:15,390 --> 00:47:18,050 And so we are essentially home. 703 00:47:18,050 --> 00:47:24,880 We have found that the magnetic field is also a standing wave. 704 00:47:24,880 --> 00:47:27,300 And this, by the way, we look at the problem, 705 00:47:27,300 --> 00:47:29,360 is what we were asked to prove. 706 00:47:29,360 --> 00:47:32,240 So we have proven the first part, 707 00:47:32,240 --> 00:47:38,050 that if this is the description of the standing 708 00:47:38,050 --> 00:47:42,910 wave of the electric field, then there must be corresponding 709 00:47:42,910 --> 00:47:46,570 a standing wave magnetic field. 710 00:47:46,570 --> 00:47:51,910 So the two-- but notice, unlike in the case 711 00:47:51,910 --> 00:47:57,280 of progressive waves, where in the progressive waves, wherever 712 00:47:57,280 --> 00:48:01,740 you had an electric field, the magnetic field 713 00:48:01,740 --> 00:48:05,200 was at right angle to it and in magnitude proportional 714 00:48:05,200 --> 00:48:08,950 to the electric field and in phase with it, et cetera. 715 00:48:08,950 --> 00:48:10,970 Here, they're not. 716 00:48:10,970 --> 00:48:13,970 Here, the electric field, when this is cosine omega t, 717 00:48:13,970 --> 00:48:15,510 this is sine omega t. 718 00:48:15,510 --> 00:48:18,590 When this is cosine kz, this is sine kz. 719 00:48:18,590 --> 00:48:22,700 These two are out of phase with each other, 720 00:48:22,700 --> 00:48:26,680 both in time and in space. 721 00:48:26,680 --> 00:48:29,280 I've tried to sketch it here, it's not very good 722 00:48:29,280 --> 00:48:30,780 sketch, but anyway. 723 00:48:30,780 --> 00:48:35,010 Suppose at some instant of time, if I look at these, 724 00:48:35,010 --> 00:48:39,720 at some instant of time, the electric vector-- 725 00:48:39,720 --> 00:48:44,510 the magnitude of it-- is represented by this curve. 726 00:48:44,510 --> 00:48:46,950 And it is in the x direction. 727 00:48:46,950 --> 00:48:53,110 So the electric vector is this, like this, and like that. 728 00:48:53,110 --> 00:48:58,300 If this is the maximum, it is, the magnetic field at that time 729 00:48:58,300 --> 00:49:00,880 will be 0, if I look at these equations. 730 00:49:00,880 --> 00:49:03,340 So there'll be no magnetic field. 731 00:49:03,340 --> 00:49:06,390 Over this distance in space, there 732 00:49:06,390 --> 00:49:09,990 will be the electric field up here, down here, 733 00:49:09,990 --> 00:49:12,160 and no magnetic field. 734 00:49:12,160 --> 00:49:16,260 Later on, half a period later, what 735 00:49:16,260 --> 00:49:24,415 you find is that when this comes to 0-- it's a quarter 736 00:49:24,415 --> 00:49:31,030 period-- when this comes to 0, the electric field is 0, 737 00:49:31,030 --> 00:49:34,110 there will be a magnetic field at its maximum. 738 00:49:34,110 --> 00:49:35,760 But it will not be this shape. 739 00:49:35,760 --> 00:49:39,110 It will be, first of all, pointing in the y direction. 740 00:49:39,110 --> 00:49:42,450 This is in the x, it will be the y direction. 741 00:49:42,450 --> 00:49:44,180 It's maximum will be in the middle, 742 00:49:44,180 --> 00:49:47,380 well here it was always 0. 743 00:49:47,380 --> 00:49:49,200 And these two oscillate. 744 00:49:49,200 --> 00:49:50,570 It's a standing wave. 745 00:49:50,570 --> 00:49:58,910 The B does this, and the E does this, all in the same place. 746 00:49:58,910 --> 00:50:03,140 But both in space and time, the two are out of phase 747 00:50:03,140 --> 00:50:04,140 with each other. 748 00:50:04,140 --> 00:50:06,430 Completely different solution. 749 00:50:06,430 --> 00:50:11,720 And both progressive waves satisfy Maxwell's equations, 750 00:50:11,720 --> 00:50:13,880 and the standard waves. 751 00:50:13,880 --> 00:50:18,880 So it's important to realize there is this difference, 752 00:50:18,880 --> 00:50:21,180 often it's easy to get confused about it. 753 00:50:21,180 --> 00:50:25,610 In a progressive wave, the electric and magnetic fields 754 00:50:25,610 --> 00:50:27,140 are right angle. 755 00:50:27,140 --> 00:50:29,180 And as if they were locked together, 756 00:50:29,180 --> 00:50:33,140 and they move forward like this. 757 00:50:33,140 --> 00:50:37,590 On the other hand, in a standing situation, 758 00:50:37,590 --> 00:50:40,030 they're still at right angle to the other. 759 00:50:40,030 --> 00:50:43,090 But when one is a maximum, the other's a minimum. 760 00:50:43,090 --> 00:50:47,160 When this one is-- They're out of phase 761 00:50:47,160 --> 00:50:52,220 with each other in both space and time. 762 00:50:52,220 --> 00:50:53,470 So that's the first part. 763 00:50:53,470 --> 00:50:57,950 And the next part we were asked, now for this standing wave, 764 00:50:57,950 --> 00:51:00,910 imagine this could be inside your microwave oven. 765 00:51:00,910 --> 00:51:05,450 Inside the microwave oven, there is a standing wave. 766 00:51:05,450 --> 00:51:09,360 Unless they specially make it so it moves a little bit in space 767 00:51:09,360 --> 00:51:11,340 so you cook your meat everywhere. 768 00:51:11,340 --> 00:51:13,680 But then the cheapo microwave oven, 769 00:51:13,680 --> 00:51:17,840 you have a stationary standing wave. 770 00:51:17,840 --> 00:51:19,740 And suppose this is it. 771 00:51:19,740 --> 00:51:24,320 At every place in space, there is an energy density 772 00:51:24,320 --> 00:51:27,280 which actually fluctuates, goes up and down in time 773 00:51:27,280 --> 00:51:29,640 and is different in every location. 774 00:51:29,640 --> 00:51:31,340 Let's calculate that. 775 00:51:31,340 --> 00:51:35,550 Well, as Professor Walter Lewin showed, 776 00:51:35,550 --> 00:51:38,260 the energy density in an electric field, 777 00:51:38,260 --> 00:51:40,640 whether it's changing with time or not, 778 00:51:40,640 --> 00:51:45,990 if I've got in space somewhere an electric field e, 779 00:51:45,990 --> 00:51:49,840 at that location, I have an energy density. 780 00:51:49,840 --> 00:51:53,030 The amount is 1 over epsilon-zero times 781 00:51:53,030 --> 00:51:55,470 the magnitude of the electric field squared. 782 00:51:55,470 --> 00:51:59,200 That is the energy density of an electric field. 783 00:51:59,200 --> 00:52:01,510 It is not a vector. 784 00:52:01,510 --> 00:52:04,780 This is E-squared, the square of the magnitude 785 00:52:04,780 --> 00:52:07,890 of the electric field energy is a scalar quantity. 786 00:52:07,890 --> 00:52:12,340 So not surprising, this is not a vector, it's a scalar quantity. 787 00:52:12,340 --> 00:52:17,570 I can now immediately go over to what we know. 788 00:52:17,570 --> 00:52:20,820 We know the electric field, we know the magnetic field. 789 00:52:20,820 --> 00:52:27,880 So I can replace E-squared by what it is at every location. 790 00:52:27,880 --> 00:52:30,820 At every position z and every x, y. 791 00:52:30,820 --> 00:52:32,070 At all times. 792 00:52:32,070 --> 00:52:35,440 And this is the energy density. 793 00:52:35,440 --> 00:52:42,590 You can see it does oscillate, but there's always [INAUDIBLE]. 794 00:52:42,590 --> 00:52:45,070 How about the magnetic field? 795 00:52:45,070 --> 00:52:50,300 The magnetic field also has energy. 796 00:52:50,300 --> 00:52:53,700 If I take it anywhere, suppose you have a bar magnet, 797 00:52:53,700 --> 00:52:56,110 one of these pocket magnets, you hold it, 798 00:52:56,110 --> 00:52:59,450 and there's a magnetic field all around the magnet. 799 00:52:59,450 --> 00:53:04,490 Take any cubic meter of the volume, 800 00:53:04,490 --> 00:53:06,860 you'll find this amount of energy. 801 00:53:06,860 --> 00:53:10,590 It's 1 over 2 Mu 0 times the magnitude of the magnetic field 802 00:53:10,590 --> 00:53:11,410 squared. 803 00:53:11,410 --> 00:53:16,550 Again, I know what B is in for my standing, wave 804 00:53:16,550 --> 00:53:20,490 so I can calculate it, and I get this answer. 805 00:53:20,490 --> 00:53:25,800 So these are the two energy densities. 806 00:53:25,800 --> 00:53:33,160 Now what one finds, if one does-- if you plot this, 807 00:53:33,160 --> 00:53:38,740 or thinks about it-- that in this standing wave, 808 00:53:38,740 --> 00:53:45,023 you find that that energy moves backwards and forwards. 809 00:53:51,180 --> 00:53:57,110 At any location in space, I can calculate 810 00:53:57,110 --> 00:54:02,550 how much energy is moving per second 811 00:54:02,550 --> 00:54:07,390 per square meter-- per unit area-- perpendicular 812 00:54:07,390 --> 00:54:11,360 to the direction of motion of that energy. 813 00:54:11,360 --> 00:54:14,895 And that is what is called the Poynting vector. 814 00:54:17,620 --> 00:54:19,180 If you think, for example, suppose 815 00:54:19,180 --> 00:54:22,680 you take an electromagnetic wave like light shining 816 00:54:22,680 --> 00:54:24,890 that the wall. 817 00:54:24,890 --> 00:54:26,640 It'll warm up to the wall, I mean 818 00:54:26,640 --> 00:54:30,860 there's heat being transmitted, there's energy comes over. 819 00:54:30,860 --> 00:54:35,180 At any instant of time, how much energy per unit area 820 00:54:35,180 --> 00:54:36,780 is hitting the wall? 821 00:54:36,780 --> 00:54:42,180 It will be equal to the Poynting vector at that instant of time. 822 00:54:42,180 --> 00:54:46,950 And the Poynting vector s is E-cross-B over Mu 0. 823 00:54:52,320 --> 00:54:58,430 By the way, this applies to any electric and magnetic fields, 824 00:54:58,430 --> 00:55:01,840 not necessarily for progressive waves or standing 825 00:55:01,840 --> 00:55:03,390 waves, et cetera. 826 00:55:03,390 --> 00:55:06,360 It's something we want to think about 827 00:55:06,360 --> 00:55:09,950 and this is very surprising. 828 00:55:09,950 --> 00:55:13,390 Even if you have static electric and magnetic fields which 829 00:55:13,390 --> 00:55:17,070 are not parallel to each other, so that this is not 0, 830 00:55:17,070 --> 00:55:19,050 there is a flow of energy. 831 00:55:19,050 --> 00:55:21,140 It's something we want to think about. 832 00:55:21,140 --> 00:55:27,460 But in our case, E and B are perpendicular to each other. 833 00:55:32,260 --> 00:55:35,910 The electric field everywhere was 834 00:55:35,910 --> 00:55:39,930 in the x direction, the magnetic in the y direction. 835 00:55:39,930 --> 00:55:46,010 And so they are right angles, so it's just the x component of E 836 00:55:46,010 --> 00:55:48,200 and the y component of B. Well, they're 837 00:55:48,200 --> 00:55:49,990 the only components that are there. 838 00:55:49,990 --> 00:55:51,640 So it's 1 over Mu 0. 839 00:55:51,640 --> 00:55:55,260 E x times B y in the z direction, 840 00:55:55,260 --> 00:55:58,220 so this if E and B are perpendicular to each other, 841 00:55:58,220 --> 00:56:00,360 z is perpendicular to both of those, 842 00:56:00,360 --> 00:56:03,120 which is in the z direction. 843 00:56:03,120 --> 00:56:08,910 If I calculate this for this, I get this equation. 844 00:56:08,910 --> 00:56:11,550 And I can rewrite it. 845 00:56:11,550 --> 00:56:15,510 And I find that the energy is some constant, 846 00:56:15,510 --> 00:56:20,220 goes in the z direction, and this looks like sine 2 omega t 847 00:56:20,220 --> 00:56:22,800 times sine 2 kz. 848 00:56:22,800 --> 00:56:27,030 Going back to our diagram, what this 849 00:56:27,030 --> 00:56:34,010 looks like is that-- if you remember that E oscillates, 850 00:56:34,010 --> 00:56:38,740 it's a maximum here, maximum here, 851 00:56:38,740 --> 00:56:43,890 and it oscillates up and down, up and down, like this. 852 00:56:43,890 --> 00:56:50,360 B is a maximum in the middle, and that's going like this. 853 00:56:50,360 --> 00:56:54,970 The product of the two, it'll be 0 here, 854 00:56:54,970 --> 00:56:57,240 because B is always 0 here. 855 00:56:57,240 --> 00:57:00,530 It'll be here because E is always 0. 856 00:57:00,530 --> 00:57:02,720 And you cross B there for 0. 857 00:57:02,720 --> 00:57:05,610 So here, here, and here is going to be 0. 858 00:57:05,610 --> 00:57:07,180 And if you look at that function, 859 00:57:07,180 --> 00:57:11,260 its actually a function which has twice the frequency 860 00:57:11,260 --> 00:57:13,410 of the electric field oscillations 861 00:57:13,410 --> 00:57:15,390 or the magnetic field oscillations, 862 00:57:15,390 --> 00:57:21,200 and also half the wavelength. 863 00:57:21,200 --> 00:57:25,015 And you will find that the maximum is somewhere here. 864 00:57:30,800 --> 00:57:41,030 So if you look at where the maximum transfer of energy is, 865 00:57:41,030 --> 00:57:45,380 it's at the quarter and 3/4 location. 866 00:57:45,380 --> 00:57:48,110 And so it's consistent with this picture. 867 00:57:48,110 --> 00:57:52,990 Energy is doing this in that situation. 868 00:57:52,990 --> 00:57:56,490 And so that answers what they were ask. 869 00:57:56,490 --> 00:58:00,300 This is the Poynting vector as a function 870 00:58:00,300 --> 00:58:06,460 of-- for all positions in space as a function of time. 871 00:58:06,460 --> 00:58:09,820 This is the energy, the electric and magnetic field, 872 00:58:09,820 --> 00:58:14,890 and we found the magnetic field corresponding 873 00:58:14,890 --> 00:58:17,050 to the electric field. 874 00:58:17,050 --> 00:58:21,840 So this is another example of a possible solution 875 00:58:21,840 --> 00:58:25,700 to Maxwell's equations, this time corresponding 876 00:58:25,700 --> 00:58:28,410 to standing waves. 877 00:58:28,410 --> 00:58:32,280 As I mentioned before, I'm repeating myself , 878 00:58:32,280 --> 00:58:36,200 there are infinite possibilities of solutions 879 00:58:36,200 --> 00:58:40,090 of Maxwell's equations. 880 00:58:40,090 --> 00:58:43,120 So to cover them all makes no sense. 881 00:58:43,120 --> 00:58:48,050 What is important, that one gets a good understanding 882 00:58:48,050 --> 00:58:50,410 of the interesting situations. 883 00:58:50,410 --> 00:58:57,390 Interesting situations are some static solutions to, say, 884 00:58:57,390 --> 00:59:00,430 magnetic fields if you need special magnets. 885 00:59:00,430 --> 00:59:03,820 Or if you have a progressive wave, 886 00:59:03,820 --> 00:59:08,510 like light, or standing waves, like in the microwave, 887 00:59:08,510 --> 00:59:09,760 for example. 888 00:59:09,760 --> 00:59:14,280 And so I've taken two cases here. 889 00:59:14,280 --> 00:59:16,780 First progressive wave solution. 890 00:59:16,780 --> 00:59:19,970 And then standing wave solution. 891 00:59:19,970 --> 00:59:23,080 And from this, we will later go on to some other problems. 892 00:59:23,080 --> 00:59:24,630 Thank you.