1 00:00:00 --> 00:00:00,599 2 00:00:00,599 --> 00:00:05,453 These are the subjects will be covered during our third exam. 3 00:00:05,453 --> 00:00:09,418 There's no way I can cover all during this review. 4 00:00:09,418 --> 00:00:13,706 Nor can I cover all of them of course during the exam. 5 00:00:13,706 --> 00:00:16,538 I can only touch upon a few of them. 6 00:00:16,538 --> 00:00:21,231 And what I cannot cover today, what I will not cover today, 7 00:00:21,231 --> 00:00:25,762 can and will be on the exam. Let's first look at magnetic 8 00:00:25,762 --> 00:00:28,675 materials. Magnetic materials come in 9 00:00:28,675 --> 00:00:32,072 dia-, para- and ferromagnetic 10 00:00:32,072 --> 00:00:35,165 materials. The molecules and the atoms in 11 00:00:35,165 --> 00:00:38,799 para- and ferromagnetic materials have intrinsic 12 00:00:38,799 --> 00:00:42,82 magnetic dipole moments. These have always -- they're 13 00:00:42,82 --> 00:00:45,758 always a multiple of the Bohr magneton. 14 00:00:45,758 --> 00:00:48,232 Has to do with quantum mechanics. 15 00:00:48,232 --> 00:00:52,407 It's not part of eight oh two. And they are going to be 16 00:00:52,407 --> 00:00:56,66 aligned by the external field, I call that um the vacuum 17 00:00:56,66 --> 00:01:00,371 field. And the degree of success 18 00:01:00,371 --> 00:01:04,848 depends on the temperature and on the strength of that external 19 00:01:04,848 --> 00:01:07,374 field. The lower the temperature the 20 00:01:07,374 --> 00:01:10,984 easier it is to align them, to overcome the thermal 21 00:01:10,984 --> 00:01:13,944 agitation. And above a certain temperature 22 00:01:13,944 --> 00:01:17,409 which we call the Curie temperature, ferromagnet- 23 00:01:17,409 --> 00:01:21,163 magnetic material loses all its qualities and becomes 24 00:01:21,163 --> 00:01:24,773 paramagnetic and I have demonstrated that during my 25 00:01:24,773 --> 00:01:27,589 lectures. Suppose we have a solenoid and 26 00:01:27,589 --> 00:01:33,229 the solenoid has N windings, and the length of the solenoid 27 00:01:33,229 --> 00:01:35,859 is N. And the current I is flowing 28 00:01:35,859 --> 00:01:39,366 through the solenoid. Then the magnetic field 29 00:01:39,366 --> 00:01:44,387 generated by that solenoid which I have called the vacuum field, 30 00:01:44,387 --> 00:01:48,612 that magnetic field can be derived using Ampere's law, 31 00:01:48,612 --> 00:01:51,162 which you see down -- down there. 32 00:01:51,162 --> 00:01:55,227 That magnetic field is approximately mu zero times I 33 00:01:55,227 --> 00:01:58,495 times N divided by L. If now I put in here 34 00:01:58,495 --> 00:02:02,48 ferromagnetic material then I have 35 00:02:02,48 --> 00:02:06,333 to include this factor kappa of M or K of M, whatever you want 36 00:02:06,333 --> 00:02:08,669 to call it. The magnetic permeability, 37 00:02:08,669 --> 00:02:11,006 and this can be huge. This can be ten, 38 00:02:11,006 --> 00:02:13,722 a hundred, even up to a thousand and higher. 39 00:02:13,722 --> 00:02:17,384 So you get an enormous increase in magnetic field strength. 40 00:02:17,384 --> 00:02:20,984 Self-inductance is defined as magnetic flux divided by the 41 00:02:20,984 --> 00:02:23,447 current I. That's just the definition of 42 00:02:23,447 --> 00:02:26,289 self-inductance. If the magnetic field goes up 43 00:02:26,289 --> 00:02:30,204 by a factor of kappa M then of course the magnetic flux will go 44 00:02:30,204 --> 00:02:33,459 up by the same factor and so the 45 00:02:33,459 --> 00:02:36,833 self-inductance will go up. And you may remember a 46 00:02:36,833 --> 00:02:41,033 demonstration that I did when I had an iron core which I moved 47 00:02:41,033 --> 00:02:44,682 inside the solenoid and depending upon how far I moved 48 00:02:44,682 --> 00:02:48,745 it in could we see that the self-inductance went up and when 49 00:02:48,745 --> 00:02:51,981 I pulled it out self-inductance went down again. 50 00:02:51,981 --> 00:02:56,25 We have an interesting problem. I think it is assignment seven, 51 00:02:56,25 --> 00:03:00,312 whereby we have iron core here and then we have somewhere an 52 00:03:00,312 --> 00:03:04,099 air gap and you may want to revisit that to refresh your 53 00:03:04,099 --> 00:03:08,625 memory. Let's now turn to transformers. 54 00:03:08,625 --> 00:03:12,153 A transformer often comes in this shape. 55 00:03:12,153 --> 00:03:15,771 Let me move it a little bit to the right. 56 00:03:15,771 --> 00:03:21,38 Often comes in this shape which is then ferromagnetic material, 57 00:03:21,38 --> 00:03:26,175 to give coupling between the left and the right sides, 58 00:03:26,175 --> 00:03:29,16 also increases the magnetic field. 59 00:03:29,16 --> 00:03:33,05 This is the -- let's call this primary side. 60 00:03:33,05 --> 00:03:38,722 N one windings, index um self-inductance L one. 61 00:03:38,722 --> 00:03:44,426 And here I put in a voltmeter to always monitor that value, 62 00:03:44,426 --> 00:03:49,246 I call that V one. And this is the secondary side. 63 00:03:49,246 --> 00:03:52,884 N two windings. Self-inductance L two. 64 00:03:52,884 --> 00:03:57,999 And I put here a voltmeter which always monitors that 65 00:03:57,999 --> 00:04:01,244 voltage and I call that one V two. 66 00:04:01,244 --> 00:04:07,145 You can show with Faraday's law as I did in class in lectures 67 00:04:07,145 --> 00:04:13,499 that V two divided by V one, let's not worry about plus or 68 00:04:13,499 --> 00:04:16,412 minus signs, is N two divided by N one. 69 00:04:16,412 --> 00:04:20,322 That's a good approximation. Depends on how well the 70 00:04:20,322 --> 00:04:23,696 coupling goes. It depends on several factors, 71 00:04:23,696 --> 00:04:28,449 but you can come very close to this and this means then that if 72 00:04:28,449 --> 00:04:33,356 you make N two larger than N one then you can step up in voltage, 73 00:04:33,356 --> 00:04:35,963 we call that a step-up transformer. 74 00:04:35,963 --> 00:04:40,409 But you can also step down if you make N two smaller than N 75 00:04:40,409 --> 00:04:42,939 one. Under very special conditions 76 00:04:42,939 --> 00:04:48,389 will the power generated on the primary side be all 77 00:04:48,389 --> 00:04:53,534 consumed for a hundred percent or nearly a hundred percent on 78 00:04:53,534 --> 00:04:57,393 the secondary side. That is very very special. 79 00:04:57,393 --> 00:05:02,452 If that's the case then the time averaged power here V one I 80 00:05:02,452 --> 00:05:06,653 one is the same as V two I two here time averaged. 81 00:05:06,653 --> 00:05:11,884 And so as a logical consequence of that you'll find that I two 82 00:05:11,884 --> 00:05:15,657 divided by I one, let's not worry about minus 83 00:05:15,657 --> 00:05:19,895 signs, is that N one divided by N two. 84 00:05:19,895 --> 00:05:23,386 That however is not so easy as you may think. 85 00:05:23,386 --> 00:05:28,304 It only can work approximately and I mentioned that on the side 86 00:05:28,304 --> 00:05:31,874 in my lectures. If the resistance here and the 87 00:05:31,874 --> 00:05:36,555 resistance there is way way smaller than the value for omega 88 00:05:36,555 --> 00:05:39,093 L. And we did try to achieve that 89 00:05:39,093 --> 00:05:43,218 during one of the demonstrations that I gave on this. 90 00:05:43,218 --> 00:05:48,216 I remember we had the induction oven whereby N two was one and N 91 00:05:48,216 --> 00:05:52,96 one was very large, I don't remember what it was 92 00:05:52,96 --> 00:05:56,473 anymore but it was of the order of several hundred, 93 00:05:56,473 --> 00:05:59,846 maybe a thousand, and we managed to get a current 94 00:05:59,846 --> 00:06:04,272 in the secondary which was huge, which was close to one thousand 95 00:06:04,272 --> 00:06:07,013 amperes. It was enough to melt that iron 96 00:06:07,013 --> 00:06:09,401 nail. And we made every effort then 97 00:06:09,401 --> 00:06:13,898 to make sure that the resistance was much much smaller than omega 98 00:06:13,898 --> 00:06:16,006 L. I think problem seven-one of 99 00:06:16,006 --> 00:06:20,292 our assignments deals with that, and very naively assumes that 100 00:06:20,292 --> 00:06:26,392 this is all true. But you should realize that it 101 00:06:26,392 --> 00:06:33,188 is not always so easy to achieve the conditions for that. 102 00:06:33,188 --> 00:06:37,678 So let's now go to RLC circuits there. 103 00:06:37,678 --> 00:06:43,26 Let's take an uh system which has a resistor R, 104 00:06:43,26 --> 00:06:48,599 it has a self-inductor, a pure self-inductor, 105 00:06:48,599 --> 00:06:51,39 L, and a capacitance, C. 106 00:06:51,39 --> 00:06:55,611 AC. And this driving power supply 107 00:06:55,611 --> 00:07:00,742 provides with a voltage V which is V zero times cosine omega T. 108 00:07:00,742 --> 00:07:05,211 Keep in mind that this can be always be sine omega T of 109 00:07:05,211 --> 00:07:08,273 course. There is nothing special about 110 00:07:08,273 --> 00:07:11,997 cosine in life. The steady state solution that 111 00:07:11,997 --> 00:07:16,632 is not when you turn the thing on but if you wait awhile, 112 00:07:16,632 --> 00:07:20,522 you get a steady state solution for the current. 113 00:07:20,522 --> 00:07:25,57 And the current that is going to flow now is V zero divided by 114 00:07:25,57 --> 00:07:32,228 the square root of R squared plus omega L minus one 115 00:07:32,228 --> 00:07:39,34 over omega C squared times the cosine of omega T minus phi. 116 00:07:39,34 --> 00:07:47,065 And the tangent of phi is omega L minus one over omega C divided 117 00:07:47,065 --> 00:07:50,989 by R. We call this the reactance. 118 00:07:50,989 --> 00:07:55,893 The upstairs. For which we give often the 119 00:07:55,893 --> 00:08:00,185 symbol X. And so this is also X then 120 00:08:00,185 --> 00:08:06,23 divided by R. And this whole square root that 121 00:08:06,23 --> 00:08:10,076 we have here, we call that the impedance. 122 00:08:10,076 --> 00:08:13,728 The units are ohms. And we call that Z. 123 00:08:13,728 --> 00:08:17,958 And so the maximum current that you can have, 124 00:08:17,958 --> 00:08:22,572 the current is of course oscillating with angular 125 00:08:22,572 --> 00:08:26,994 frequency omega, the maximum value that you can 126 00:08:26,994 --> 00:08:30,839 have for the current, which I call I max, 127 00:08:30,839 --> 00:08:38,575 is then V zero divided by Z. Then the cosine term is either 128 00:08:38,575 --> 00:08:44,494 plus or minus one. I can plot now this I max as a 129 00:08:44,494 --> 00:08:50,782 function of frequency. So here is frequency and here 130 00:08:50,782 --> 00:08:55,714 is I max. If the frequency is very low or 131 00:08:55,714 --> 00:09:03,112 near zero then this term here becomes infinitely high because 132 00:09:03,112 --> 00:09:06,913 the impedance is infinitely high 133 00:09:06,913 --> 00:09:10,133 and so the current is zero. I max is zero. 134 00:09:10,133 --> 00:09:12,725 There's no current flowing at all. 135 00:09:12,725 --> 00:09:16,809 When we go to very high frequencies it is the omega L 136 00:09:16,809 --> 00:09:21,364 term that goes to infinity. And so again Z goes to infinity 137 00:09:21,364 --> 00:09:25,84 so again I max goes to zero. And for other values of omega 138 00:09:25,84 --> 00:09:30,63 you get an I max which is not zero and so you get a curve like 139 00:09:30,63 --> 00:09:33,929 this which has the name of resonance curve. 140 00:09:33,929 --> 00:09:37,855 This I max reaches a maximum value 141 00:09:37,855 --> 00:09:41,911 when the system is at resonance, that's what we call 142 00:09:41,911 --> 00:09:44,853 resonance. And that's the case clearly 143 00:09:44,853 --> 00:09:49,306 when the reactance is zero. Because when the reactance is 144 00:09:49,306 --> 00:09:53,361 zero this part vanishes. And if the reactance is not 145 00:09:53,361 --> 00:09:57,098 zero then the maximum current can only be lower, 146 00:09:57,098 --> 00:10:00,676 can never be higher. And so when X equals zero 147 00:10:00,676 --> 00:10:05,686 you'll find that omega L is one divided by omega C and so the -- 148 00:10:05,686 --> 00:10:09,821 the frequency for which that happens, 149 00:10:09,821 --> 00:10:14,355 I call that omega zero, reminds me that it is the -- 150 00:10:14,355 --> 00:10:18,09 the resonance, is one divided by the square 151 00:10:18,09 --> 00:10:21,38 root of LC. When I am at resonance phi 152 00:10:21,38 --> 00:10:24,848 becomes zero. So there is no phase delay 153 00:10:24,848 --> 00:10:28,316 between current and the driving voltage. 154 00:10:28,316 --> 00:10:31,25 They are in phase with each other. 155 00:10:31,25 --> 00:10:36,496 And the value for I max now simply becomes V zero divided by 156 00:10:36,496 --> 00:10:40,586 R. Because the impedance itself 157 00:10:40,586 --> 00:10:42,422 becomes R. Very boring, 158 00:10:42,422 --> 00:10:45,758 very simple, you're looking here at Ohm's 159 00:10:45,758 --> 00:10:47,843 law. When the system is at 160 00:10:47,843 --> 00:10:50,929 resonance, forget the self-inductance, 161 00:10:50,929 --> 00:10:54,266 forget the capacitor, they are not there, 162 00:10:54,266 --> 00:10:59,104 they annihilate each other, and so the system behaves as if 163 00:10:59,104 --> 00:11:03,942 there were only a resistor, and that's exactly what you see 164 00:11:03,942 --> 00:11:06,861 here. I have here some numbers which 165 00:11:06,861 --> 00:11:11,496 you have seen before. During my lectures. 166 00:11:11,496 --> 00:11:16,126 You can download this from the Web but you have to go back to 167 00:11:16,126 --> 00:11:18,672 the lecture when I discussed that. 168 00:11:18,672 --> 00:11:21,604 And you see here s- some numbers for R, 169 00:11:21,604 --> 00:11:25,924 L and C and also for V zero. And I calculate for you here 170 00:11:25,924 --> 00:11:30,091 the resonance frequency. I calculate the frequency also 171 00:11:30,091 --> 00:11:34,103 in terms of kilohertz. And here you see the impedance 172 00:11:34,103 --> 00:11:38,346 and here you see the reactance. If I'm ten percent below 173 00:11:38,346 --> 00:11:42,822 resonance notice that the one over omega C 174 00:11:42,822 --> 00:11:45,38 term is always larger than omega L. 175 00:11:45,38 --> 00:11:49,894 So your reactance in this case becomes minus eighty-six ohms. 176 00:11:49,894 --> 00:11:54,258 The minus sign has of course no consequence for the current 177 00:11:54,258 --> 00:11:56,816 because you have an X squared here. 178 00:11:56,816 --> 00:12:01,481 But notice that Z is now almost exclusively determined by X and 179 00:12:01,481 --> 00:12:04,942 not by R anymore. Because the ten ohms of the R 180 00:12:04,942 --> 00:12:07,425 here play no role, almost no role, 181 00:12:07,425 --> 00:12:09,908 in comparison with the eighty-six. 182 00:12:09,908 --> 00:12:15,697 Z becomes eighty-seven and the maximum current is one-tenth of 183 00:12:15,697 --> 00:12:18,239 an ampere. When you're on resonance, 184 00:12:18,239 --> 00:12:21,363 and that is characteristic for on resonance, 185 00:12:21,363 --> 00:12:25,286 the two omega L and one over omega C eat each other up. 186 00:12:25,286 --> 00:12:29,645 They annihilate each other and so the reactance becomes zero. 187 00:12:29,645 --> 00:12:32,115 So Z now is just pure R. X is zero. 188 00:12:32,115 --> 00:12:35,457 And so the maximum current in this case is one. 189 00:12:35,457 --> 00:12:38,871 Because I chose V zero at ten and I chose R ten. 190 00:12:38,871 --> 00:12:43,956 And then when I'm ten percent above resonance then the 191 00:12:43,956 --> 00:12:49,2 omega L term is larger than the reactance of the capacitor and 192 00:12:49,2 --> 00:12:52,725 accordingly you get a lower current again, 193 00:12:52,725 --> 00:12:56,077 about one-eighth of the -- of an ampere. 194 00:12:56,077 --> 00:13:01,493 And so you see this curve being formed in a very natural way and 195 00:13:01,493 --> 00:13:05,447 that's quantitative you see there some numbers. 196 00:13:05,447 --> 00:13:10,605 So now comes the question which of course in practice is very 197 00:13:10,605 --> 00:13:13,785 important. And that has to do with the 198 00:13:13,785 --> 00:13:17,949 power that is generated by the power 199 00:13:17,949 --> 00:13:20,719 supply. That power comes out in the 200 00:13:20,719 --> 00:13:24,222 form of heat. Heat in the resistor and so if 201 00:13:24,222 --> 00:13:29,192 you time average the power then the time average value you can 202 00:13:29,192 --> 00:13:33,999 take the -- the voltage of the power supply multiply that by 203 00:13:33,999 --> 00:13:37,258 the current. You could also take the time 204 00:13:37,258 --> 00:13:41,902 average value of I squared R. Because all that energy will 205 00:13:41,902 --> 00:13:45,812 ultimately come out in the form of 206 00:13:45,812 --> 00:13:49,901 heat of the resistor. Either one will be fine. 207 00:13:49,901 --> 00:13:54,898 I've decided to take this one. So I will get then V zero 208 00:13:54,898 --> 00:14:00,35 cosine omega T -- the I becomes V zero divided by Z times the 209 00:14:00,35 --> 00:14:05,438 cosine omega T minus phi. This is the power at any moment 210 00:14:05,438 --> 00:14:08,891 in time. I will do the time averaging a 211 00:14:08,891 --> 00:14:12,889 little later. When I see cosine omega T minus 212 00:14:12,889 --> 00:14:17,523 phi, that reminds me of my high school 213 00:14:17,523 --> 00:14:22,723 days, cosine alpha minus cosine -- no, cosine alpha minus beta 214 00:14:22,723 --> 00:14:27,241 is cosine alpha cosine beta plus sine alpha sine beta. 215 00:14:27,241 --> 00:14:30,309 That was drilled into my memory here. 216 00:14:30,309 --> 00:14:33,037 I will never forget that I think. 217 00:14:33,037 --> 00:14:38,237 And so I will write down here -- my math teacher will be proud 218 00:14:38,237 --> 00:14:43,437 of me -- cosine omega T cosine phi plus sine omega T sine phi. 219 00:14:43,437 --> 00:14:47,614 So this is this term. If I'm going to time average 220 00:14:47,614 --> 00:14:52,133 it, I have a cosine omega T multiplied 221 00:14:52,133 --> 00:14:56,769 by sine omega T, that time average is zero. 222 00:14:56,769 --> 00:15:02,288 So this term vanishes. So the time average value of 223 00:15:02,288 --> 00:15:07,586 the power I get a V zero squared, I get a Z here, 224 00:15:07,586 --> 00:15:13,658 and now I have here cosine omega T times cosine omega T. 225 00:15:13,658 --> 00:15:19,177 The time average value of cosine squared omega T is 226 00:15:19,177 --> 00:15:23,702 one-half. So I get a two here. 227 00:15:23,702 --> 00:15:27,205 And then I still have my cosine phi there. 228 00:15:27,205 --> 00:15:30,879 And I'm done. If you like to get rid of this 229 00:15:30,879 --> 00:15:33,442 uh cosine phi, you can do that. 230 00:15:33,442 --> 00:15:37,542 Because remember, the way that phi is defined the 231 00:15:37,542 --> 00:15:41,301 tangent of phi is the reactance divided by R. 232 00:15:41,301 --> 00:15:45,829 You still see it there. So that means if this angle is 233 00:15:45,829 --> 00:15:49,161 ninety degrees that this side must be Z. 234 00:15:49,161 --> 00:15:54,799 That's the square root of X squared plus R squared. 235 00:15:54,799 --> 00:15:58,559 That's this part. And so the cosine of phi is 236 00:15:58,559 --> 00:16:02,659 also R divided by Z. And so if you prefer that -- 237 00:16:02,659 --> 00:16:07,871 there's no particular advantage but if you prefer that you can 238 00:16:07,871 --> 00:16:11,288 write down for cosine phi R divided by Z. 239 00:16:11,288 --> 00:16:16,329 And so you get V zero squared times R divided by two and now 240 00:16:16,329 --> 00:16:20,515 you get Z squared. And so there you see the power, 241 00:16:20,515 --> 00:16:23,676 time averaged power in an RLC circuit. 242 00:16:23,676 --> 00:16:27,692 So now we can look at resonance. 243 00:16:27,692 --> 00:16:30,329 It's always a very special situation. 244 00:16:30,329 --> 00:16:32,967 When we are at resonance, Z equals R. 245 00:16:32,967 --> 00:16:35,458 So you replace this capital Z by R. 246 00:16:35,458 --> 00:16:39,049 And then you find V zero squared divided by two R. 247 00:16:39,049 --> 00:16:42,932 That's utterly trivial. You could have predicted that. 248 00:16:42,932 --> 00:16:46,229 It's really Ohm's law staring you in the face. 249 00:16:46,229 --> 00:16:50,259 There is no self-inductance and there is no capacitor at 250 00:16:50,259 --> 00:16:52,824 resonance. So you might as well have 251 00:16:52,824 --> 00:16:55,828 treated it as a simple system only with R. 252 00:16:55,828 --> 00:17:00,298 And you find immediately then that answer. 253 00:17:00,298 --> 00:17:05,514 At any other frequency than omega zero Z would always be 254 00:17:05,514 --> 00:17:09,688 larger than R. You see that immediately here. 255 00:17:09,688 --> 00:17:14,904 And so that means that the average power would always be 256 00:17:14,904 --> 00:17:18,319 lower. So it's only at resonance that 257 00:17:18,319 --> 00:17:22,018 you generate the highest power possible. 258 00:17:22,018 --> 00:17:25,717 All right. Let's go back to our subjects 259 00:17:25,717 --> 00:17:29,511 and see what's next. We did LRC circuits. 260 00:17:29,511 --> 00:17:35,941 Oh yes, we're getting now to traveling waves and standing 261 00:17:35,941 --> 00:17:38,796 waves. Let's start with a traveling 262 00:17:38,796 --> 00:17:42,658 wave in a string. That's always very nice to do 263 00:17:42,658 --> 00:17:47,781 that because the parallel with electromagnetic waves is nearly 264 00:17:47,781 --> 00:17:51,895 a hundred percent. I have a string that oscillates 265 00:17:51,895 --> 00:17:56,01 in the Y direction. And let's say it propagates in 266 00:17:56,01 --> 00:17:59,789 the X direction. Y zero cosine K X minus omega 267 00:17:59,789 --> 00:18:02,56 T. When I see this traveling wave, 268 00:18:02,56 --> 00:18:09,152 so the traveling wave in the Y direction and it propagates in 269 00:18:09,152 --> 00:18:13,338 the X direction, I see immediately that it goes 270 00:18:13,338 --> 00:18:18,161 into the plus X direction, because I have a minus sign 271 00:18:18,161 --> 00:18:21,892 here, it tells me in the plus X direction. 272 00:18:21,892 --> 00:18:26,26 K gives me all the information on the wavelength. 273 00:18:26,26 --> 00:18:31,538 It's two pi divided by lambda. Omega equals two pi times F, 274 00:18:31,538 --> 00:18:37,18 F being the frequency in hertz. It would also be two pi divided 275 00:18:37,18 --> 00:18:40,547 by capital T, capital T 276 00:18:40,547 --> 00:18:45,622 now being the period of one complete oscillation. 277 00:18:45,622 --> 00:18:51,332 The speed of propagation, the disturbance propagates in 278 00:18:51,332 --> 00:18:55,35 the X direction, is omega divided by K. 279 00:18:55,35 --> 00:19:01,271 And so if I draw here at a particular moment in time this 280 00:19:01,271 --> 00:19:07,404 string and this would be Y zero, that would also be Y zero, 281 00:19:07,404 --> 00:19:14,912 and it would propagate in this direction with that velocity, 282 00:19:14,912 --> 00:19:18,461 and this would be the wavelength, lambda, 283 00:19:18,461 --> 00:19:23,431 and that lambda is V times T. It's immediately obvious if 284 00:19:23,431 --> 00:19:28,578 something propagates with a speed V and it has T seconds to 285 00:19:28,578 --> 00:19:33,37 go for one oscillation it moves over a distance lambda. 286 00:19:33,37 --> 00:19:37,63 That's eight oh one. Do not confuse this speed of 287 00:19:37,63 --> 00:19:42,245 propagation with which the disturbance moves with the 288 00:19:42,245 --> 00:19:47,776 actual speed of the atoms, of the particles in the string. 289 00:19:47,776 --> 00:19:51,515 If you were a particle in the string and you were sitting 290 00:19:51,515 --> 00:19:54,053 here, you never move in this direction. 291 00:19:54,053 --> 00:19:56,19 That's the same with water waves. 292 00:19:56,19 --> 00:19:59,061 If a water wave comes by all you do is this. 293 00:19:59,061 --> 00:20:02,2 You go up and down. Your motion is only in the Y 294 00:20:02,2 --> 00:20:04,804 direction. You go from here to there and 295 00:20:04,804 --> 00:20:08,01 you oscillate back and forth with that frequency, 296 00:20:08,01 --> 00:20:11,148 angular frequency omega. And so if you're really 297 00:20:11,148 --> 00:20:14,821 interested in the speed with which you are moving up and 298 00:20:14,821 --> 00:20:17,644 down, that of course, 299 00:20:17,644 --> 00:20:21,798 that speed which we call the transverse speed is dY dT, 300 00:20:21,798 --> 00:20:26,106 and you have to do that then for a particular location X, 301 00:20:26,106 --> 00:20:29,799 which you can choose, wherever you want to sit on 302 00:20:29,799 --> 00:20:32,876 that string. And I think we had a problem 303 00:20:32,876 --> 00:20:37,337 once where we asked you in the homework what the transverse 304 00:20:37,337 --> 00:20:39,414 speed was. So if now move to 305 00:20:39,414 --> 00:20:42,953 electromagnetic waves then very little changes. 306 00:20:42,953 --> 00:20:46,414 Uh take an electromagnetic wave, a plain wave, 307 00:20:46,414 --> 00:20:51,107 whereby the E vector is in the Y direction, 308 00:20:51,107 --> 00:20:54,728 so E is E zero, there's the unit vector in the 309 00:20:54,728 --> 00:20:57,704 Y direction, cosine K X minus omega T. 310 00:20:57,704 --> 00:21:02,371 This could be a sine of course. There is nothing special in 311 00:21:02,371 --> 00:21:06,313 life with a cosine. I come to the same conclusion. 312 00:21:06,313 --> 00:21:10,496 The wave is traveling in the plus X direction and the 313 00:21:10,496 --> 00:21:15,243 velocity, speed of propagation, is omega divided by K and if 314 00:21:15,243 --> 00:21:18,622 this is vacuum which will I assume for now, 315 00:21:18,622 --> 00:21:22,644 then that is C and as we have seen 316 00:21:22,644 --> 00:21:27,688 from Maxwell's equations C is one divided by the square root 317 00:21:27,688 --> 00:21:32,134 of epsilon zero mu zero. Surprising as that is that's 318 00:21:32,134 --> 00:21:34,869 comes out of Maxwell's equations. 319 00:21:34,869 --> 00:21:39,913 And so if now you were asked, which would be a natural thing 320 00:21:39,913 --> 00:21:44,102 for me to ask you, what is the associated magnetic 321 00:21:44,102 --> 00:21:48,718 field, well, the magnetic field is perpendicular to the 322 00:21:48,718 --> 00:21:53,335 direction of propagation. It's also perpendicular to E. 323 00:21:53,335 --> 00:21:58,479 And B zero is E zero divided by C in vacuum. 324 00:21:58,479 --> 00:22:03,082 And so if I make a coordinate system, this is X, 325 00:22:03,082 --> 00:22:07,88 this is Y and this is Z, notice that my coordinate 326 00:22:07,88 --> 00:22:13,854 system always is chosen so that X roof cross Y roof is Z roof, 327 00:22:13,854 --> 00:22:18,652 this is called the right-handed coordinate system. 328 00:22:18,652 --> 00:22:23,156 If you choose any other system you're an idiot. 329 00:22:23,156 --> 00:22:27,586 You always get yourself into trouble. 330 00:22:27,586 --> 00:22:30,917 Make sure that you always choose this as a coordinate 331 00:22:30,917 --> 00:22:33,287 system. And notice I have X cross Y is 332 00:22:33,287 --> 00:22:36,041 Z coming out of the blackboard. At a moment, 333 00:22:36,041 --> 00:22:39,628 a particular moment in time, let's say the E vector is in 334 00:22:39,628 --> 00:22:41,806 this direction, in the Y direction, 335 00:22:41,806 --> 00:22:45,778 I pick a random moment in time and I pick a random location for 336 00:22:45,778 --> 00:22:47,892 X. Now I must be sure that E cross 337 00:22:47,892 --> 00:22:50,198 B is in the direction of propagation. 338 00:22:50,198 --> 00:22:53,721 So E cross B must be in this case in the direction of X. 339 00:22:53,721 --> 00:22:57,051 Because it's going in the plus X 340 00:22:57,051 --> 00:23:00,469 direction. And so my problem is solved. 341 00:23:00,469 --> 00:23:04,695 I know that that can only happen if B is in this 342 00:23:04,695 --> 00:23:08,292 direction. At this moment in time at this 343 00:23:08,292 --> 00:23:11,08 location. And that's all I need. 344 00:23:11,08 --> 00:23:15,846 Because now I can write down that the B vector that is 345 00:23:15,846 --> 00:23:21,062 associated with that electric field is E zero divided by C, 346 00:23:21,062 --> 00:23:27,537 that is the largest value that the magnetic field can have, 347 00:23:27,537 --> 00:23:31,405 times the same cosine K X minus omega T. 348 00:23:31,405 --> 00:23:36,86 And now I must have here Z roof and now I'm in business. 349 00:23:36,86 --> 00:23:42,81 So now I have the E vector that goes with the -- the B vector 350 00:23:42,81 --> 00:23:48,662 that goes with the E vector. We would often be interested in 351 00:23:48,662 --> 00:23:54,613 energy, how much energy per unit area per unit time is in the 352 00:23:54,613 --> 00:23:59,77 plain wave. That is given by the pointing 353 00:23:59,77 --> 00:24:03,279 vector S which is E cross B divided by mu zero. 354 00:24:03,279 --> 00:24:07,093 We're still in vacuum. I still assume that it's all 355 00:24:07,093 --> 00:24:10,221 vacuum, so this is watts per square meter. 356 00:24:10,221 --> 00:24:14,95 Of course we are in general not interested in the instantaneous 357 00:24:14,95 --> 00:24:18,764 value of the pointing vector, who cares about that, 358 00:24:18,764 --> 00:24:22,731 it oscillates like mad, I'm more interested in a time 359 00:24:22,731 --> 00:24:26,087 average value. And so the time average value, 360 00:24:26,087 --> 00:24:29,902 this has a cosine omega T, this has 361 00:24:29,902 --> 00:24:33,667 a cosine omega T, the product average out to be 362 00:24:33,667 --> 00:24:36,368 one-half of cosine square omega T. 363 00:24:36,368 --> 00:24:38,905 I get an E zero. I get a B zero. 364 00:24:38,905 --> 00:24:42,998 And I get a mu zero. And if now you want to get rid 365 00:24:42,998 --> 00:24:48,072 of your B zero because you want to get everything in terms of E 366 00:24:48,072 --> 00:24:52,247 zero, you can replace B zero by E zero divided by C. 367 00:24:52,247 --> 00:24:57,076 And so this would also be fine. E zero squared divided by mu 368 00:24:57,076 --> 00:24:59,859 zero C. And so now we have the time 369 00:24:59,859 --> 00:25:04,278 average value of the pointing vector. 370 00:25:04,278 --> 00:25:09,616 The moment that you move to vacuum -- from vacuum to -- to 371 00:25:09,616 --> 00:25:14,392 matter, so we now go from vacuum, we move to matter, 372 00:25:14,392 --> 00:25:19,917 and the matter has dielectric constant K and it has magnetic 373 00:25:19,917 --> 00:25:24,319 permeability K of M. That's only important if we 374 00:25:24,319 --> 00:25:29,375 deal with ferromagnetism because with paramagnetism and 375 00:25:29,375 --> 00:25:34,807 diamagnetism K of M is always practically for all practical 376 00:25:34,807 --> 00:25:39,349 purposes one. But kappa can vary a great 377 00:25:39,349 --> 00:25:42,023 deal. From the various substances. 378 00:25:42,023 --> 00:25:46,237 And so now all you have to do, if you go to Maxwell's 379 00:25:46,237 --> 00:25:49,803 equations, if they were given only in vacuum, 380 00:25:49,803 --> 00:25:54,098 then you have to replace epsilon zero by kappa epsilon 381 00:25:54,098 --> 00:25:56,611 zero. That's already done there. 382 00:25:56,611 --> 00:26:01,149 And you have to replace mu zero by kappa M times mu zero. 383 00:26:01,149 --> 00:26:06,093 And therefore you have to place C by V because the velocity is 384 00:26:06,093 --> 00:26:11,361 different in matter of electromagnetic radiation. 385 00:26:11,361 --> 00:26:17,438 And you see immediately how it changes because epsilon zero and 386 00:26:17,438 --> 00:26:23,516 mu zero have to be replaced by epsilon zero K and mu zero kappa 387 00:26:23,516 --> 00:26:29,496 of M and so you see here that the velocity following my recipe 388 00:26:29,496 --> 00:26:33,614 becomes epsilon zero mu zero kappa kappa M. 389 00:26:33,614 --> 00:26:39,594 And so my B zero becomes now E zero divided by V and no longer 390 00:26:39,594 --> 00:26:44,887 divided by C. So this B zero becomes E zero 391 00:26:44,887 --> 00:26:47,904 divided by V. Kappa can be a very strong 392 00:26:47,904 --> 00:26:50,766 function of frequency. So can kappa M, 393 00:26:50,766 --> 00:26:55,097 but kappa M of course is only important for ferromagnetic 394 00:26:55,097 --> 00:26:57,805 materials. I've shown you an example 395 00:26:57,805 --> 00:27:01,285 before that kappa, the dielectric constant for 396 00:27:01,285 --> 00:27:05,462 water, was eighty at low frequencies, even at a hundred 397 00:27:05,462 --> 00:27:07,628 megahertz. Radio frequencies. 398 00:27:07,628 --> 00:27:11,495 It was still eighty. But at the visible light where 399 00:27:11,495 --> 00:27:16,711 you're dealing with frequencies of a few times ten 400 00:27:16,711 --> 00:27:20,762 to the fourteen hertz, that value for kappa was way 401 00:27:20,762 --> 00:27:23,355 lower, was one point seven seven. 402 00:27:23,355 --> 00:27:27,65 And so kappa is a small function, a strong function of 403 00:27:27,65 --> 00:27:30,973 um of frequency, and we introduce index of 404 00:27:30,973 --> 00:27:33,809 refraction, which is C divided by V. 405 00:27:33,809 --> 00:27:38,022 And since V itself is a strong function of frequency, 406 00:27:38,022 --> 00:27:42,641 the index of refraction can also be a very strong function 407 00:27:42,641 --> 00:27:47,585 of frequency. Was one point three roughly for 408 00:27:47,585 --> 00:27:50,737 water. But it is a little different 409 00:27:50,737 --> 00:27:54,167 from -- for red light from blue light. 410 00:27:54,167 --> 00:27:58,153 If not, we wouldn't be able to see rainbows. 411 00:27:58,153 --> 00:28:01,12 OK. Let's now talk about standing 412 00:28:01,12 --> 00:28:04,55 waves. Let's start with strings again. 413 00:28:04,55 --> 00:28:10,206 This is a string with length L and I generate in this string a 414 00:28:10,206 --> 00:28:13,914 standing wave. Standing waves can only be 415 00:28:13,914 --> 00:28:20,403 generated at very discrete frequencies for very special 416 00:28:20,403 --> 00:28:24,117 wavelengths. It's a resonance phenomenon. 417 00:28:24,117 --> 00:28:29,595 And the lowest frequency for which this occurs will make the 418 00:28:29,595 --> 00:28:32,566 string oscillate in this fashion. 419 00:28:32,566 --> 00:28:37,672 The string will do this. This is called the fundamental. 420 00:28:37,672 --> 00:28:42,778 Also the first harmonic. Then there is a second harmonic 421 00:28:42,778 --> 00:28:48,07 which is the next frequency up for which it will resonate, 422 00:28:48,07 --> 00:28:54,012 which adds an extra node, there is already a node here 423 00:28:54,012 --> 00:28:57,546 and a node there, and the system will then 424 00:28:57,546 --> 00:29:01,166 oscillate like so. Fweet fweet fweet fweet. 425 00:29:01,166 --> 00:29:04,527 And then so this is the second harmonic. 426 00:29:04,527 --> 00:29:09,44 And then I can go to the third harmonic and I can continue 427 00:29:09,44 --> 00:29:13,75 forever, not quite, but I can continue quite a bit, 428 00:29:13,75 --> 00:29:17,198 and this now would be the third harmonic. 429 00:29:17,198 --> 00:29:23,317 And the frequencies that are the resonant frequencies are 430 00:29:23,317 --> 00:29:25,755 given by F of N, N being Nancy, 431 00:29:25,755 --> 00:29:29,41 that means either one or two or three or four, 432 00:29:29,41 --> 00:29:33,96 one being the fundamental, two being the second harmonic, 433 00:29:33,96 --> 00:29:38,996 this frequency is given N times V divided by two L whereby V is 434 00:29:38,996 --> 00:29:43,871 the speed of propagation of a disturbance along the direction 435 00:29:43,871 --> 00:29:47,039 of the string. And L in this case is the 436 00:29:47,039 --> 00:29:49,557 length. And the asso- associated 437 00:29:49,557 --> 00:29:53,944 wavelength lambda for that particular harmonic is two L 438 00:29:53,944 --> 00:29:59,531 divided by N. If you put in N equals one you 439 00:29:59,531 --> 00:30:05,41 see that the wavelength is indeed twice the length and if 440 00:30:05,41 --> 00:30:11,815 you put in N equals two you see that the wavelength is exactly 441 00:30:11,815 --> 00:30:14,964 L. So let us write down for the 442 00:30:14,964 --> 00:30:19,584 fundamental the equation for a standing wave. 443 00:30:19,584 --> 00:30:25,043 Y one, one refers to the fundamental, would be Y zero 444 00:30:25,043 --> 00:30:30,083 one, times the cosine of omega one T 445 00:30:30,083 --> 00:30:36,78 times the sine of K one X. This is very different from a 446 00:30:36,78 --> 00:30:42,016 traveling wave. All the time information now 447 00:30:42,016 --> 00:30:47,618 here is decoupled from the spatial information. 448 00:30:47,618 --> 00:30:55,046 K one is again as before -- let me write down a nicer K one -- 449 00:30:55,046 --> 00:31:01,866 two pi divided by lambda. And omega one is two pi times F 450 00:31:01,866 --> 00:31:06,291 one. What is so special here is that 451 00:31:06,291 --> 00:31:09,224 there are points here, values for X, 452 00:31:09,224 --> 00:31:11,822 for which the sine becomes zero. 453 00:31:11,822 --> 00:31:16,683 In the case of the fundamental put in X equals zero and you 454 00:31:16,683 --> 00:31:19,532 find that that sine is always zero. 455 00:31:19,532 --> 00:31:23,555 But if you put in X equals L, you can check that, 456 00:31:23,555 --> 00:31:27,075 it's also zero. And if you go to the second 457 00:31:27,075 --> 00:31:33,192 harmonic you will find that the sine is zero there as well. 458 00:31:33,192 --> 00:31:36,906 And so there are points now which never move, 459 00:31:36,906 --> 00:31:41,125 we call those nodes. And that's very characteristic 460 00:31:41,125 --> 00:31:44,838 for a standing wave. I could go to the second 461 00:31:44,838 --> 00:31:50,154 harmonic and all I would have to do is put here a two and here a 462 00:31:50,154 --> 00:31:52,939 two and here a two and here a two. 463 00:31:52,939 --> 00:31:56,146 It would have its own little amplitude. 464 00:31:56,146 --> 00:31:59,606 Which would be this. It would have its own 465 00:31:59,606 --> 00:32:03,066 frequency. But that frequency omega two is 466 00:32:03,066 --> 00:32:07,77 nonnegotiable. That must be two omega one. 467 00:32:07,77 --> 00:32:11,871 And if I go to the third harmonic, then omega three is 468 00:32:11,871 --> 00:32:16,358 going to be three omega one. But it has its own wavelength, 469 00:32:16,358 --> 00:32:20,768 you'll get K of two would be two pi divided by lambda two, 470 00:32:20,768 --> 00:32:24,713 and the lambdas are -- the lambdas are given by this 471 00:32:24,713 --> 00:32:27,498 relationship. I want you to get a few 472 00:32:27,498 --> 00:32:32,063 minutes' rest now so that you can digest this and then we'll 473 00:32:32,063 --> 00:32:36,009 go to electromagnetic standing waves. 474 00:32:36,009 --> 00:32:38,252 And I want you to see this again. 475 00:32:38,252 --> 00:32:42,527 You've seen that before but I want you to see it in a way that 476 00:32:42,527 --> 00:32:46,453 you have not seen it yet. We have here a -- a rubber hose 477 00:32:46,453 --> 00:32:50,869 which we can oscillate and we're going to oscillate it in such a 478 00:32:50,869 --> 00:32:54,794 way, at least that's our goal, to get the third harmonic, 479 00:32:54,794 --> 00:32:58,158 and it's not so easy to get exactly on resonance, 480 00:32:58,158 --> 00:33:01,663 but we will try that, and Marcos decided to make it 481 00:33:01,663 --> 00:33:06,359 very beautiful for you, so he's going to put up a -- 482 00:33:06,359 --> 00:33:10,426 a black screen there because once we are at resonance we're 483 00:33:10,426 --> 00:33:14,913 going to strobe the s- string so that you will in fact be able to 484 00:33:14,913 --> 00:33:17,998 follow the motion. Your eyes can't see what's 485 00:33:17,998 --> 00:33:21,363 going up and what's going down. It goes too fast. 486 00:33:21,363 --> 00:33:24,378 But when we strobe it we can slow that down, 487 00:33:24,378 --> 00:33:26,832 make the thing actually stand still. 488 00:33:26,832 --> 00:33:30,688 And that's our objective, so you'll be able to see that. 489 00:33:30,688 --> 00:33:33,983 And so let me turn on the strobe, and this black 490 00:33:33,983 --> 00:33:37,349 background will help, the strobe frequency is not 491 00:33:37,349 --> 00:33:42,121 exactly the same as the frequency of the -- of 492 00:33:42,121 --> 00:33:46,598 the uh oscillating string, so that allows you to actually 493 00:33:46,598 --> 00:33:50,596 see the slow motion. And no one is arguing with me, 494 00:33:50,596 --> 00:33:55,473 right, when the center portion is down, the left and the right 495 00:33:55,473 --> 00:33:57,871 portion are up, and vice versa. 496 00:33:57,871 --> 00:34:02,748 So you can really see now the characteristics of this standing 497 00:34:02,748 --> 00:34:05,307 wave. Notice you see in this case 498 00:34:05,307 --> 00:34:07,625 four nodes. One at either end, 499 00:34:07,625 --> 00:34:11,782 and you see two nodes in the middle. 500 00:34:11,782 --> 00:34:16,334 That is where the s- sine of that curve is always zero. 501 00:34:16,334 --> 00:34:20,802 I can add another strobe light at a slightly different 502 00:34:20,802 --> 00:34:24,173 frequency. I collaborated for a few years 503 00:34:24,173 --> 00:34:27,208 with uh an artist, his name was Tsai, 504 00:34:27,208 --> 00:34:32,097 he worked at the Center for Advanced Visual Studies here at 505 00:34:32,097 --> 00:34:37,491 MIT, and he actually made art by oscillating objects at resonance 506 00:34:37,491 --> 00:34:43,054 frequency, rods and strings, and he strobed them in a 507 00:34:43,054 --> 00:34:49,437 way that I'm doing, so I actually learned this from 508 00:34:49,437 --> 00:34:53,266 him. Quite pretty but also very 509 00:34:53,266 --> 00:34:58,755 instructive. You can really see what's going 510 00:34:58,755 --> 00:35:02,968 on. In green you see the s- string 511 00:35:02,968 --> 00:35:09,989 at a s- different moment in time than you see it in red. 512 00:35:09,989 --> 00:35:17,775 And as I said I purposely made the frequencies of the strobe a 513 00:35:17,775 --> 00:35:21,436 little different. 514 00:35:21,436 --> 00:35:26,255 Thanks, Marcos. As you perhaps remember, 515 00:35:26,255 --> 00:35:32,557 you can also generate a standing wave in air itself. 516 00:35:32,557 --> 00:35:39,6 Wind instruments are nothing but air columns which go into 517 00:35:39,6 --> 00:35:43,43 resonance. If I have here a wind 518 00:35:43,43 --> 00:35:49,237 instrument which is open and open on both sides, 519 00:35:49,237 --> 00:35:55,416 this is the length of that instrument, 520 00:35:55,416 --> 00:35:59,605 then the frequencies that I can generate are given by this 521 00:35:59,605 --> 00:36:02,546 equation. The only fundamental difference 522 00:36:02,546 --> 00:36:07,25 between the strings and the wind instruments is that with strings 523 00:36:07,25 --> 00:36:11,807 you can change V at will more or less, you can choose different 524 00:36:11,807 --> 00:36:15,482 materials through which the speed of propagation is 525 00:36:15,482 --> 00:36:18,864 different, and you can also change the tension. 526 00:36:18,864 --> 00:36:21,951 If you increase the tension this V goes up. 527 00:36:21,951 --> 00:36:25,626 So you can give a s- violin four different strings, 528 00:36:25,626 --> 00:36:30,207 they'll give you four different fundamentals. 529 00:36:30,207 --> 00:36:34,905 With a wind instrument you cannot manipulate V because V is 530 00:36:34,905 --> 00:36:38,712 the speed of sound. And so that V of air at room 531 00:36:38,712 --> 00:36:42,924 temperature is three hundred forty meters per second. 532 00:36:42,924 --> 00:36:46,568 That's nonnegotiable. And so you can with wind 533 00:36:46,568 --> 00:36:51,428 instruments, you can very easily predict the frequencies that 534 00:36:51,428 --> 00:36:55,316 you're going to hear. I have here a pipe which is 535 00:36:55,316 --> 00:36:57,989 open and open, open on both sides. 536 00:36:57,989 --> 00:37:03,384 One-and-a-half meters long. And if I apply that equation I 537 00:37:03,384 --> 00:37:06,892 will find that the fundamental would be at a hundred and 538 00:37:06,892 --> 00:37:09,57 thirteen hertz. Perhaps you remember that I 539 00:37:09,57 --> 00:37:13,268 told you that resonances can occur when you sometimes least 540 00:37:13,268 --> 00:37:16,01 expect them. Just by blowing air you can get 541 00:37:16,01 --> 00:37:18,561 resonances. If you take a wind instrument 542 00:37:18,561 --> 00:37:21,558 and you start to blow air it excites resonances. 543 00:37:21,558 --> 00:37:25,256 Remember the Tacoma Bridge. There was wind and it went into 544 00:37:25,256 --> 00:37:27,424 resonance. It was very destructive. 545 00:37:27,424 --> 00:37:30,931 Well here we have a system that we can also make go into 546 00:37:30,931 --> 00:37:34,438 resonance when you least expect it. 547 00:37:34,438 --> 00:37:37,101 We have a -- a copper grid here. 548 00:37:37,101 --> 00:37:39,763 And I'm going to heat that grid. 549 00:37:39,763 --> 00:37:45,172 And when I heat the grid -- get an air flow going through there. 550 00:37:45,172 --> 00:37:49,724 That all by itself will not make it go into resonance. 551 00:37:49,724 --> 00:37:54,704 But when I take my heat source away and that grid starts to 552 00:37:54,704 --> 00:37:59,599 cool, it goes into resonance. So I'm going to heat it now. 553 00:37:59,599 --> 00:38:03,034 Will take a while. If I heat it too long, 554 00:38:03,034 --> 00:38:09,302 the copper grid will melt. Many years ago when I was doing 555 00:38:09,302 --> 00:38:17,97 this with a very short burner the molten copper came down on 556 00:38:17,97 --> 00:38:23,994 my hands, which was not funny, believe me. 557 00:38:23,994 --> 00:38:32,221 If I don't do it long enough, it won't go into resonance. 558 00:38:32,221 --> 00:38:37,656 So I'm sort of guessing this a little. 559 00:38:37,656 --> 00:38:43,748 A hundred thirteen hertz. All right. 560 00:38:43,748 --> 00:38:49,56 Standing electromagnetic waves. When we go to standing 561 00:38:49,56 --> 00:38:55,919 electromagnetic waves let me stay on the center board here. 562 00:38:55,919 --> 00:39:01,95 The situation is almost identical to standing waves on a 563 00:39:01,95 --> 00:39:06,117 string. Again we have -- we have nodes. 564 00:39:06,117 --> 00:39:12,916 We have locations where the electric field is always 565 00:39:12,916 --> 00:39:16,124 zero. Very very different from a 566 00:39:16,124 --> 00:39:19,954 traveling wave. I refer you to problem 567 00:39:19,954 --> 00:39:26,06 nine-four where you will see a standing electromagnetic wave 568 00:39:26,06 --> 00:39:31,235 which just like we had with the -- with the string, 569 00:39:31,235 --> 00:39:36,721 it has exactly this form. The time domain is decoupled 570 00:39:36,721 --> 00:39:42,31 from the spatial domain. The only complication that you 571 00:39:42,31 --> 00:39:45,793 have with a standing wave, 572 00:39:45,793 --> 00:39:49,405 electromagnetic, is that it is not so easy to 573 00:39:49,405 --> 00:39:52,196 find the associated magnetic field. 574 00:39:52,196 --> 00:39:56,875 And therefore I refer you to that problem nine-four if you 575 00:39:56,875 --> 00:39:59,666 want to revisit that. Polarization. 576 00:39:59,666 --> 00:40:02,703 Oh, we want to see the subjects again. 577 00:40:02,703 --> 00:40:07,464 See where we are on the list. But I think the time has come 578 00:40:07,464 --> 00:40:10,666 to talk a little bit about polarization. 579 00:40:10,666 --> 00:40:12,8 That's right. Polarization. 580 00:40:12,8 --> 00:40:17,79 Let's take electromagnetic waves that come 581 00:40:17,79 --> 00:40:21,002 straight out of the blackboard to you. 582 00:40:21,002 --> 00:40:26,472 And I call this the Y direction and I call this the Z direc- the 583 00:40:26,472 --> 00:40:29,511 -- the X direction. And Z is to you. 584 00:40:29,511 --> 00:40:34,98 Notice X cross Y is Z always in my case, right-handed coordinate 585 00:40:34,98 --> 00:40:37,759 system. And so the electric field 586 00:40:37,759 --> 00:40:43,055 vector of the plain wave that is coming out of the blackboard, 587 00:40:43,055 --> 00:40:49,192 let us assume that the electric vector is oscillating like so. 588 00:40:49,192 --> 00:40:52,685 Fweet fweet fweet fweet fweet fweet fweet fweet. 589 00:40:52,685 --> 00:40:56,997 With angular frequency omega. If this is a straight line we 590 00:40:56,997 --> 00:40:59,822 call that linearly polarized radiation. 591 00:40:59,822 --> 00:41:03,836 Could be radio emission as we did with the seventy-five 592 00:41:03,836 --> 00:41:07,627 megahertz transmitter. It can also be visible light. 593 00:41:07,627 --> 00:41:11,939 As long as the E vector stays along a straight line we call 594 00:41:11,939 --> 00:41:15,581 that linearly polarized electromagnetic radiation. 595 00:41:15,581 --> 00:41:20,211 Electromagnetic radiation including radio 596 00:41:20,211 --> 00:41:25,698 waves, including visible light, can be circularly polarized. 597 00:41:25,698 --> 00:41:30,813 In which case the electric vector doesn't oscillate like 598 00:41:30,813 --> 00:41:36,394 you see here but always has the same strength and is rotating 599 00:41:36,394 --> 00:41:40,951 around in a circle, either in this direction or in 600 00:41:40,951 --> 00:41:44,857 this direction. And it's very easy to make, 601 00:41:44,857 --> 00:41:48,949 actually. I could have done that 602 00:41:48,949 --> 00:41:51,416 here in lectures but I never did. 603 00:41:51,416 --> 00:41:55,731 Suppose we have an antenna in the Y direction and we have 604 00:41:55,731 --> 00:42:00,432 another one in the X direction like our seventy-five megahertz 605 00:42:00,432 --> 00:42:04,208 transmitter was a s- copper bar in this direction. 606 00:42:04,208 --> 00:42:08,832 And suppose each one radiates with exactly the same value for 607 00:42:08,832 --> 00:42:13,225 E zero, with exactly the same frequency but they're ninety 608 00:42:13,225 --> 00:42:16,847 degrees out of phase. That's not so difficult to 609 00:42:16,847 --> 00:42:21,018 arrange. Then I would get an EX which 610 00:42:21,018 --> 00:42:25,044 would be E zero. And if I pick my value for Z -- 611 00:42:25,044 --> 00:42:29,841 let's take Z equals zero, who cares where you are in this 612 00:42:29,841 --> 00:42:34,638 line, so there is no K Z term, so we simply have a cosine 613 00:42:34,638 --> 00:42:38,407 omega T here. So this is the component of the 614 00:42:38,407 --> 00:42:41,577 electric vector in the direction of X. 615 00:42:41,577 --> 00:42:45,945 And let the one in the Y direction -- must be ninety 616 00:42:45,945 --> 00:42:51,672 degrees out of phase, but must have exactly the same 617 00:42:51,672 --> 00:42:57,31 amplitude, so ninety degrees out of phase would for instance be 618 00:42:57,31 --> 00:43:00,675 sine omega T. Omegas must be the same. 619 00:43:00,675 --> 00:43:04,04 To get circularly polarized radiation. 620 00:43:04,04 --> 00:43:09,314 And so the net electric field vector, the one that you will 621 00:43:09,314 --> 00:43:14,498 experience sitting here on the Z axis, will be EX in the X 622 00:43:14,498 --> 00:43:17,772 direction plus EY in the Y direction. 623 00:43:17,772 --> 00:43:23,604 And so the magnitude of this vector will be the 624 00:43:23,604 --> 00:43:30,032 square root of EX squared plus EY squared, and that is E zero. 625 00:43:30,032 --> 00:43:36,039 Because sine squared omega T plus cosine square omega T is 626 00:43:36,039 --> 00:43:39,727 one. And so you get under the square 627 00:43:39,727 --> 00:43:45,102 root E zero squared times one, and so that's E zero. 628 00:43:45,102 --> 00:43:51,53 And so you see that the amplitude is always E zero. 629 00:43:51,53 --> 00:43:56,101 You see that here in front of you and so it's going to rotate 630 00:43:56,101 --> 00:43:59,986 around when it's maximum in Y, then it is zero in X, 631 00:43:59,986 --> 00:44:03,262 and when it is maximum in X it is zero in Y, 632 00:44:03,262 --> 00:44:07,604 and so therefore you get this rotation, either this way or 633 00:44:07,604 --> 00:44:10,423 that way. Depending upon how the phase 634 00:44:10,423 --> 00:44:13,775 delay is arranged. And you can turn this into 635 00:44:13,775 --> 00:44:18,498 elliptical polarized radiation by simply for instance putting a 636 00:44:18,498 --> 00:44:21,392 two here. If you put a two there or let 637 00:44:21,392 --> 00:44:25,277 me a put a two in the X direction, 638 00:44:25,277 --> 00:44:30,567 because I have more room in the X direction on the blackboard, 639 00:44:30,567 --> 00:44:35,596 so if I put a two here that means that in the X direction I 640 00:44:35,596 --> 00:44:40,799 can go twice as far as I can go in the Y direction and so now 641 00:44:40,799 --> 00:44:45,915 the E vector will go like this. You see this now is twice as 642 00:44:45,915 --> 00:44:50,424 much as this and so now I have elliptically polarized 643 00:44:50,424 --> 00:44:53,112 radiation. OK, so far is for the 644 00:44:53,112 --> 00:44:57,447 polarization is concerned. Let's talk 645 00:44:57,447 --> 00:45:02,407 about Snell's law. Snell's law was discovered two 646 00:45:02,407 --> 00:45:05,92 hundred fifty years before Maxwell. 647 00:45:05,92 --> 00:45:11,19 It's quite an amazing accomplishment even though you 648 00:45:11,19 --> 00:45:16,15 can derive it from Maxwell's equations of course. 649 00:45:16,15 --> 00:45:22,143 But it was derived by this Dutchman two hundred fifty years 650 00:45:22,143 --> 00:45:27,206 earlier. And what Maxwell's e- what uh 651 00:45:27,206 --> 00:45:32,376 Snell's law tells us is that if we have light going from a 652 00:45:32,376 --> 00:45:37,457 medium one with index of refraction N one going to medium 653 00:45:37,457 --> 00:45:43,081 two with index of refraction N two, and this angle of incidence 654 00:45:43,081 --> 00:45:46,528 is theta one, this is the normal to the 655 00:45:46,528 --> 00:45:51,427 surface, and we get some reflection, this angle is also 656 00:45:51,427 --> 00:45:56,689 theta one, and then some of that light will enter into this 657 00:45:56,689 --> 00:46:00,952 medium and this angle will then be 658 00:46:00,952 --> 00:46:04,481 theta two. And Snell's law says that the 659 00:46:04,481 --> 00:46:09,728 sine of theta one divided by the sine of theta two is N one 660 00:46:09,728 --> 00:46:13,98 divided by N two. Always the medium where you're 661 00:46:13,98 --> 00:46:19,589 going is up, the medium where -- oh, it's the other way around. 662 00:46:19,589 --> 00:46:24,927 Ha, good that I caught that, this is N two divided by N one, 663 00:46:24,927 --> 00:46:30,446 so the medium that you're going is up and the medium where you 664 00:46:30,446 --> 00:46:33,112 came from is down. 665 00:46:33,112 --> 00:46:38,41 And so it's immediately obvious that if N two is larger than N 666 00:46:38,41 --> 00:46:43,534 one that the angle of theta two is always smaller than theta 667 00:46:43,534 --> 00:46:46,399 one. Let us assume now that you go 668 00:46:46,399 --> 00:46:51,349 from air to glass but that somehow you come out in the air 669 00:46:51,349 --> 00:46:54,128 again. So right here this is your 670 00:46:54,128 --> 00:46:57,168 angle of incidence now, I call it I, 671 00:46:57,168 --> 00:47:02,378 it's clear that I is theta two. And here you're coming out of 672 00:47:02,378 --> 00:47:05,601 the medium in air again, 673 00:47:05,601 --> 00:47:09,688 and I call this angle R, and I'm going to ask you the 674 00:47:09,688 --> 00:47:14,405 question what is that angle R, now some of you may have great 675 00:47:14,405 --> 00:47:18,885 insight and immediately say oh well it's obvious that it's 676 00:47:18,885 --> 00:47:21,479 going to be the same as theta one. 677 00:47:21,479 --> 00:47:26,038 And that is indeed correct. You can easily see that because 678 00:47:26,038 --> 00:47:30,833 the sine of theta one divided by the sine of theta two at this 679 00:47:30,833 --> 00:47:35,786 transition at point A -- so this is at point A -- 680 00:47:35,786 --> 00:47:41,888 would be N of glass divided by N of air, and now we come here 681 00:47:41,888 --> 00:47:46,464 at this point B, so at point B we get the sine 682 00:47:46,464 --> 00:47:52,159 of this angle of incidence I, which we know is theta two, 683 00:47:52,159 --> 00:47:56,838 that's obvious, divided by the sine of R is now 684 00:47:56,838 --> 00:48:02,838 the one where we're going to which is air divided by the one 685 00:48:02,838 --> 00:48:07,211 where we were, so this is N air divided by N 686 00:48:07,211 --> 00:48:11,277 glass. And we already agreed that I is 687 00:48:11,277 --> 00:48:13,891 theta two. And so when I multiply these 688 00:48:13,891 --> 00:48:16,988 two equations, on the right side I get exactly 689 00:48:16,988 --> 00:48:19,396 one. Independent of the color of the 690 00:48:19,396 --> 00:48:21,804 light. If blue light has a different 691 00:48:21,804 --> 00:48:25,589 index of refraction than red light, that doesn't matter, 692 00:48:25,589 --> 00:48:29,029 because I have the same one here that I have there. 693 00:48:29,029 --> 00:48:32,057 And so you get exactly one on the right side, 694 00:48:32,057 --> 00:48:35,085 so you must get exactly one on the left side, 695 00:48:35,085 --> 00:48:39,833 and so the consequence is that theta one must be R, 696 00:48:39,833 --> 00:48:42,71 and so this angle here is the same as that angle, 697 00:48:42,71 --> 00:48:44,748 which is perhaps not so surprising. 698 00:48:44,748 --> 00:48:47,806 Because these two planes are parallel to each other. 699 00:48:47,806 --> 00:48:51,103 If they were not parallel, as they were with one of your 700 00:48:51,103 --> 00:48:54,759 problems where we had a prism, then you would get a separation 701 00:48:54,759 --> 00:48:57,637 of the colors here, and then the red and the blue 702 00:48:57,637 --> 00:48:59,915 would come out in different directions. 703 00:48:59,915 --> 00:49:03,572 In this case red and blue and green and yellow all come out in 704 00:49:03,572 --> 00:49:07,049 the same direction and so you see white light when you look 705 00:49:07,049 --> 00:49:09,985 through plain parallel glass. 706 00:49:09,985 --> 00:49:13,81 When it comes to total reflection, I refer you to 707 00:49:13,81 --> 00:49:17,317 problem nine-eight if you can spare the time. 708 00:49:17,317 --> 00:49:20,027 You're going to have five problems. 709 00:49:20,027 --> 00:49:24,888 Two of them have one question. Two of them have two questions. 710 00:49:24,888 --> 00:49:27,837 And one has four true-false questions. 711 00:49:27,837 --> 00:49:31,264 For each correct answer you get four points. 712 00:49:31,264 --> 00:49:35,408 For each wrong answer I have to subtract four points. 713 00:49:35,408 --> 00:49:38,038 However, you don't have to answer. 714 00:49:38,038 --> 00:49:42,262 If you don't answer you don't gain, 715 00:49:42,262 --> 00:49:45,369 you don't lose points. Now before you hate me for 716 00:49:45,369 --> 00:49:48,863 subtracting four points, think about this for a minute. 717 00:49:48,863 --> 00:49:51,775 If you give true-false questions to a class of 718 00:49:51,775 --> 00:49:55,723 five-year-olds they will have half on average correct and half 719 00:49:55,723 --> 00:49:57,535 wrong. Yet they deserve zero. 720 00:49:57,535 --> 00:50:01,547 So clearly the only reasonable thing is that for a wrong answer 721 00:50:01,547 --> 00:50:05,041 you must subtract points. But you don't have to answer. 722 00:50:05,041 --> 00:50:08,924 So if you know to two for sure out of four you could consider 723 00:50:08,924 --> 00:50:12,03 not to answer the other two. That is your choice. 724 00:50:12,03 --> 00:50:16,591 I'll give you an example. The Benham top consists of 725 00:50:16,591 --> 00:50:19,624 several colors. When you rotate it fast you see 726 00:50:19,624 --> 00:50:21,272 white light. That's wrong. 727 00:50:21,272 --> 00:50:24,107 That's false. Because the Benham top did not 728 00:50:24,107 --> 00:50:28,128 have several colors and when we rotated it we didn't see white 729 00:50:28,128 --> 00:50:30,501 light. I'll give you another example. 730 00:50:30,501 --> 00:50:34,325 One of two tails of comets is due to radiation pressure and 731 00:50:34,325 --> 00:50:36,566 the other is due to the solar wind. 732 00:50:36,566 --> 00:50:39,532 That's correct. We discussed that in lectures. 733 00:50:39,532 --> 00:50:41,906 Let me end with some fatherly advice. 734 00:50:41,906 --> 00:50:45,333 Read each problem at least twice 735 00:50:45,333 --> 00:50:48,632 and do those problems first that you like the best. 736 00:50:48,632 --> 00:50:52,195 Those that suit you the best. Never spend more than ten 737 00:50:52,195 --> 00:50:55,098 minutes on one problem. Then move to another. 738 00:50:55,098 --> 00:50:58,727 There is another review tomorrow evening for three hours 739 00:50:58,727 --> 00:51:01,498 by Ali Nayeri. You may want to attend that. 740 00:51:01,498 --> 00:51:05,06 And we also will provide you with tutoring this Sunday. 741 00:51:05,06 --> 00:51:07,963 Look at the Web. Because we will update it as 742 00:51:07,963 --> 00:51:09,942 the time comes. See you Monday. 743 00:51:09,942 --> 51:15 Have a good weekend.