WEBVTT 00:02.410 --> 00:07.160 Prof: I thought I would go back a little bit to LCR 00:07.162 --> 00:08.082 circuits. 00:08.080 --> 00:11.000 I think that's one of the more useful things you learn in this 00:10.999 --> 00:11.429 course. 00:11.430 --> 00:16.310 I want to do it carefully so everyone's on top of that. 00:16.309 --> 00:21.579 So remember that the problem you want to solve looks like 00:21.581 --> 00:22.241 this. 00:22.240 --> 00:24.880 That's an AC source. 00:24.880 --> 00:27.540 That is a resistor. 00:27.540 --> 00:32.470 There's an inductor and a capacitor. 00:32.470 --> 00:38.350 And that's the circuit. 00:38.350 --> 00:43.380 And this voltage we take to be V_0 00:43.384 --> 00:46.124 cosωt. 00:46.120 --> 00:48.930 That's the kind of voltage that will come out of any AC 00:48.926 --> 00:49.546 generator. 00:49.550 --> 00:53.170 The omega is controlled by how the turbine is spinning. 00:53.170 --> 00:55.910 That frequency is the same as this frequency. 00:55.910 --> 01:01.330 Our job is to find the current in the circuit. 01:01.329 --> 01:03.729 Now what makes it difficult is that, 01:03.728 --> 01:07.758 unlike DC circuits where you take a voltage and divide by 01:07.762 --> 01:11.582 some effective resistance, the equations here are not 01:11.581 --> 01:14.781 algebraic equations but differential equations. 01:14.780 --> 01:17.660 So we'll write it down then I'll tell you how we handle it. 01:17.659 --> 01:20.749 So the equation for this circuit, you begin as usual, 01:20.750 --> 01:23.910 you go around the whole loop and you add everything to 0, 01:23.909 --> 01:27.949 then you will find V_0 01:27.945 --> 01:34.585 cosωt = R times I L 01:34.593 --> 01:41.123 times dI/dt 1 over integral of I up to time t 01:41.123 --> 01:44.333 times 1/C. 01:44.330 --> 01:46.840 That's the voltage. 01:46.840 --> 01:49.700 Everybody with me on that part? 01:49.700 --> 01:54.130 That determines the condition, that determines the current at 01:54.128 --> 01:55.898 every instant in time. 01:55.900 --> 01:58.520 So if you can solve this equation, you solve for the 01:58.516 --> 02:01.486 current, but what's difficult is that it's not an algebraic 02:01.492 --> 02:02.162 equation. 02:02.159 --> 02:04.749 It's not like V = IR, where you divide by R, 02:04.745 --> 02:05.875 you've got the current. 02:05.879 --> 02:08.699 This has got integrals, it's got derivatives. 02:08.699 --> 02:11.989 But it turns out that if the voltage is V_0 02:11.985 --> 02:15.425 cosωt, there's a very clever way to 02:15.425 --> 02:19.075 solve this problem, which takes it back almost to 02:19.079 --> 02:21.859 the good old days of V = IR. 02:21.860 --> 02:23.810 No differentials, no integrals, 02:23.810 --> 02:24.460 nothing. 02:24.460 --> 02:25.160 That's the magic. 02:25.158 --> 02:28.088 That was invented by an engineer from GE called 02:28.086 --> 02:28.846 Steinmetz. 02:28.848 --> 02:32.188 He put to work some ideas that may seem esoteric, 02:32.187 --> 02:35.937 because they involve complex numbers, but it makes life 02:35.941 --> 02:39.141 tolerable for people doing circuit theory. 02:39.139 --> 02:43.089 And here is the trick: I ignore this problem, 02:43.091 --> 02:47.851 and I solve the following problem, purely mathematical 02:47.852 --> 02:48.842 device. 02:48.840 --> 02:53.600 The problem I solve has a voltage V_0 02:53.604 --> 02:55.554 e^(Iωt). 02:55.550 --> 02:58.410 We all realize that you cannot get a source that does that, 02:58.408 --> 02:59.788 but it's pure mathematics. 02:59.788 --> 03:07.718 Then the solution to that problem, that voltage, 03:07.717 --> 03:17.667 I assume, will drive some current I call I˜. 03:17.669 --> 03:21.509 So I˜ is the answer to the problem where the 03:21.506 --> 03:26.986 driving voltage is not real, but this, if you mathematically 03:26.990 --> 03:29.700 write the same equation. 03:29.699 --> 03:34.339 But now we realize that I˜ can be complex. 03:34.340 --> 03:36.650 I˜ is not real. 03:36.650 --> 03:38.350 It can be a complex number. 03:38.348 --> 03:40.458 Just because the voltage is a complex number, 03:40.460 --> 03:42.600 it's an equation, it's got imaginary and real 03:42.595 --> 03:45.155 parts on the left hand side, it will have imaginary and real 03:45.160 --> 03:50.400 parts on the right hand side, and so you'll need both to make 03:50.396 --> 03:51.606 it work. 03:51.610 --> 03:58.360 But if you have solved this problem, I claim you have also 03:58.355 --> 04:00.955 solved this problem. 04:00.960 --> 04:01.880 Why? 04:01.878 --> 04:03.568 Because if you take this equation, 04:03.568 --> 04:06.668 which is a complex equation, then the real part on the left 04:06.665 --> 04:09.865 hand side will match the real part on the right hand side. 04:09.870 --> 04:12.760 Imaginary part will match the imaginary part. 04:12.758 --> 04:14.518 That's because if you have two complex numbers, 04:14.520 --> 04:17.240 z_1 and you say they're equal, 04:17.240 --> 04:20.030 you mean x_1 Iy_1 is 04:20.026 --> 04:22.326 x_2 Iy_2. 04:22.329 --> 04:25.969 And they can be equal only if these guys are equal, 04:25.973 --> 04:27.653 those guys are equal. 04:27.649 --> 04:29.839 The real part has to match, the imaginary part has to. 04:29.839 --> 04:32.009 You cannot borrow from the real part and give to the imaginary 04:32.009 --> 04:32.259 part. 04:32.259 --> 04:35.519 They're apples and they're oranges, so you've got to match 04:35.519 --> 04:36.549 them separately. 04:36.550 --> 04:39.130 The closest analog is a vector equation. 04:39.129 --> 04:41.799 If you have a vector equation and you equate two vectors, 04:41.797 --> 04:44.387 so you say vector F is m times 04:44.386 --> 04:46.526 vector a, then F_x will 04:46.526 --> 04:47.896 be m times a_x and 04:47.904 --> 04:49.324 F_y will be m times 04:49.315 --> 04:50.185 a_y. 04:50.190 --> 04:52.980 So there are two equations in one single vector equation, 04:52.980 --> 04:55.370 three in 3D and a complex equation, there are two 04:55.372 --> 04:56.072 equations. 04:56.069 --> 04:59.429 So let's take the equivalent of the real part in the left hand 04:59.434 --> 05:01.924 side and the real part in the right hand side, 05:01.915 --> 05:03.235 and what do you get? 05:03.240 --> 05:05.640 The real part of this is the most important thing. 05:05.639 --> 05:08.879 The real part of this is cosωt. 05:08.879 --> 05:10.789 that's because e^(iωt) is 05:10.790 --> 05:13.320 cosωt isinωt. 05:13.319 --> 05:16.029 The real part is just cosωt. 05:16.028 --> 05:19.218 On the right hand side, I should take R times the real 05:19.223 --> 05:20.793 part of I˜. 05:20.790 --> 05:23.480 Let me just call that guy I. 05:23.480 --> 05:26.710 Then I should take L times the derivative of the real 05:26.714 --> 05:29.844 part of I˜, which I want to call I. 05:29.839 --> 05:33.129 Then 1/C times the integral of the real part of 05:33.125 --> 05:35.105 I˜, the real part of 05:35.110 --> 05:37.840 I˜ I want to call I. 05:37.839 --> 05:42.169 In other words, let the real part of 05:42.166 --> 05:46.736 I˜ be called I. 05:46.740 --> 05:51.080 Then we notice that this I is exactly the I 05:51.077 --> 05:52.237 that we want. 05:52.240 --> 05:55.910 It satisfies exactly the equation we want to solve. 05:55.910 --> 06:00.290 So we have some good news and some bad news. 06:00.290 --> 06:03.670 The good news is that if you solve this equation, 06:03.666 --> 06:06.126 you've also solved this equation. 06:06.129 --> 06:09.039 The bad news is, if you cannot even solve the 06:09.036 --> 06:11.806 real equation, what makes you think you can 06:11.812 --> 06:14.392 solve this complex equation, right? 06:14.389 --> 06:16.799 I cooked up another problem that looks even more difficult 06:16.800 --> 06:18.490 and I say, "Hey, if you can do that, 06:18.492 --> 06:19.552 we can do this." 06:19.550 --> 06:24.070 Well, it turns out this complex problem is actually easier to 06:24.071 --> 06:25.731 solve than this guy. 06:25.730 --> 06:28.810 I'll try to tell you why it's easier to solve. 06:28.810 --> 06:32.380 If you come to this equation, I've told you repeatedly, 06:32.382 --> 06:35.292 any equation involving derivatives and so on, 06:35.293 --> 06:38.143 you guess the answer and you put it in. 06:38.139 --> 06:40.699 You try to make a guess. 06:40.699 --> 06:43.649 I'm trying to guess, what kind of function I(t) 06:43.648 --> 06:47.258 has the property that when I multiply it by R, 06:47.259 --> 06:49.259 I should get a cosωt, 06:49.259 --> 06:51.309 when I differentiate, I should get something like 06:51.310 --> 06:53.240 cosωt, when I integrate, 06:53.242 --> 06:55.172 I should get cosωt? 06:55.170 --> 06:57.840 That's the only way left and right will match, 06:57.839 --> 07:00.689 and neither cosine nor sine will do the trick. 07:00.689 --> 07:03.199 If you pick the sine, then when you differentiate it, 07:03.196 --> 07:04.976 you will get a cosine that matches. 07:04.980 --> 07:07.640 This one integrated, you will get a cosine that 07:07.642 --> 07:10.252 matches this, but this will contain a sine. 07:10.250 --> 07:15.180 On the other hand here, since this exponential, 07:15.180 --> 07:19.370 if you make the choice I˜(t), 07:19.370 --> 07:24.720 it's I˜_0 e^(iωt), 07:24.720 --> 07:26.830 it is going to work. 07:26.829 --> 07:29.769 It is going to work because if you take R times that, 07:29.769 --> 07:31.659 you get R times I_0 07:31.663 --> 07:32.763 e^(iωt). 07:32.759 --> 07:35.869 If you take the derivative of this guy, that's Iω 07:35.865 --> 07:37.135 times the same current. 07:37.139 --> 07:40.489 Integral is 1 over Iω times the same 07:40.490 --> 07:41.050 thing. 07:41.050 --> 07:43.760 That's the beautiful property of the exponential, 07:43.757 --> 07:46.857 that when you differentiate it, just like multiplying by 07:46.860 --> 07:47.990 iω. 07:47.990 --> 07:49.960 When you integrate it, it's like dividing by 07:49.956 --> 07:51.786 iω, and when you multiply by 07:51.785 --> 07:53.975 R, it's like multiplying by R. 07:53.980 --> 07:57.600 So if you take this assumed solution and put it into that 07:57.601 --> 07:59.801 equation, let's see what we get. 07:59.800 --> 08:03.040 We get e_0 e^(iωt) = (now 08:03.043 --> 08:06.353 allow me to jump one or two steps, because I did it last 08:06.348 --> 08:06.888 time. 08:06.889 --> 08:09.129 Try to do this in your head.) 08:09.129 --> 08:15.049 R times I˜ is going to contain R 08:15.048 --> 08:20.148 times I˜_0 e^(iωt). 08:20.149 --> 08:22.429 How about L times dI˜/dt, 08:22.430 --> 08:24.820 then I come with the d/dt, this guy is a 08:24.815 --> 08:25.485 constant. 08:25.490 --> 08:28.000 This brings me an Iω, 08:28.000 --> 08:31.880 so I write IωL times the same thing. 08:31.879 --> 08:36.899 When I integrate, I get 1 over Iω 08:36.904 --> 08:41.824 times the 1/C that looks like that. 08:41.820 --> 08:44.800 So the time dependence, this is a time dependent 08:44.799 --> 08:48.349 problem, but miraculously, time dependence matches on the 08:48.351 --> 08:49.241 two sides. 08:49.240 --> 08:59.180 And now we get this equation, V_0 = R iωL 08:59.178 --> 09:07.938 1/iωC times I˜_0. 09:07.940 --> 09:11.340 And our goal was to find I˜_0, 09:11.340 --> 09:15.680 so I˜_0 = V_0 divided by 09:15.677 --> 09:18.077 this complex number Z. 09:18.080 --> 09:23.010 This complex number Z is called the impedance, 09:23.009 --> 09:26.709 and it = R iωL 1/iωC, 09:26.707 --> 09:28.697 for this problem. 09:28.700 --> 09:31.960 I'm not telling you the answer for every problem, 09:31.960 --> 09:34.950 but for this problem, life is very simple. 09:34.950 --> 09:39.400 You find that current as a voltage divided by a number, 09:39.399 --> 09:42.119 just like dividing by R. 09:42.120 --> 09:44.020 The only trick is, there are two catches now. 09:44.019 --> 09:46.849 First catch is, this number is not real, 09:46.847 --> 09:51.197 so you've got to get used to the fact that you will work with 09:51.198 --> 09:52.648 complex numbers. 09:52.649 --> 09:55.619 If you're willing to work with them, you get this. 09:55.620 --> 09:58.180 Second thing is, this is not the current we were 09:58.176 --> 09:59.206 supposed to find. 09:59.210 --> 10:01.600 What's the current we were supposed to find? 10:01.600 --> 10:07.000 The current that I wanted was the real part of this 10:06.996 --> 10:08.936 I˜. 10:08.940 --> 10:14.630 That means it's the real part of I˜_0 10:14.629 --> 10:19.739 e^(iωt) divided by--sorry. 10:19.740 --> 10:20.310 That's correct. 10:20.309 --> 10:22.549 That's what I want. 10:22.548 --> 10:28.968 And that's going to be then the real part of V_0 10:28.971 --> 10:32.611 over Z e^(iωt). 10:32.610 --> 10:34.640 We always pick V_0 to be a real number. 10:34.639 --> 10:38.419 The amplitude on the voltage you can take it to be a real 10:38.422 --> 10:39.032 number. 10:39.029 --> 10:43.419 Then you write it as a real part of V_0 10:43.418 --> 10:46.848 divided by Z e^(i)^(Φ). 10:46.850 --> 10:48.580 So I should tell you what this is. 10:48.580 --> 10:52.900 I'm saying, imagine the complex number Z, plotted this way. 10:52.899 --> 10:57.809 This is R, this is iωL − 1/i − 10:57.807 --> 11:02.217 i/ωC, and this is the complex number 11:02.224 --> 11:03.504 Z. 11:03.500 --> 11:05.800 This is the angle Φ. 11:05.798 --> 11:11.238 Then you write the same complex number in polar form as a 11:11.240 --> 11:16.880 modulus and e^(i)^ (Φ). So now you 11:16.875 --> 11:21.145 find I of t is the real part of 11:21.150 --> 11:25.820 V_0 over mod Z e^(iωt - 11:25.815 --> 11:30.425 )^(Φ), and that we know is 11:30.433 --> 11:35.283 V_0 over mod Z cos(ωt − 11:35.278 --> 11:39.688 Φ). So that's the final 11:39.692 --> 11:40.232 result. 11:40.230 --> 11:45.280 That's the answer to the original question. 11:45.279 --> 11:48.449 Now you know trigonometry well enough to know that 11:48.445 --> 11:51.675 cos(ωt − Φ) is 11:51.676 --> 11:54.256 really cosωt cosΦ 11:54.259 --> 11:56.909 sinωt sinΦ. 11:56.908 --> 11:59.548 So the answer cannot be just a cos and cannot be a sine. 11:59.548 --> 12:03.398 It's a suitably chosen admixture of sines and cosines 12:03.399 --> 12:05.029 that does the trick. 12:05.028 --> 12:06.858 So if you want to, if you say, "I don't want 12:06.863 --> 12:08.123 to deal with complex numbers" 12:08.123 --> 12:10.533 you can take a guess like this and put it in the equation, 12:10.528 --> 12:11.978 and after a lot of manipulation, 12:11.975 --> 12:14.725 you will find there is a Φ satisfying this 12:14.726 --> 12:15.376 condition. 12:15.379 --> 12:16.989 What's the property of Φ? 12:16.990 --> 12:21.480 Tan of Φ is ωL − 12:21.484 --> 12:25.544 1/ωC divided by R. 12:25.538 --> 12:29.238 But the beauty of the complex numbers is, it just comes out in 12:29.239 --> 12:32.089 one package as the phase of a complex number. 12:32.090 --> 12:36.410 I also told you to think about how you could do this with real 12:36.410 --> 12:37.120 numbers. 12:37.120 --> 12:40.180 It's not going to be easy to take a voltage, 12:40.179 --> 12:42.949 which is cosωt, divided by anything, 12:42.946 --> 12:45.306 to get a current which is cos(ωt − 12:45.313 --> 12:46.353 Φ). 12:46.350 --> 12:49.450 There's nothing you can do to a cosine which will shift its 12:49.452 --> 12:49.882 phase. 12:49.879 --> 12:52.099 But if you're working with complex numbers, 12:52.100 --> 12:54.600 you can take e^(iωt) divided by 12:54.602 --> 12:58.152 e^(i)^(Φ) and turn it into e^(iωt - 12:58.154 --> 12:59.614 )^(Φ). 12:59.610 --> 13:02.740 So in the imaginary world, it's very easy to shift the 13:02.738 --> 13:04.558 phase, because a complex number, 13:04.557 --> 13:06.977 when it multiplies or divides another number, 13:06.980 --> 13:09.430 rescales and rotates it. 13:09.428 --> 13:15.668 So in the complex plane, it's very easy to attach the 13:15.668 --> 13:22.508 phi, and at the end of the day, you take the real part. 13:22.509 --> 13:25.389 So you can imagine doing any realistic problem. 13:25.389 --> 13:31.169 Suppose I tell you R is 10 ohms, where ω is 13:31.168 --> 13:33.218 some 100Π. 13:33.220 --> 13:39.330 L is 3 henries and the capacitance is 2 microfarads, 13:39.330 --> 13:45.420 you can find Z. Z for this circuit will be 100-- 13:45.418 --> 13:49.448 I'm sorry Z will be 10 ωL, 13:49.450 --> 13:54.280 will be 100Π times 3 henries − 13:54.279 --> 14:01.369 1/ωC, is 100Π times 14:01.370 --> 14:03.870 2�10^(-6). 14:03.870 --> 14:05.270 That's some complex number. 14:05.269 --> 14:08.969 I forgot the i here. 14:08.970 --> 14:09.980 That is a complex number. 14:09.980 --> 14:12.260 I don't want to calculate it. 14:12.259 --> 14:13.939 It's whatever it is. 14:13.940 --> 14:17.050 Then it's got a real part which is 10, an imaginary part, 14:17.051 --> 14:19.441 which is this algebraic sum of these two. 14:19.440 --> 14:22.770 You can actually plot the real values you get here in this 14:22.765 --> 14:23.345 problem. 14:23.350 --> 14:25.110 I'm just going to show you the 10. 14:25.110 --> 14:29.260 The other guy's whatever it is, and you can get the phase. 14:29.259 --> 14:30.579 That tells you that in this problem, 14:30.580 --> 14:34.770 the current will have an amplitude which is the volts you 14:34.774 --> 14:38.384 apply, maybe you applied 100 volts, 14:38.376 --> 14:43.706 and you divide by this absolute value of Z. 14:43.710 --> 14:46.680 So if you want, I'll complete the last part. 14:46.678 --> 14:50.558 The absolute value of Z, like for any complex number, 14:50.558 --> 14:56.318 is R^(2) ωL - 1 over ωC squared, 14:56.320 --> 14:57.610 right? 14:57.610 --> 15:01.800 Any complex number, the absolute value is real 15:01.798 --> 15:05.428 squared imaginary squared under root. 15:05.428 --> 15:08.588 So let's write in all its glory the current that we want. 15:08.590 --> 15:12.490 I(t) is V_0/R^(2) 15:12.485 --> 15:18.005 ωL - 1 over ωC squared cosine 15:18.005 --> 15:22.115 of ωt - Φ where 15:22.119 --> 15:28.499 tanΦ = ωL − 1/ωC divided 15:28.504 --> 15:30.674 by R. 15:30.668 --> 15:33.488 Here is the picture, here's R. 15:33.490 --> 15:36.310 ωL minus 1/ωC. 15:36.309 --> 15:37.139 There's Φ. 15:37.139 --> 15:46.519 There's a complex number Z. Okay. 15:46.519 --> 15:48.799 So you can imagine now putting numbers and getting what you 15:48.803 --> 15:49.083 want. 15:49.080 --> 15:53.760 Notice that the current lags the applied voltage. 15:53.759 --> 15:56.129 For example, if cosωt is a 15:56.125 --> 15:59.795 maximum of t = 0 and Φ was 45 degrees, 15:59.798 --> 16:02.468 you'll have to wait till ωt is 45 degrees 16:02.466 --> 16:04.476 before the current reaches a maximum, 16:04.480 --> 16:06.270 so it will be lagging the voltage. 16:06.269 --> 16:08.739 But sometimes Φ can come out negative. 16:08.740 --> 16:17.380 Can you see how Φ could come out negative? 16:17.379 --> 16:20.609 This angle Φ need not be positive, 16:20.605 --> 16:22.435 do you understand that? 16:22.440 --> 16:27.180 What would make it negative? 16:27.178 --> 16:32.838 Look at the formula there, ωL − 16:32.841 --> 16:34.981 1/ωC. 16:34.980 --> 16:36.070 Yes? 16:36.070 --> 16:38.470 Student: If 1/ωC is bigger than 16:38.471 --> 16:39.321 ωL. 16:39.320 --> 16:40.040 Prof: Right. 16:40.038 --> 16:42.638 If 1 over ωC is bigger than ωL, 16:42.635 --> 16:44.625 this will be more negative than positive. 16:44.629 --> 16:46.489 The complex number may end up like that. 16:46.490 --> 16:50.780 So I've written it for one possible sign. 16:50.779 --> 16:52.709 If that flips sign, the Φ itself will 16:52.706 --> 16:53.276 change sign. 16:53.279 --> 16:55.599 It will become ωt 32 degrees. 16:55.600 --> 16:57.090 It can go either way. 16:57.090 --> 17:01.050 It depends on who is dominating it, whether the inductor or the 17:01.048 --> 17:02.898 capacitor is dominating it. 17:02.899 --> 17:08.189 So let's look at the answer for one other interesting feature. 17:08.190 --> 17:10.480 I want you to know two things. 17:10.480 --> 17:15.100 This trick works only if the voltage is a pure oscillatory 17:15.101 --> 17:17.941 function like cosωt. 17:17.940 --> 17:19.970 But then you can write it as a real part of 17:19.973 --> 17:21.673 e^(iωt) and do this. 17:21.670 --> 17:25.020 Secondly, the impedance is not a constant. 17:25.019 --> 17:29.159 Whereas the resistance is just 10 ohms, impedance varies with 17:29.161 --> 17:29.991 frequency. 17:29.990 --> 17:34.680 It's a frequency dependent number that you divide the 17:34.682 --> 17:37.572 voltage by to get the current. 17:37.568 --> 17:39.758 So the consequence of this is the following. 17:39.759 --> 17:43.949 Suppose you plot here the magnitude of the current, 17:43.954 --> 17:47.064 let's call it I_0. 17:47.058 --> 17:48.738 This whole thing, let me call it 17:48.741 --> 17:49.991 I_0. 17:49.990 --> 17:53.680 That's the amplitude of the current, as a function of 17:53.684 --> 17:58.024 frequency for a given applied voltage V_0. 17:58.019 --> 18:01.279 So V_0 is fixed, but when ω 18:01.275 --> 18:03.775 varies, these numbers are varying. 18:03.778 --> 18:09.628 When ω goes to 0, you've got a 1 over 0 in the 18:09.627 --> 18:11.037 denominator. 18:11.039 --> 18:12.979 That's going to beat everything. 18:12.980 --> 18:16.740 And 1 over 0 squared in the denominator means the whole 18:16.737 --> 18:17.917 thing vanishes. 18:17.920 --> 18:21.290 The current starts out as 0. 18:21.288 --> 18:24.978 That corresponds to the fact that if your voltage had been a 18:24.977 --> 18:27.287 DC source instead of an AC source-- 18:27.288 --> 18:29.478 that's what ω = 0 means-- 18:29.480 --> 18:31.820 the capacitor will charge to some point and then stop the 18:31.820 --> 18:32.240 current. 18:32.240 --> 18:34.430 That's it. 18:34.430 --> 18:36.490 Then it will go up, then come down, 18:36.492 --> 18:38.682 because at very large ω, 18:38.678 --> 18:41.408 the ωL is going to dominate. 18:41.410 --> 18:43.750 You get 1 over ωL squared under root, 18:43.750 --> 18:45.470 that looks like 1/ωL. 18:45.470 --> 18:49.050 It will fall like 1/ω at very large 18:49.051 --> 18:50.121 frequencies. 18:50.118 --> 18:56.728 And it will be the maximum when these two guys cancel each 18:56.730 --> 18:57.660 other. 18:57.660 --> 18:59.160 So you've got R^(2). 18:59.160 --> 19:00.590 If you're trying to get the maximum current, 19:00.586 --> 19:02.176 there's nothing you can do about R^(2). 19:02.180 --> 19:04.080 It is what it is. 19:04.078 --> 19:08.558 But you can play these two guys against each other and find a 19:08.561 --> 19:12.221 frequency ω and R so that this 19:12.221 --> 19:15.581 is true and that ω_0 is 19:15.584 --> 19:20.594 just related to the resonant frequency of the LC combination. 19:20.588 --> 19:25.548 So at that frequency, the current amplitude will be 19:25.549 --> 19:30.609 simply--I_0 maximum, will be simply 19:30.611 --> 19:33.291 V_0/R. 19:33.288 --> 19:35.838 It's as if L and C are not there. 19:35.838 --> 19:38.088 You've got them there but they're not there. 19:38.089 --> 19:38.979 They're not doing anything. 19:38.980 --> 19:42.420 They neutralize each other. 19:42.420 --> 19:44.000 So you can ask, why put something in that 19:44.000 --> 19:44.870 doesn't do anything? 19:44.868 --> 19:46.468 Well, first of all, it does something on other 19:46.471 --> 19:46.971 frequencies. 19:46.970 --> 19:50.830 Only at one magical frequency, they go away. 19:50.828 --> 19:57.498 But what is interesting is the very sharp response you have. 19:57.500 --> 20:00.630 So do you know where that comes into play? 20:00.630 --> 20:04.790 Student: Resonance? 20:04.788 --> 20:08.108 Prof: Resonance, yeah, but which part of your 20:08.105 --> 20:08.555 life? 20:08.559 --> 20:09.289 Pardon me? 20:09.289 --> 20:10.009 Student: Radio. 20:10.009 --> 20:10.989 Prof: Radio. 20:10.990 --> 20:13.070 You guys know what that is, because I don't. 20:13.068 --> 20:15.378 You're always carrying some recorded medium. 20:15.380 --> 20:18.930 But if you listen to radio, like in the old days, 20:18.929 --> 20:22.109 you have this room full of radio signals. 20:22.109 --> 20:24.229 Everyone wants your attention. 20:24.230 --> 20:28.250 All the radio stations are all sending signals right now in 20:28.247 --> 20:32.747 this room, and you want to pick just one station that you like. 20:32.750 --> 20:36.220 So what happens if that station sends that information at a 20:36.217 --> 20:38.427 certain ω_0? 20:38.430 --> 20:40.630 And if that's all you want, you go to the store, 20:40.630 --> 20:43.250 buy an LCR circuit, with L and C 20:43.248 --> 20:46.208 chosen so that LC is 1/ω_0 20:46.214 --> 20:48.884 ^(2), then you will get a huge 20:48.880 --> 20:52.890 response when you get the signal from that station. 20:52.890 --> 20:56.050 Now there's another station with a different frequency. 20:56.048 --> 20:58.978 You don't want to listen to them, but you have to listen to 20:58.977 --> 21:01.097 some of them, because if their frequency is 21:01.096 --> 21:03.566 here, your response to that station is not 0. 21:03.568 --> 21:06.468 It's a lot smaller than this, but it's not 0, 21:06.471 --> 21:10.101 and there may be yet another station you can hear in the 21:10.099 --> 21:11.089 background. 21:11.088 --> 21:13.658 If you don't want to hear them, you would like to get 0 signal 21:13.661 --> 21:15.691 from them, but you cannot kill this 21:15.685 --> 21:17.835 function except at one frequency, 21:17.839 --> 21:20.299 but you can make it very sharp. 21:20.298 --> 21:22.828 If R is very, very small, this function will 21:22.829 --> 21:25.509 be very large at resonance, because V_0/R 21:25.510 --> 21:27.030 will be a huge number. 21:27.028 --> 21:30.608 It will also be very narrow, so you can make this thing very 21:30.605 --> 21:31.085 sharp. 21:31.088 --> 21:36.408 And that's really controlled by the ratio of R to 21:36.405 --> 21:37.465 L. 21:37.470 --> 21:41.730 So other stations do give you weak signal, but you can make 21:41.733 --> 21:45.563 them weaker and weaker by picking a very finely tuned 21:45.557 --> 21:46.657 oscillator. 21:46.660 --> 21:52.120 Now the question is, what if you changed your mind 21:52.115 --> 21:56.455 and you want to listen to these guys? 21:56.460 --> 22:01.720 What should you do? 22:01.720 --> 22:04.230 Can you go and buy one radio for this station, 22:04.230 --> 22:05.850 one radio for that station? 22:05.848 --> 22:07.808 You know you do something, right? 22:07.808 --> 22:13.048 You fiddle with the dial, but what do you think it does? 22:13.049 --> 22:13.639 Yes? 22:13.640 --> 22:14.760 Student: Changes the capacitance. 22:14.759 --> 22:16.419 Prof: Changes the capacitance. 22:16.420 --> 22:19.540 It's not easy to change the inductance, but it changes the 22:19.541 --> 22:20.311 capacitance. 22:20.308 --> 22:25.228 Nowadays if you open a radio, I don't know what you will see. 22:25.230 --> 22:27.960 It's all glued on to something. 22:27.960 --> 22:30.680 But in the old days, when all the parts were big, 22:30.680 --> 22:35.110 you can open the radio and look inside and you have a capacitor 22:35.112 --> 22:37.902 which is called a variable capacitor. 22:37.900 --> 22:39.890 You show that with an arrow. 22:39.890 --> 22:41.850 You can also have variable inductors, I suppose, 22:41.846 --> 22:42.926 but this is very common. 22:42.930 --> 22:44.910 Now how do you vary the capacitance? 22:44.910 --> 22:48.920 Capacitance you know is ε_0 22:48.924 --> 22:49.784 A/d. 22:49.779 --> 22:50.519 Yes? 22:50.519 --> 22:52.009 Student: Can you change the surface area? 22:52.009 --> 22:53.509 Prof: You can change the surface area, 22:53.510 --> 22:55.490 but how do you think you can go change the surface area? 22:55.490 --> 22:57.760 Student: Having two half circles and then the 22:57.759 --> 22:58.849 current rotates around. 22:58.849 --> 22:59.309 Prof: That's right. 22:59.308 --> 23:03.018 You've got these interlocking plates and you can have them 23:03.015 --> 23:05.415 fully overlapped or not overlapped. 23:05.420 --> 23:10.660 If you draw the plates like this, it's got many things. 23:10.660 --> 23:14.160 And these plates can be either really overlapping with the 23:14.159 --> 23:15.939 other plates or pulled out. 23:15.940 --> 23:18.820 When you turn the dial, the two things move in and out, 23:18.824 --> 23:20.644 and that varies the capacitance. 23:20.640 --> 23:24.610 That will give you a range of frequencies and that's the range 23:24.613 --> 23:25.593 you can hear. 23:25.589 --> 23:27.089 That's one thing. 23:27.088 --> 23:29.258 Another thing I want to mention is, 23:29.259 --> 23:33.109 this solution I wrote down, I(t) is 23:33.114 --> 23:38.854 V_0 over mod Z cos(ωt − 23:38.846 --> 23:44.296 Φ) has no free parameters in it. 23:44.298 --> 23:47.048 You tell me the time, I tell you the current. 23:47.048 --> 23:49.358 Whatever the voltage is, you take that, 23:49.364 --> 23:53.084 shift the phase by Φ and divide by mod Z. 23:53.078 --> 23:58.328 But you know that a second order equation in time must have 23:58.329 --> 24:02.219 two free parameters, so where are those free 24:02.221 --> 24:05.301 parameters going to come from? 24:05.298 --> 24:09.598 Have you seen this before in your math whatever? 24:09.599 --> 24:12.879 No? 24:12.880 --> 24:18.710 Okay, I'm going to give you a clue and you have to think about 24:18.709 --> 24:23.009 this clue and see what you get out of that. 24:23.009 --> 24:26.469 V_0 cosωt = 24:26.467 --> 24:30.267 V_0 cosωt 0. 24:30.269 --> 24:35.469 That's your clue. 24:35.470 --> 24:44.770 What can you do with that clue? 24:44.769 --> 24:45.789 Yes? 24:45.788 --> 24:48.588 Student: Set the original equation equal to 0 to 24:48.588 --> 24:50.318 find the complementary solution. 24:50.318 --> 24:53.178 Prof: Yes, so I will translate. 24:53.180 --> 24:56.460 If you have a problem where you apply one voltage, 24:56.460 --> 24:58.170 V_1 and you get a current 24:58.174 --> 24:59.914 I_1, and a second voltage 24:59.912 --> 25:02.332 V_2 and you get a current I_2, 25:02.328 --> 25:04.468 just by adding the two equations, left hand side to 25:04.473 --> 25:06.923 left hand side and right hand side to right hand side, 25:06.920 --> 25:08.610 you can check that V_1 25:08.613 --> 25:11.103 V_2 drives a current I_1 25:11.104 --> 25:12.204 I_2. 25:12.200 --> 25:14.690 If you don't see that, you should do that. 25:14.690 --> 25:17.030 Take that equation, but any V of t, 25:17.028 --> 25:19.208 get an answer due to V_1, 25:19.210 --> 25:20.350 call it I_1, another due to 25:20.352 --> 25:22.522 V_2, call it I_2 25:22.519 --> 25:23.449 and add them up. 25:23.450 --> 25:25.100 Left hand side is clearly V_1 25:25.104 --> 25:26.924 V_2 and the right hand side, 25:26.920 --> 25:29.100 R times I_1 R times I_2 25:29.095 --> 25:30.225 is R times I_1 25:30.233 --> 25:31.343 I_2 and so on. 25:31.338 --> 25:33.698 So I_1 I_2 will satisfy 25:33.695 --> 25:36.475 the equation when the driving voltage is the sum of the two 25:36.483 --> 25:37.113 voltages. 25:37.108 --> 25:40.228 It all comes because there's nothing nonlinear. 25:40.230 --> 25:43.030 For example, for the right hand side at 25:43.032 --> 25:46.322 R times I^(2), then the first equation will 25:46.319 --> 25:47.999 have R times I_1^(2), 25:48.000 --> 25:50.090 second will have R times I_2^(2). 25:50.088 --> 25:51.968 When you add them, you'll get R times 25:51.965 --> 25:54.275 I_1^(2) I_2^(2), 25:54.279 --> 25:56.019 but what you want is really R times I_1 25:56.019 --> 25:58.779 I_2, the whole thing squared. 25:58.779 --> 26:01.339 So if you have nonlinear terms, you cannot add solutions, 26:01.339 --> 26:03.809 but if you have linear terms, you can add solutions. 26:03.809 --> 26:05.309 Therefore 0 is a hint. 26:05.308 --> 26:08.308 It says, add the solution for 0 voltage. 26:08.308 --> 26:11.108 You might say, "There is no solution for 26:11.111 --> 26:13.981 0 voltage," but I will have to remind you 26:13.977 --> 26:17.797 that we did solve the problem the other day of an LCR circuit 26:17.797 --> 26:21.997 connected to nothing that had a current flowing for a while. 26:22.000 --> 26:24.340 But it won't do it if it's completely inert. 26:24.338 --> 26:27.008 But if you have put some charge on this guy to begin with, 26:27.009 --> 26:29.679 then you saw that the current and the charge, 26:29.680 --> 26:33.290 everything starts and oscillates exponentially like 26:33.288 --> 26:36.748 e to the - some number t times some 26:36.752 --> 26:39.932 cosω't - some other phase, 26:39.930 --> 26:42.990 call it χ. 26:42.990 --> 26:44.760 And this thing had two free parameters. 26:44.759 --> 26:48.509 There's an area here and a χ here. 26:48.509 --> 26:51.429 Or you can write it as e^(-)^(γ) 26:51.431 --> 26:54.851 ^(t) times Acosω't 26:54.849 --> 26:56.839 Bsinω't. 26:56.838 --> 26:59.088 No matter how we write it, this function, 26:59.088 --> 27:01.878 this solution to 0 external voltage is called a 27:01.881 --> 27:05.411 complementary solution, and the final solution for 27:05.413 --> 27:09.293 I is really what I wrote, V over mod Z 27:09.286 --> 27:12.156 cos(ωt − Φ), 27:12.160 --> 27:17.600 this complementary function. 27:17.598 --> 27:21.068 Because if you take this complementary function and add 27:21.067 --> 27:23.437 it to the equation, you might think it will mess up 27:23.442 --> 27:24.672 the equation, but it won't, 27:24.669 --> 27:27.179 because the I that I wrote down will give me 27:27.182 --> 27:28.822 V_0cosωt. 27:28.818 --> 27:31.338 The complementary function, when you take RI 27:31.336 --> 27:34.656 LdI/dt 1 over C I dt for the complementary part, 27:34.660 --> 27:38.530 you will get 0, because the complementary guy 27:38.534 --> 27:43.384 obeys the following equation: it obeys the equation L 27:43.376 --> 27:48.656 dI/dt R times IC 1 over C integral IC dt 27:48.661 --> 27:51.941 is 0, is the equation satisfied in 27:51.935 --> 27:52.665 that loop. 27:52.670 --> 27:57.130 If it has a non-zero solution, you can add it to your 27:57.126 --> 27:58.066 equation. 27:58.068 --> 28:01.348 See, in elementary calculus, you learn that if you solve an 28:01.351 --> 28:03.681 equation, say dF/dx = 28:03.680 --> 28:06.610 x_2, the answer is 28:06.606 --> 28:08.996 x_2/2 [x^(3)/3] 28:08.996 --> 28:10.766 a constant, because when you add a 28:10.772 --> 28:12.442 constant, you don't screw up the solution, 28:12.440 --> 28:17.040 because the d/dx kills the constant. 28:17.038 --> 28:19.008 Question is, what can you add to this 28:19.012 --> 28:20.332 differential equation? 28:20.328 --> 28:22.548 You can add anything, any function, 28:22.554 --> 28:25.894 that's annihilated by that differential equation. 28:25.890 --> 28:28.490 If you put the function in, you get 0 from all the 28:28.486 --> 28:31.556 derivatives and integrals, you can add that to anything. 28:31.558 --> 28:35.628 Therefore the true answer is this complementary function. 28:35.630 --> 28:39.200 It will contain these two parameters A and B and you pick 28:39.195 --> 28:41.675 them depending on initial conditions. 28:41.680 --> 28:44.380 In other words, if you took an LCR circuit and 28:44.380 --> 28:46.840 took a generator and you hooked it up, 28:46.838 --> 28:51.148 from the instant you hooked it up, it won't immediately assume 28:51.150 --> 28:54.160 this form that I have here, because for example, 28:54.163 --> 28:56.833 at t = 0, maybe the current is 0 in the 28:56.830 --> 28:57.730 real problem. 28:57.730 --> 29:01.090 This current doesn't vanish, but you should add to this, 29:01.090 --> 29:02.070 this function. 29:02.068 --> 29:06.078 Choose A and B so that the current vanishes at the = 0. 29:06.078 --> 29:09.628 You can fit the actual experiment to the solution by 29:09.625 --> 29:13.515 choosing A and B to match the initial current and say the 29:13.517 --> 29:16.017 initial charge in the capacitor. 29:16.019 --> 29:18.439 But we don't usually care about this guy. 29:18.440 --> 29:22.980 Do you know why we don't talk about this too much? 29:22.980 --> 29:23.530 Yes? 29:23.529 --> 29:25.039 Student: It dies off. 29:25.038 --> 29:26.718 Prof: It dies off exponentially, 29:26.721 --> 29:28.361 so in real life, it doesn't matter. 29:28.358 --> 29:30.178 If you really want to know what's happening, 29:30.182 --> 29:31.672 in other words, take an LCR circuit, 29:31.666 --> 29:32.596 take an AC source. 29:32.598 --> 29:35.708 Put a switch, keep the switch open. 29:35.710 --> 29:38.110 The minute you close it, the current will be 0 by 29:38.108 --> 29:38.758 continuity. 29:38.759 --> 29:40.569 Current cannot jump. 29:40.568 --> 29:43.308 So this function obviously is not the whole answer. 29:43.308 --> 29:45.788 It doesn't vanish at t = 0. 29:45.788 --> 29:49.568 But this extra part you add in with suitably chosen A and B 29:49.571 --> 29:51.921 will fit those initial conditions. 29:51.920 --> 29:54.700 But after oscillating for a while, e to the - 29:54.696 --> 29:57.906 something t for a long t will just go away. 29:57.910 --> 30:00.550 Then it will settle down only to this function. 30:00.548 --> 30:04.148 This is called a steady state current and this is called a 30:04.154 --> 30:05.424 transient current. 30:05.420 --> 30:07.670 So transients are important; you cannot ignore them. 30:07.670 --> 30:10.110 Maybe a transient will burn your circuit, 30:10.113 --> 30:12.683 but it doesn't matter after a long time. 30:12.680 --> 30:15.310 If you survive the early time, it doesn't matter for the long 30:15.311 --> 30:15.621 time. 30:15.619 --> 30:18.179 It's a lot like this course. 30:18.180 --> 30:23.170 Okay, all right, so I want to take a minute to 30:23.166 --> 30:30.256 tell you a little more about the use of these complex numbers. 30:30.259 --> 30:32.099 I do it for a lot of reasons. 30:32.098 --> 30:34.058 Now I don't know what field you guys are in. 30:34.059 --> 30:35.699 You could be in art history. 30:35.700 --> 30:39.180 You just don't know when complex numbers are going to be 30:39.176 --> 30:42.146 relevant to what you do, you just don't know. 30:42.150 --> 30:44.140 But I'm serious, if you're doing engineering, 30:44.140 --> 30:46.030 electrical engineering, mechanical engineering, 30:46.029 --> 30:48.849 it's all about complex numbers, because everybody writes a 30:48.851 --> 30:50.041 differential equation. 30:50.038 --> 30:53.168 If it's a linear differential equation, you solve it with 30:53.172 --> 30:54.182 complex numbers. 30:54.180 --> 30:58.160 And I like to also show you why sometimes we should not resist 30:58.157 --> 31:02.027 the mathematics so much, because you might think in this 31:02.030 --> 31:04.620 circuit I don't need complex numbers. 31:04.618 --> 31:06.638 After all, in the end, the answer was a 31:06.636 --> 31:09.396 cos(ωt − Φ) and that is 31:09.397 --> 31:11.997 cos times some number sine times some number. 31:12.000 --> 31:15.130 I can put the combination and fiddle with it till it works, 31:15.133 --> 31:15.893 that's true. 31:15.890 --> 31:18.910 But suppose I gave you a more complicated circuit. 31:18.910 --> 31:20.840 Here's a more complicated circuit. 31:20.838 --> 31:26.658 I have maybe a resistor, an inductor. 31:26.660 --> 31:29.500 Then I come here, I have a capacitor, 31:29.498 --> 31:31.468 maybe another inductor. 31:31.470 --> 31:34.050 Go like that, let's call it R_1, 31:34.048 --> 31:36.728 L_1, C_2, 31:36.727 --> 31:41.507 L_3 and this is some V_0 31:41.508 --> 31:44.088 cosωt. 31:44.088 --> 31:46.408 How are you going to solve this problem by guessing? 31:46.410 --> 31:49.650 You have no idea how to guess. 31:49.650 --> 31:52.770 You cannot solve this problem by guesswork. 31:52.769 --> 31:57.789 But I claim that if you replace this fellow by V_0 31:57.788 --> 32:00.858 e^(iωt), you'll be able to guess the 32:00.855 --> 32:02.585 answer, you'll be able to solve the 32:02.592 --> 32:03.022 problem. 32:03.019 --> 32:08.249 So I want to take a minute to tell you why that works. 32:08.250 --> 32:09.730 So do you know what I'm trying to do now? 32:09.730 --> 32:13.400 I'm trying to tell you that an arbitrarily complex AC circuit 32:13.403 --> 32:16.383 built of resistors, inductors, capacitors, 32:16.384 --> 32:20.194 connected any way you like, add whatever you want, 32:20.190 --> 32:23.640 if you go back to the differential equation, 32:23.640 --> 32:25.870 it will drive you insane, because there is a derivative 32:25.873 --> 32:27.853 of this current, derivative of that current, 32:27.845 --> 32:30.285 integral of that current, all coupled in some way. 32:30.289 --> 32:32.759 How are you going to solve it? 32:32.759 --> 32:35.579 You cannot guess anything cosine and anything sine. 32:35.579 --> 32:37.519 It's going to be a madness. 32:37.519 --> 32:40.579 But I will show you that if you use complex numbers, 32:40.578 --> 32:44.118 you can reduce the problem to something that looks like a DC 32:44.118 --> 32:44.838 circuit. 32:44.839 --> 32:46.839 That's all I want to do. 32:46.838 --> 32:50.498 So I'm going to give you the idea, but I don't want to do the 32:50.503 --> 32:54.053 whole thing, because I don't want to spend too much time on 32:54.047 --> 32:54.777 algebra. 32:54.779 --> 32:57.349 For those of you who want to know where it comes from, 32:57.353 --> 32:58.813 I want to give you a chance. 32:58.808 --> 33:01.478 Those of you who don't want to know exactly where everything 33:01.484 --> 33:03.524 comes from, in the end, I will give you the 33:03.517 --> 33:05.847 so called bottom line, stuff you must know, 33:05.853 --> 33:07.673 then you'll forget all this. 33:07.670 --> 33:10.210 So let's take that problem. 33:10.210 --> 33:15.400 Let me take an easier one that this one, because it's got too 33:15.396 --> 33:17.296 many wrinkles in it. 33:17.298 --> 33:23.928 But the point I'm making, I can illustrate just as easily 33:23.933 --> 33:25.833 with this guy. 33:25.829 --> 33:26.739 So what did I call it? 33:26.740 --> 33:29.050 R_1, L_1, 33:29.046 --> 33:31.646 C_2, L_3. 33:31.650 --> 33:34.120 Let the current here be I_1. 33:34.118 --> 33:36.568 Let the current here be I_2, 33:36.574 --> 33:38.754 current here is I_3. 33:38.750 --> 33:44.570 And this is V_0 cosωt. 33:44.568 --> 33:47.618 Our job is to find those currents if they're all 33:47.615 --> 33:51.435 oscillatory functions of time, in magnitude and in phase. 33:51.440 --> 33:53.390 If you do that, you're done. 33:53.390 --> 33:55.250 So how do we do it normally? 33:55.250 --> 33:58.350 What are the fundamental equations for a circuit? 33:58.348 --> 34:02.058 The fundamental equations are that at every branch, 34:02.058 --> 34:05.728 the incoming current should be equal to the outgoing current, 34:05.730 --> 34:12.040 which for us means I_1 = I_2 34:12.043 --> 34:14.573 I_3. 34:14.570 --> 34:16.200 So I_2 and I_3 come down 34:16.202 --> 34:17.922 here and they join and become I_1 again, 34:17.920 --> 34:20.550 so I'm going to assume that and write I_1 34:20.550 --> 34:20.920 there. 34:20.920 --> 34:23.010 So I've got two unknown currents, I_2 34:23.012 --> 34:24.082 and I_3. 34:24.079 --> 34:26.509 That's the first thing to understand, how many free, 34:26.514 --> 34:28.094 independent currents there are. 34:28.090 --> 34:30.040 Generally, if it's a simple problem, it's equal to the 34:30.036 --> 34:31.026 number of loops you have. 34:31.030 --> 34:34.040 That's a loop here, that's a loop there. 34:34.039 --> 34:37.739 You can draw many other loops, like that one and that one, 34:37.744 --> 34:40.674 but they will not give you any new results. 34:40.670 --> 34:44.140 So you've got to write voltage equations that say if you go 34:44.144 --> 34:46.484 around a loop, and you come to where you 34:46.481 --> 34:48.101 start, the change is 0. 34:48.099 --> 34:53.589 The equation I would write would be V_0 34:53.588 --> 34:59.288 cosωt = (I'm going to take this loop) 34:59.293 --> 35:03.603 R_1 I_1 35:03.599 --> 35:08.339 L_1 dI_1/dt 1 over 35:08.335 --> 35:13.385 C integral I_2 dt'. Back 35:13.393 --> 35:15.873 to here is 0. 35:15.869 --> 35:20.889 That's one equation. 35:20.889 --> 35:24.069 For the second loop, I can go around the other way 35:24.065 --> 35:27.885 if you like, going round this inductor, or I can take a loop 35:27.891 --> 35:28.801 like this. 35:28.800 --> 35:32.520 I just want to write that to show you what it may look like. 35:32.518 --> 35:36.198 That equation will say L_3 35:36.199 --> 35:41.909 dI_3/dt - 1 over C_2 integral 35:41.911 --> 35:45.011 I_2 dt = 0. 35:45.010 --> 35:48.750 The - sign comes because if the current is assumed to flow this 35:48.746 --> 35:50.836 way, and your loop goes counter to 35:50.835 --> 35:52.525 the current, you're going uphill, 35:52.527 --> 35:54.167 so everything should be subtracted, 35:54.170 --> 35:57.920 but if you go down, you can write it going down 35:57.922 --> 35:58.822 this way. 35:58.820 --> 36:01.830 So these are the three equations to solve. 36:01.829 --> 36:05.219 They're complicated because there are differentials, 36:05.217 --> 36:07.207 integrals and you know what. 36:07.210 --> 36:12.750 But if I solve the following problem: V_0 36:12.753 --> 36:16.843 e^(iωt)-- so don't bother to write all of 36:16.842 --> 36:18.532 this, because you're not responsible 36:18.527 --> 36:19.237 for this detail. 36:19.239 --> 36:21.439 I just want you to know what's going on. 36:21.440 --> 36:25.040 So I scrap this problem and I solve this problem, 36:25.039 --> 36:28.719 and the current for that I take to be the twiddle, 36:28.715 --> 36:29.385 okay? 36:29.389 --> 36:31.789 I say, let the I˜'s be the answer 36:31.791 --> 36:34.141 to this problem with e^(iωt). 36:34.139 --> 36:40.129 Can you see that if I took the real part of the first equation, 36:40.132 --> 36:45.452 I get the correct equation for the physical currents; 36:45.449 --> 36:47.639 that if I took the real part of the second equation, 36:47.639 --> 36:49.789 everywhere I replace I˜ by the real part 36:49.793 --> 36:51.613 which I call I, and this becomes 36:51.605 --> 36:53.635 V_0 cosωt, 36:53.639 --> 36:56.519 I get a second voltage equation. 36:56.518 --> 36:59.388 And here again you drop the twiddles by taking the real 36:59.393 --> 37:01.043 part, that's the equation satisfied 37:01.041 --> 37:03.331 by the physical currents, I_2 and 37:03.327 --> 37:04.377 I_3. 37:04.380 --> 37:06.730 So it is still true, even in this complicated 37:06.731 --> 37:08.731 problem, that once you've solved it, 37:08.731 --> 37:11.761 you may take the real part of the voltage and take the real 37:11.764 --> 37:15.084 part of the current anywhere, that's the answer to the 37:15.081 --> 37:16.411 primordial problem. 37:16.409 --> 37:19.239 So why does this make life easy for you? 37:19.239 --> 37:21.409 Because now you may take the current. 37:21.409 --> 37:24.169 I'll just take one of these as an example. 37:24.170 --> 37:28.430 Let the current be current I˜_1, 37:28.429 --> 37:33.829 let it be I˜_1,0 e^(iωt), 37:33.829 --> 37:37.799 and I˜_2 = I˜_2,0 37:37.797 --> 37:42.477 e^(iωt), and I˜_3 37:42.483 --> 37:46.053 is I˜_3,0 e^(iωt). 37:46.050 --> 37:49.420 Make that guess, and what does this mean? 37:49.420 --> 37:49.930 Think about it? 37:49.929 --> 37:52.889 Go to the first equation. 37:52.889 --> 37:57.329 Equate these twiddle guys and all the e^(iωt)'s 37:57.327 --> 38:01.907 cancel on all the three terms, so you get an equation that 38:01.909 --> 38:06.109 says I˜_1,0 = I˜_2,0 38:06.106 --> 38:08.436 I˜_3,0. 38:08.440 --> 38:10.640 No time dependence anywhere. 38:10.639 --> 38:14.829 It looks like a DC equation. 38:14.829 --> 38:18.019 And let's go to this equation here. 38:18.018 --> 38:21.968 You will find V_0 = R times 38:21.969 --> 38:26.609 I˜_1_ iωL_1 38:26.608 --> 38:31.588 I˜_1 1/iωC (I should 38:31.590 --> 38:34.770 have called this C_2) 38:34.766 --> 38:37.516 ωC_2 times 38:37.516 --> 38:40.776 I˜_2. 38:40.780 --> 38:45.470 e^(iωt)'s have been canceled everywhere. 38:45.469 --> 38:46.709 So what do you find? 38:46.710 --> 38:53.470 This again looks like an Ohm's law problem where R_1 38:53.465 --> 38:57.965 is the impedance of this guy, iωL_1 is 38:57.965 --> 39:00.465 the impedance of that guy, so the whole thing looks like 39:00.472 --> 39:03.762 some Z_1 times I˜_1 39:03.755 --> 39:07.035 Z_2 times I˜_2. 39:07.039 --> 39:12.049 In other words, let me just write it below this 39:12.047 --> 39:12.807 here. 39:12.809 --> 39:15.679 The equation we get is, V_0 = 39:15.675 --> 39:19.045 Z_1 times I˜_1 39:19.052 --> 39:22.812 Z_2 times I˜_2. 39:22.809 --> 39:26.649 So the problem will look like this: just put some black box 39:26.650 --> 39:28.420 here, Z_1, 39:28.418 --> 39:30.798 another black box, Z_2, 39:30.802 --> 39:33.592 another black box, Z_3, 39:33.585 --> 39:35.685 join them, bring them out. 39:35.690 --> 39:38.540 Call the current here as I˜_1,0, 39:38.539 --> 39:41.189 the current here is I˜_2,0, 39:41.190 --> 39:44.650 the current here is I˜_3,0. 39:44.650 --> 39:49.370 And these constant numbers, they have no time dependence, 39:49.371 --> 39:54.351 obey the same equations as the real currents did in the real 39:54.347 --> 39:55.357 problem. 39:55.360 --> 39:57.040 So it's like a DC problem. 39:57.039 --> 40:01.099 The only subtlety is, the impedances are complex, 40:01.103 --> 40:04.833 and the rule for impedance is very simple. 40:04.829 --> 40:09.249 If you open the box and you find a resistor an inductor, 40:09.246 --> 40:12.536 the impedance for that is R_1 40:12.539 --> 40:14.949 iωL_1. 40:14.949 --> 40:15.689 That's it. 40:15.690 --> 40:21.550 Apart from that variation, it is just like DC circuits. 40:21.550 --> 40:23.920 Yes? 40:23.920 --> 40:25.920 Student: Is there any physical significance to the 40:25.920 --> 40:27.080 imaginary part of the solution? 40:27.079 --> 40:28.229 Prof: No. 40:28.230 --> 40:30.930 If you want, the imaginary part of the 40:30.929 --> 40:34.649 solution is the answer to a person whose voltage was 40:34.650 --> 40:37.860 V_0 sinωt. 40:37.860 --> 40:40.720 So you are solving two problems, but we believe that 40:40.715 --> 40:41.775 it's nothing new. 40:41.780 --> 40:43.260 If you can do cosωt, 40:43.259 --> 40:45.289 the answer to sinωt is the same, 40:45.293 --> 40:47.053 except you shift by Π/2. 40:47.050 --> 40:48.040 Everything looks the same. 40:48.039 --> 40:51.819 But it answers that question. 40:51.820 --> 40:53.980 Okay, so this is what I want you to know. 40:53.980 --> 40:58.020 Given an AC circuit, here's what you do. 40:58.018 --> 41:01.548 You make this rule: whenever you see this guy, 41:01.547 --> 41:02.877 it is R. 41:02.880 --> 41:07.440 Whenever you see this guy, it is iωL and 41:07.436 --> 41:12.076 whenever you see this guy, it's 1/iωC. 41:12.079 --> 41:15.929 Then you can combine impedances in series by adding them, 41:15.927 --> 41:19.707 or in parallel by doing the reciprocals and adding them, 41:19.706 --> 41:22.176 then taking the inverse of that. 41:22.179 --> 41:24.229 It's all like DC circuits. 41:24.230 --> 41:25.810 What will you have at the end? 41:25.809 --> 41:28.909 At the end, you will have found out every current 41:28.909 --> 41:31.429 I˜_ 0,α. 41:31.429 --> 41:36.529 α is 1,2 or 3 in this problem. 41:36.530 --> 41:40.040 You solve these DC like equations and you will find 41:40.039 --> 41:43.479 these DC like currents, but they'll be complex. 41:43.480 --> 41:46.230 What is the relation of this guy to the actual current 41:46.226 --> 41:48.556 flowing through the actual circuit element? 41:48.559 --> 41:51.909 The answer is I_α(t) 41:51.905 --> 41:55.645 will be I˜_ α times 41:55.648 --> 42:02.648 e^(iωt), then take the real part. 42:02.650 --> 42:06.040 That's the algorithm. 42:06.039 --> 42:08.159 That's the relation between these time independent 42:08.164 --> 42:09.774 functions, because what did you do? 42:09.768 --> 42:11.538 You stripped their time dependence. 42:11.539 --> 42:13.849 You wrote every current as e^(iωt) times a 42:13.853 --> 42:15.443 number that doesn't depend on time. 42:15.440 --> 42:17.450 It's those numbers that we have solved for. 42:17.449 --> 42:19.519 But to go back to the original current, 42:19.518 --> 42:22.268 first you've got to reinstate the exponential you canceled 42:22.268 --> 42:25.208 everywhere and remember to take the real part because that was 42:25.211 --> 42:25.841 the deal. 42:25.840 --> 42:27.620 It's called the real deal, real deal. 42:27.619 --> 42:30.769 You take the real part of your answer. 42:30.769 --> 42:32.979 How about following? 42:32.980 --> 42:34.660 You've solved a very complicated circuit. 42:34.659 --> 42:36.139 I don't even know what's going on. 42:36.139 --> 42:37.389 Here's one guy. 42:37.389 --> 42:39.079 It's part of a big mess. 42:39.079 --> 42:42.209 Its impedance is Z. 42:42.210 --> 42:45.390 There is a current, I˜_0 42:45.391 --> 42:48.231 going through it, meaning it's the complex 42:48.228 --> 42:49.748 amplitude of that. 42:49.750 --> 42:52.440 If I ask you, what is the actual voltage 42:52.443 --> 42:53.413 across this? 42:53.409 --> 42:57.459 If I put a volt meter here, what will I measure? 42:57.460 --> 42:59.510 Answer will be, again, 42:59.509 --> 43:03.899 I˜_0 times Z. 43:03.900 --> 43:06.960 That's in the fake Ohm's law calculation, but you've got to 43:06.958 --> 43:10.278 remember that these all had time dependence, which I removed. 43:10.280 --> 43:14.110 Then you've got to remember that I should take the real part 43:14.110 --> 43:15.150 of the answer. 43:15.150 --> 43:20.310 That will be the actual instantaneous voltage across 43:20.313 --> 43:24.163 that circuit element at that time t. 43:24.159 --> 43:26.149 But remember, I˜_0 43:26.153 --> 43:27.553 could be a complex number. 43:27.550 --> 43:30.120 The Z inside the box could be a complex number. 43:30.119 --> 43:32.229 When you multiply them, you've got lots of complex 43:32.228 --> 43:32.658 numbers. 43:32.659 --> 43:37.859 You add on to get this guy and take the real part. 43:37.860 --> 43:40.000 I would recommend, when you come to such problems, 43:40.000 --> 43:42.550 that every complex number like Z, 43:42.550 --> 43:46.570 you write as an absolute value times e to the i times some 43:46.565 --> 43:47.125 phase. 43:47.130 --> 43:49.920 That makes your life easier, because then the whole thing 43:49.920 --> 43:52.960 will be the absolute value of this times the absolute value of 43:52.960 --> 43:55.990 that, times e^(iωt) the 43:55.987 --> 43:58.767 phase of this the phase of that. 43:58.768 --> 44:02.068 And the real part of it is the cosine of ωt these 44:02.067 --> 44:02.767 two phases. 44:02.769 --> 44:04.079 You understand? 44:04.079 --> 44:08.109 So if you wrote it as mod times e^(i)^(Φ) 44:08.112 --> 44:11.222 _1, absolutely value of Z 44:11.221 --> 44:13.361 e^(i)^(Φ) _2, 44:13.360 --> 44:17.280 times e^(iωt), what is the real part of this 44:17.280 --> 44:18.150 crazy number? 44:18.150 --> 44:19.720 These are all real numbers? 44:19.719 --> 44:22.639 You pull them out, combine the exponentials, 44:22.639 --> 44:24.339 you'll get e^(iωt i)^(Φ) 44:24.344 --> 44:25.974 _1 from this one, 44:25.969 --> 44:27.339 iΦ _2 from 44:27.338 --> 44:27.678 that one. 44:27.679 --> 44:30.379 So the full current will be cos(ωt Φ 44:30.382 --> 44:32.122 _1 Φ 44:32.123 --> 44:33.273 _2). 44:33.268 --> 44:37.908 So the answer will be mod I_0 mod 44:37.911 --> 44:42.461 Z cos(ωt Φ 44:42.460 --> 44:46.060 _1 Φ 44:46.061 --> 44:48.621 _2). 44:48.619 --> 44:55.149 So I don't mind waiting to give you some time to digest this. 44:55.150 --> 44:58.530 You should be able to at least solve simple DC circuits with 44:58.527 --> 44:59.727 more than one loop. 44:59.730 --> 45:03.170 If you've got only one loop, just do what we did earlier, 45:03.166 --> 45:04.636 V over Z and all that. 45:04.639 --> 45:07.289 But if you've got two loops, you should learn how to handle 45:07.293 --> 45:09.173 that problem, because it's very useful. 45:09.170 --> 45:13.270 You're going to deal with circuits no matter what you do. 45:13.269 --> 45:14.129 Yes? 45:14.130 --> 45:17.560 Student: In the initial problem, if you were just trying 45:17.556 --> 45:19.946 to find the total current through the system, 45:19.951 --> 45:21.421 impedances add, like-- 45:21.420 --> 45:22.130 Prof: That's what I said. 45:22.130 --> 45:22.920 That's correct. 45:22.920 --> 45:24.440 So let us ask the following question. 45:24.440 --> 45:26.740 His question was, what if I'd like to know the 45:26.740 --> 45:28.430 current coming out of this guy? 45:28.429 --> 45:29.469 Right? 45:29.469 --> 45:30.759 Everybody with me? 45:30.760 --> 45:32.310 That was his question? 45:32.309 --> 45:34.909 So we do it as if everything were real. 45:34.909 --> 45:39.329 First we combine these two guys into a single impedance, 45:39.329 --> 45:41.969 which will be Z_2 Z_3 divided by 45:41.969 --> 45:43.699 Z_2 Z_3, 45:43.699 --> 45:46.069 just like you combine resistors in parallel. 45:46.070 --> 45:47.790 To that guy, I will add 45:47.786 --> 45:49.576 Z_1. 45:49.579 --> 45:53.589 That whole thing is the impedance seen by this thing. 45:53.590 --> 45:57.480 Divide the voltage by this total effective impedance, 45:57.481 --> 46:00.851 that will give you the current coming here. 46:00.849 --> 46:02.819 Yes? 46:02.820 --> 46:04.130 Take that current, multiply by 46:04.132 --> 46:06.262 Z_1 times e^(iωt), 46:06.259 --> 46:07.299 take the real part. 46:07.300 --> 46:10.230 That's the instantaneous voltage on this one. 46:10.230 --> 46:13.230 Here's one more thing: when that current comes here, 46:13.226 --> 46:15.926 how is it going to branch between these two? 46:15.929 --> 46:17.569 In DC circuits, if this was 46:17.568 --> 46:20.928 R_2, this is R_1, 46:20.929 --> 46:24.689 and the current comes here, its propensity to go on this 46:24.686 --> 46:27.406 side is proportional to that resistor, 46:27.409 --> 46:30.469 because if that is more resistive, it's more likely to 46:30.472 --> 46:31.052 go here. 46:31.050 --> 46:32.700 So the fraction going to this side will be 46:32.701 --> 46:34.591 R_2 divided by R_1 46:34.594 --> 46:35.484 R_2. 46:35.480 --> 46:37.190 The fraction going to this branch will be 46:37.192 --> 46:39.252 R_1 divided by R_1 46:39.248 --> 46:40.188 R_2. 46:40.190 --> 46:42.660 You replace all of the R's with Z's and 46:42.663 --> 46:43.903 you get the same result. 46:43.900 --> 46:47.050 So you can do everything like before, except you've got to get 46:47.047 --> 46:48.387 used to complex numbers. 46:48.389 --> 46:52.569 Here's the problem: take the circuit assigned to 46:52.567 --> 46:55.497 each element, these impedances. 46:55.500 --> 46:58.530 Add them like in the old days, combine them like in the old 46:58.530 --> 47:00.100 days, solve for the current. 47:00.099 --> 47:03.879 To get the physical current, multiply by e^(iωt) 47:03.875 --> 47:05.825 and take the real part. 47:05.829 --> 47:07.879 To find the drop against any entry, 47:07.880 --> 47:10.030 any element, take the current there times 47:10.034 --> 47:12.244 the impedance, times e^(iωt), 47:12.239 --> 47:14.329 then take the real part. 47:14.329 --> 47:14.869 That's it. 47:14.869 --> 47:17.249 I cannot say it anymore because I don't know another way to say 47:17.248 --> 47:17.438 it. 47:17.440 --> 47:21.230 You've just got to do problems so you get a feeling for how 47:21.233 --> 47:22.023 it's done. 47:22.018 --> 47:26.388 I wanted you to understand, sometimes you take a problem, 47:26.389 --> 47:29.509 you embed it in a bigger family of problems that looks more 47:29.510 --> 47:32.460 difficult, but is actually easier. 47:32.460 --> 47:34.210 It's even true, there are many integrals you 47:34.211 --> 47:35.721 try to do which are very difficult. 47:35.719 --> 47:38.679 But if you think of it as an integral in the complex plane, 47:38.682 --> 47:41.902 sometimes that problem's easier to solve than the real integral, 47:41.900 --> 47:43.230 so this happens a lot. 47:43.230 --> 47:44.910 You take a problem, you generalize it. 47:44.909 --> 47:47.949 Sometimes the generalized problem is easier than the 47:47.945 --> 47:48.835 original one. 47:48.840 --> 47:52.760 Now I'm going to give you one counter-example, 47:52.764 --> 47:57.654 let me see where to pick it, where this rule about taking 47:57.648 --> 48:01.658 the real part at the end of the day fails. 48:01.659 --> 48:05.029 So far the rule was, do everything with a complex 48:05.027 --> 48:05.587 thing. 48:05.590 --> 48:07.990 At the end of the day when you've found something, 48:07.987 --> 48:09.207 you take the real part. 48:09.210 --> 48:12.170 Here is a place where it does not work, so let me tell you 48:12.172 --> 48:12.902 where it is. 48:12.900 --> 48:20.220 Take the LCR circuit, where we found that the voltage 48:20.215 --> 48:26.825 was V_0 cosωt, 48:26.829 --> 48:29.739 and the current was V_0 over 48:29.735 --> 48:32.505 absolute value of Zcos(ωt - 48:32.510 --> 48:34.990 Φ), where Φ is all that 48:34.994 --> 48:35.224 stuff. 48:35.219 --> 48:38.119 I don't want to repeat it. 48:38.119 --> 48:44.109 What is the instantaneous power generated by the power supply? 48:44.110 --> 48:47.440 The instantaneous power for any voltage source equals the 48:47.440 --> 48:51.010 voltage at that instant times the current at that instant. 48:51.010 --> 48:53.530 And all of these are real voltages and real currents. 48:53.530 --> 48:54.930 I'm not doing any games right now. 48:54.929 --> 48:56.989 This is the real thing. 48:56.989 --> 48:58.779 So what do you get here? 48:58.780 --> 49:02.640 You get V_0^(2) over 49:02.643 --> 49:08.443 mod Z cosωt times cos(ωt - 49:08.438 --> 49:10.798 Φ). 49:10.800 --> 49:12.390 That's what it is, right? 49:12.389 --> 49:15.639 One voltage going like cos and the current is going like 49:15.641 --> 49:17.891 cos(ωt - Φ). 49:17.889 --> 49:21.839 So I want to write this out for you as follows: 49:21.835 --> 49:25.695 this is cos^(2)ωt cosΦ 49:25.695 --> 49:28.435 sinωt cosωt 49:28.438 --> 49:30.668 cosΦ. 49:30.670 --> 49:34.840 This is just some high school trig identities. 49:34.840 --> 49:39.720 Cos of A B is cos A cos B sine 49:39.719 --> 49:41.859 A sine B. 49:41.860 --> 49:43.280 So what do you notice? 49:43.280 --> 49:46.470 CosΦ and sinΦ are some 49:46.471 --> 49:50.411 constants, but these are functions of time and they're 49:50.405 --> 49:51.885 all oscillating. 49:51.889 --> 49:54.779 The reason they are oscillating is that when you drive a current 49:54.784 --> 49:56.824 in these circuits, sometimes the L and 49:56.822 --> 49:58.722 C are drawing energy from the source, 49:58.719 --> 50:01.069 sometimes they're giving it back. 50:01.070 --> 50:03.430 So sometimes it's not monotonic, it's not always 50:03.434 --> 50:04.294 drawing energy. 50:04.289 --> 50:06.819 Sometimes it's taking it, sometimes it's giving it back. 50:06.820 --> 50:09.090 That's all the oscillatory terms. 50:09.090 --> 50:14.870 So what one likes to study is called the average power over a 50:14.867 --> 50:16.117 full cycle. 50:16.119 --> 50:19.259 That means you take this time dependent function, 50:19.260 --> 50:24.130 you integrate it over time, over a full cycle at frequency 50:24.128 --> 50:27.288 ω and you divide by the time. 50:27.289 --> 50:29.659 And I ask what you get. 50:29.659 --> 50:33.619 So let me write the obvious parts, V_0^(2) 50:33.621 --> 50:34.511 over mod Z. 50:34.510 --> 50:37.830 Cosine square is a positive definite number. 50:37.829 --> 50:41.199 Its average over a cycle is 1 half. 50:41.199 --> 50:43.339 I don't know how you want me to show this. 50:43.340 --> 50:47.280 One is to say cos^(2)θ is 1 50:47.284 --> 50:49.614 cos2θ/2. 50:49.610 --> 50:52.060 And if you integrate that guy over a full cycle, 50:52.061 --> 50:54.201 the cosine 2θ is periodic. 50:54.199 --> 50:56.759 It gives you 0, you get 1 half. 50:56.760 --> 50:59.030 Sinωt cosωt, 50:59.030 --> 51:02.040 if you put a 2 and divide by 2, is proportional to 51:02.036 --> 51:03.506 sin2ωt. 51:03.510 --> 51:06.070 That guy completes 2 cycles in one period. 51:06.070 --> 51:08.230 Its average is definitely 0. 51:08.230 --> 51:11.690 So the only thing that survives then from all of this is the 51:11.690 --> 51:14.330 cosine Φ term and the 1 over 2, 51:14.329 --> 51:18.439 because average of cosine squared is half. 51:18.440 --> 51:22.700 So the average power in the circuit looks like 51:22.697 --> 51:27.047 V^(2)/2 mod Z cosΦ. 51:27.050 --> 51:31.960 That's the correct result. 51:31.960 --> 51:33.750 In fact, we can see it in another way. 51:33.750 --> 51:37.990 Let's write it as V_0 over mod 51:37.992 --> 51:42.972 Z squared times mod Z cosine Φ 51:42.974 --> 51:43.994 over 2. 51:43.989 --> 51:46.029 What is mod Z cosΦ? 51:46.030 --> 51:51.080 Can you tell from some picture I drew of Z? 51:51.079 --> 51:52.139 I don't know if I have any picture. 51:52.139 --> 51:53.399 I've hidden all of them. 51:53.400 --> 51:55.800 But here is what Z looks like. 51:55.800 --> 51:57.750 That is Z, that's Φ, 51:57.753 --> 52:00.293 this is R, this is ωL − 52:00.288 --> 52:01.448 1/ωC. 52:01.449 --> 52:02.919 What is mod Z cosΦ? 52:02.920 --> 52:03.540 Yes? 52:03.539 --> 52:04.759 Student: Shouldn't it just be R? 52:04.760 --> 52:06.380 Prof: It is just the resistance. 52:06.380 --> 52:08.520 So this is a fancy way for R. 52:08.519 --> 52:11.779 So this is just 1 over 2. 52:11.780 --> 52:14.540 If you want, it's just the amplitude of the 52:14.539 --> 52:16.709 current squared times R. 52:16.710 --> 52:20.570 That means the real energy loss is taking place only in the 52:20.568 --> 52:21.298 resistor. 52:21.300 --> 52:24.160 You can write that as mod Z cosine Φ if 52:24.157 --> 52:24.717 you like. 52:24.719 --> 52:26.629 The 1 half is new to AC circuits. 52:26.630 --> 52:29.020 In a DC circuit, it's just I^(2)R, 52:29.016 --> 52:30.446 because it's constant. 52:30.449 --> 52:33.859 Here it's oscillating like a cosine squared and the average 52:33.862 --> 52:35.042 of that is 1 half. 52:35.039 --> 52:38.349 So what people sometimes like to do is to find something 52:38.353 --> 52:40.483 called the RMS current, which is 52:40.476 --> 52:45.116 I_0/√2, and to define the RMS voltage, 52:45.121 --> 52:48.351 which is V_0/√2. 52:48.349 --> 52:51.659 Then the whole thing looks like I_0I 52:51.664 --> 52:54.674 _rms^(2)R, or if you like, 52:54.666 --> 52:58.756 V_rms times I_rms, 52:58.760 --> 53:02.710 period. 53:02.710 --> 53:04.510 Now RMS means root mean squared. 53:04.510 --> 53:07.360 There's a reason why for a trigonometric function, 53:07.360 --> 53:10.760 it's got a number which is 1 over root 2 times the amplitude, 53:10.760 --> 53:13.380 but I don't want to go into that, because we don't use it 53:13.376 --> 53:13.746 again. 53:13.750 --> 53:15.410 The main thing to know is that, if you want, 53:15.409 --> 53:16.899 you can redefine a new quantity, 53:16.902 --> 53:18.442 V_0/√2 and 53:18.443 --> 53:21.693 I_0/√2, so that these factors of 2 53:21.690 --> 53:26.300 disappear, and things look like the good 53:26.302 --> 53:30.602 old days of resistive circuits. 53:30.599 --> 53:35.349 So very often when they say this is a 110 volt power supply, 53:35.353 --> 53:38.823 they really mean the RMS value 110 volts. 53:38.820 --> 53:41.240 That means V_0/√2 is 53:41.242 --> 53:41.592 110. 53:41.590 --> 53:43.700 V_0 itself is 110 √2. 53:43.699 --> 53:49.139 So the actual voltage goes up and down from 110 √2 to 53:49.139 --> 53:50.639 -110 √2. 53:50.639 --> 53:53.249 Because for computation of power, the peak amplitude 53:53.246 --> 53:54.316 doesn't control it. 53:54.320 --> 53:59.730 It's this RMS value that controls the average power 53:59.726 --> 54:01.236 consumption. 54:01.239 --> 54:08.449 Now there is a way to get the power using the complex numbers, 54:08.454 --> 54:10.114 if you like. 54:10.110 --> 54:13.320 I'm going to assign that as a homework problem, 54:13.317 --> 54:15.057 rather than do it here. 54:15.059 --> 54:19.029 But I want you to think about why does it fail here? 54:19.030 --> 54:21.860 Why is it that when you come to the power, 54:21.860 --> 54:23.810 this is not simply--in other words, 54:23.809 --> 54:28.589 what I'm telling you is the power is not simply the real 54:28.588 --> 54:32.758 part of I˜t and V˜t. 54:32.760 --> 54:35.090 V˜ is a 1 with V_0 54:35.094 --> 54:36.184 e^(iωt). 54:36.179 --> 54:41.469 Why is it that taking the real part of this guy doesn't work? 54:41.469 --> 54:42.919 Everywhere else, we took the real part of the 54:42.918 --> 54:44.478 current, we got the right answer, 54:44.483 --> 54:48.313 real part of the voltage, we got the right answer, 54:48.306 --> 54:52.926 but not when you take the real part of this. 54:52.929 --> 54:56.419 Yes? 54:56.420 --> 54:57.990 Student: Yes, he's saying you're 54:57.985 --> 54:59.175 multiplying but not adding. 54:59.179 --> 55:02.119 So let me say the following: suppose you had a complex 55:02.119 --> 55:04.959 number Z_1, it's x_1 55:04.960 --> 55:07.330 Iy_1, and another complex number 55:07.331 --> 55:10.231 Z_2, which is x_2 55:10.233 --> 55:11.673 Iy_2. 55:11.670 --> 55:14.080 If you take the real part of Z_1 you get 55:14.079 --> 55:15.069 x_1. 55:15.070 --> 55:17.340 You take the real part of Z_2 you get 55:17.335 --> 55:18.315 x_2. 55:18.320 --> 55:21.160 But suppose you wanted x_1x 55:21.164 --> 55:23.724 _2 for some reason. 55:23.719 --> 55:26.289 That is the real part of Z_1 times the 55:26.288 --> 55:27.998 real part of Z_2. 55:28.000 --> 55:31.830 That does not = the real part of Z_1Z 55:31.833 --> 55:33.323 _2. 55:33.320 --> 55:34.680 Because the real part of Z_1Z 55:34.684 --> 55:36.054 _2 has got x_1x 55:36.047 --> 55:37.657 _2, that's pretty obvious, 55:37.661 --> 55:39.111 but it's always got -y_1 55:39.106 --> 55:41.016 y_2, because when these two 55:41.016 --> 55:45.586 imaginaries multiply, they can contribute a real part. 55:45.590 --> 55:50.730 But what you wanted is the analog of this one. 55:50.730 --> 55:52.360 So you can fix that. 55:52.360 --> 55:54.280 I mean, it's not that you can never extract x_1x 55:54.280 --> 55:54.890 _2. 55:54.889 --> 55:58.909 The correct solution is x_1 is Z 55:58.905 --> 56:01.735 Z* over 2 x2 Z_2 56:01.742 --> 56:05.932 Z_2* over 2, and you can multiply it all out 56:05.934 --> 56:07.014 and see what happens. 56:07.010 --> 56:11.200 Then you can write the answer in terms of these guys, 56:11.199 --> 56:13.049 if you like, but I'd rather just leave it 56:13.052 --> 56:15.312 this way, because it's not going to be 56:15.307 --> 56:16.397 used extensively. 56:16.400 --> 56:18.310 I just want you to be careful. 56:18.309 --> 56:21.379 This is an example of something that is quadratic in the 56:21.376 --> 56:22.766 interesting quantities. 56:22.768 --> 56:25.548 You cannot take the real part at the end of the calculation, 56:25.550 --> 56:27.200 only for things which are linear. 56:27.199 --> 56:31.159 Every time, the real and imaginary parts do not talk to 56:31.163 --> 56:34.543 each other throughout the whole calculation. 56:34.539 --> 56:37.909 You can follow them through and at the end take the real part as 56:37.907 --> 56:39.027 what's interesting. 56:39.030 --> 56:41.350 But if you're going to multiply two things, 56:41.349 --> 56:43.999 where you have added on a complex part to this guy and a 56:44.000 --> 56:46.240 complex part to this guy, in the product, 56:46.239 --> 56:49.069 you've added on something real, and that was not your 56:49.072 --> 56:50.862 intention, so you've got to take that out. 56:50.860 --> 56:53.500 That's what makes it complicated. 56:53.500 --> 56:56.790 Therefore I tell you that when you do these AC circuits, 56:56.789 --> 56:59.959 when it comes to power, forget any of the gimmicks you 56:59.960 --> 57:00.680 learned. 57:00.679 --> 57:03.879 You will be usually asked to find the power only for LCR 57:03.876 --> 57:07.356 circuit, the single loop like that, then just go back to this 57:07.364 --> 57:07.834 one. 57:07.829 --> 57:11.759 It's just I^(2)R written in this language. 57:11.760 --> 57:17.710 And cosine Φ is called the power factor. 57:17.710 --> 57:24.720 Okay, so this completes one chunk of the course. 57:24.719 --> 57:26.879 I'm really going to the finishing line for 57:26.875 --> 57:28.185 electromagnetic theory. 57:28.190 --> 57:32.130 There's only one big stuff left, then we'll do optics and 57:32.134 --> 57:34.534 then we'll do quantum mechanics. 57:34.530 --> 57:38.720 Now the nice thing about quantum mechanics is that you 57:38.719 --> 57:42.589 don't have to worry about whether you will get it, 57:42.594 --> 57:44.734 because nobody gets it. 57:44.730 --> 57:47.810 My idea is to take a class where only I don't get it, 57:47.806 --> 57:51.116 and turn it into a class where everybody don't get it. 57:51.119 --> 57:53.499 That's the plan. 57:53.500 --> 57:56.430 When I say "get it," I think I talked to one of you 57:56.425 --> 57:59.345 guys who called me and said something which is quite true. 57:59.349 --> 58:02.339 They said, "In the first part of the course, 58:02.336 --> 58:04.886 we could visualize what you were doing. 58:04.889 --> 58:07.699 There were masses rolling down and things colliding. 58:07.699 --> 58:11.339 With electricity and magnetism, we're not afraid of the math, 58:11.338 --> 58:15.218 but we don't have an intuitive feeling for these things." 58:15.219 --> 58:16.729 So I agree. 58:16.730 --> 58:20.190 But if you work long enough with electricity and magnetism, 58:20.188 --> 58:21.558 you get used to them. 58:21.559 --> 58:25.109 The only things born with the knowledge of electricity and 58:25.105 --> 58:27.775 magnetism are some creatures, like ducks. 58:27.780 --> 58:30.820 Apparently, ducks can feel the earth's magnetic field and they 58:30.824 --> 58:33.574 are just going through and they know which way to go. 58:33.570 --> 58:37.660 They don't solve Maxwell's equation, but they know how to 58:37.661 --> 58:38.321 travel. 58:38.320 --> 58:41.240 And some bees are sensitive to the polarization of light, 58:41.242 --> 58:44.112 so they do respond to certain things in an intrinsic way 58:44.112 --> 58:45.002 people don't. 58:45.000 --> 58:47.630 So you have to get used to it only by doing more problems. 58:47.630 --> 58:51.450 But in quantum theory, any intuitive feeling you have 58:51.449 --> 58:53.359 is actually a detriment. 58:53.360 --> 58:55.740 It will get you in trouble, because nothing is the way you 58:55.744 --> 58:56.294 imagine it. 58:56.289 --> 58:59.179 So no one has an advantage over anybody else. 58:59.179 --> 59:01.529 So it's better not to be well informed. 59:01.530 --> 59:04.090 So you may be saying, "Hey, here is the course 59:04.085 --> 59:05.205 for me," right? 59:05.210 --> 59:07.890 So the less you know, the better off you are in 59:07.885 --> 59:09.045 quantum mechanics. 59:09.050 --> 59:10.300 So don't worry about that. 59:10.300 --> 59:14.050 It's a very strange world and I want to give you an introduction 59:14.045 --> 59:14.635 to that. 59:14.639 --> 59:16.439 Anyway, they are the two things. 59:16.440 --> 59:18.980 So what's left now is just electromagnetic waves, 59:18.976 --> 59:21.986 and we're sort of building up to electromagnetic waves. 59:21.989 --> 59:27.479 So I'm going to start by writing here the equations that 59:27.483 --> 59:31.783 we know about electromagnetism as of now. 59:31.780 --> 59:36.260 So the surface integral of the electric field on any surface is 59:36.260 --> 59:40.740 the charge enclosed divided by ε_0. 59:40.739 --> 59:44.459 That's called Gauss's law. 59:44.460 --> 59:48.430 The line integral of the electric field on a closed loop 59:48.434 --> 59:51.184 used to be 0 when things are static. 59:51.179 --> 59:56.929 But when things are changing with time, it's the rate of 59:56.925 --> 1:00:00.055 change of the magnetic flux. 1:00:00.059 --> 1:00:02.889 The surface integral of the magnetic field, 1:00:02.889 --> 1:00:06.069 B⋅dA is just 0 all the time, 1:00:06.074 --> 1:00:08.604 because there are no magnetic monopoles. 1:00:08.599 --> 1:00:11.339 There's nothing from which lines come out. 1:00:11.340 --> 1:00:13.300 So if you integrate any configuration, 1:00:13.300 --> 1:00:16.540 the lines don't start and end anywhere, so whatever enters the 1:00:16.536 --> 1:00:18.176 surface leaves the surface. 1:00:18.179 --> 1:00:20.209 The surface integral of B is always 0. 1:00:20.210 --> 1:00:21.830 B⋅dA is 0. 1:00:21.829 --> 1:00:23.559 The line integral of B⋅ 1:00:23.563 --> 1:00:25.813 dA--b ⋅dl is 1:00:25.811 --> 1:00:27.171 μ_0I. 1:00:27.170 --> 1:00:32.790 That's the old Ampere's law. 1:00:32.789 --> 1:00:37.969 These are the four equations we have. 1:00:37.969 --> 1:00:42.439 Now it turns out this is not still the end. 1:00:42.440 --> 1:00:43.030 It's not the end. 1:00:43.030 --> 1:00:45.910 There's one more fiddling you have to do. 1:00:45.909 --> 1:00:48.309 That's the last part I want to talk about. 1:00:48.309 --> 1:00:52.069 Remember, this itself came from new experiments where you 1:00:52.070 --> 1:00:54.490 started moving magnets and so on, 1:00:54.489 --> 1:00:56.749 found a current whenever there's a changing magnetic 1:00:56.748 --> 1:00:57.058 flux. 1:00:57.059 --> 1:01:01.139 That's the induced electric field due to changing flux. 1:01:01.139 --> 1:01:06.039 So here is the thought process that led Mr. Maxwell to a 1:01:06.036 --> 1:01:07.726 little paradox. 1:01:07.730 --> 1:01:08.830 So here is some circuit. 1:01:08.829 --> 1:01:10.219 We don't know where it begins or ends; 1:01:10.219 --> 1:01:10.969 we don't care. 1:01:10.969 --> 1:01:12.069 Here's the circuit. 1:01:12.070 --> 1:01:15.720 There's a current flowing through a capacitor. 1:01:15.719 --> 1:01:18.419 When I say flowing through a capacitor, I hope you know what 1:01:18.416 --> 1:01:18.916 it means. 1:01:18.920 --> 1:01:21.380 I want you to be very clear about this. 1:01:21.380 --> 1:01:23.490 Suppose you connect it to some AC source. 1:01:23.489 --> 1:01:25.599 I want you all to know what's going on. 1:01:25.599 --> 1:01:28.139 Nothing really flows through the gap in the capacitor. 1:01:28.139 --> 1:01:29.549 It's not possible. 1:01:29.550 --> 1:01:31.300 What happens is, for a while, 1:01:31.300 --> 1:01:34.550 charges rush here and - charges rush the other plate, 1:01:34.550 --> 1:01:37.550 because that's the polarity of my AC source. 1:01:37.550 --> 1:01:41.950 Later on it's reversed, then - charges go here and 1:01:41.945 --> 1:01:43.645 charges go there. 1:01:43.650 --> 1:01:46.560 So there is an alternating current in the circuit, 1:01:46.557 --> 1:01:50.177 but there is no current flowing right through in one sense. 1:01:50.179 --> 1:01:53.119 But there's nothing to keep it from going this way and that way 1:01:53.117 --> 1:01:55.817 and this way and that way, because they don't have to jump 1:01:55.818 --> 1:01:56.908 the gap to do that. 1:01:56.909 --> 1:01:58.069 Yes? 1:01:58.070 --> 1:02:00.250 Student: It's sort of a random question, 1:02:00.251 --> 1:02:02.901 but you know how we were talking about jumping the gap. 1:02:02.900 --> 1:02:06.910 When would it actually jump the gap? 1:02:06.909 --> 1:02:10.529 Prof: It will jump the gap if the--here is what 1:02:10.534 --> 1:02:11.224 happens. 1:02:11.219 --> 1:02:13.729 It depends on what's in the medium. 1:02:13.730 --> 1:02:16.110 If you put air, for example, 1:02:16.108 --> 1:02:19.368 the air molecules are all neutral, 1:02:19.369 --> 1:02:24.319 so there is no way for the charges to ride them and go to 1:02:24.315 --> 1:02:27.395 the - charge, to the - terminal. 1:02:27.400 --> 1:02:29.350 But if the electric fields, this certainly will produce 1:02:29.349 --> 1:02:32.429 electric field, they're so strong that they rip 1:02:32.429 --> 1:02:35.579 out the and - charges, that suddenly you've got a 1:02:35.583 --> 1:02:37.833 bunch of free carriers, then the - charges can 1:02:37.826 --> 1:02:39.716 immediately go and neutralize this one, 1:02:39.719 --> 1:02:42.149 can go and neutralize that one. 1:02:42.150 --> 1:02:45.280 That's like a lightning strike inside the little world. 1:02:45.280 --> 1:02:47.570 That's what happens to us during lightning. 1:02:47.570 --> 1:02:48.910 We are in a capacitor. 1:02:48.909 --> 1:02:52.269 The lower plate is the earth, upper plate is the clouds. 1:02:52.268 --> 1:02:54.848 And after a while, there's a heavy charging. 1:02:54.849 --> 1:02:56.609 At some point, they cannot take it anymore, 1:02:56.610 --> 1:02:59.440 and they ionize the air and they create a little path and 1:02:59.440 --> 1:03:05.070 the current flows through that, and the clouds are discharged. 1:03:05.070 --> 1:03:08.340 Okay, so here is a problem that Maxwell had. 1:03:08.340 --> 1:03:10.700 He says, let's look at this equation, 1:03:10.699 --> 1:03:14.179 B⋅dl in a closed loop = 1:03:14.179 --> 1:03:18.529 μ_0I, where I is the current passing 1:03:18.525 --> 1:03:22.535 through any surface with that loop as the boundary. 1:03:22.539 --> 1:03:24.929 Remember that? 1:03:24.929 --> 1:03:31.959 So if you draw a loop here like that, then some current is 1:03:31.956 --> 1:03:35.776 crossing that shaded surface. 1:03:35.780 --> 1:03:37.040 And the line integral of B, 1:03:37.039 --> 1:03:39.649 you know B fields go around the current and the line 1:03:39.650 --> 1:03:42.030 integral of B around a loop will be the current 1:03:42.034 --> 1:03:43.524 crossing that shaded surface. 1:03:43.519 --> 1:03:46.179 That's how we get Ampere's law. 1:03:46.179 --> 1:03:49.259 But now you say, that's not the only surface 1:03:49.257 --> 1:03:51.617 with that loop as the boundary. 1:03:51.619 --> 1:03:54.429 I should be able to draw any surface with the loop as the 1:03:54.427 --> 1:03:54.977 boundary. 1:03:54.980 --> 1:03:59.670 So you say let me take that surface. 1:03:59.670 --> 1:04:03.080 You are still okay, because whatever current passes 1:04:03.079 --> 1:04:06.079 this phase also passes through that phase. 1:04:06.079 --> 1:04:09.619 The law is still good. 1:04:09.619 --> 1:04:14.799 So giddy with success, you say, why not this? 1:04:14.800 --> 1:04:16.400 That's my new surface. 1:04:16.400 --> 1:04:19.280 It goes all the way around one of the plates of the capacitor 1:04:19.275 --> 1:04:20.085 and comes back. 1:04:20.090 --> 1:04:22.590 Now I have a problem. 1:04:22.590 --> 1:04:27.240 I have a problem because there is no current passing that 1:04:27.239 --> 1:04:28.069 surface. 1:04:28.070 --> 1:04:31.640 Nothing going on between the plates in terms of current. 1:04:31.639 --> 1:04:34.089 If I draw an even bigger one like this one, 1:04:34.085 --> 1:04:36.705 I am again okay, because now that same current 1:04:36.706 --> 1:04:38.566 is passing through this one. 1:04:38.570 --> 1:04:44.550 It's this surface in between that things don't work. 1:04:44.550 --> 1:04:45.190 You follow that? 1:04:45.190 --> 1:04:47.310 We have a crisis where if you say it's 1:04:47.313 --> 1:04:50.763 μ_0I and you say it's on any surface with 1:04:50.759 --> 1:04:53.889 this loop as the boundary, it's the boundary of the 1:04:53.889 --> 1:04:55.969 surface, then this kind of surface, 1:04:55.969 --> 1:04:58.839 half in and half out, has a problem. 1:04:58.840 --> 1:05:03.980 So what will you do? 1:05:03.980 --> 1:05:07.220 You realize you have to modify your equations. 1:05:07.219 --> 1:05:09.319 It's quite often how people modify equations. 1:05:09.320 --> 1:05:12.500 Sometimes they do experiments, sometimes they do thought 1:05:12.500 --> 1:05:13.310 experiments. 1:05:13.309 --> 1:05:16.329 Einstein loved doing these experiments, which are called 1:05:16.327 --> 1:05:17.587 Gedanken experiments. 1:05:17.590 --> 1:05:19.580 You don't really do the experiment, but you say, 1:05:19.581 --> 1:05:21.531 "If I did this, what will happen?" 1:05:21.530 --> 1:05:24.970 There's a little paradox, and you have to modify your 1:05:24.972 --> 1:05:25.572 theory. 1:05:25.570 --> 1:05:27.070 So there are two reasons to modify the theory. 1:05:27.070 --> 1:05:28.910 One is experiments tell you is wrong; 1:05:28.909 --> 1:05:31.989 other is, theory tells you there's a problem. 1:05:31.989 --> 1:05:36.979 So you have to add something to this. 1:05:36.980 --> 1:05:42.760 That something I add should not have any contribution from this 1:05:42.755 --> 1:05:44.775 surface, but on this surface, 1:05:44.780 --> 1:05:47.740 it should make exactly the same contribution as the physical 1:05:47.737 --> 1:05:50.757 current made on this surface, so that no matter which surface 1:05:50.760 --> 1:05:53.510 I take, I get the same answer. 1:05:53.510 --> 1:05:56.450 And I'm going to do that by the--there are many ways to do 1:05:56.449 --> 1:05:56.809 this. 1:05:56.809 --> 1:05:59.409 If you know more math, there are more ways to get 1:05:59.414 --> 1:06:01.754 this, but this is one that's good enough. 1:06:01.750 --> 1:06:02.690 I'm going to do the following? 1:06:02.690 --> 1:06:07.660 I'm going to rewrite this as follows. 1:06:07.659 --> 1:06:10.039 You see in the region between the plates, I have no current, 1:06:10.036 --> 1:06:10.436 I agree. 1:06:10.440 --> 1:06:12.700 I gave up. 1:06:12.699 --> 1:06:15.739 I do have something between the plates I don't have in the wire. 1:06:15.739 --> 1:06:18.129 You know what that is? 1:06:18.130 --> 1:06:18.960 Pardon me? 1:06:18.960 --> 1:06:20.120 Student: Electric field. 1:06:20.119 --> 1:06:21.419 Prof: You have an electric field. 1:06:21.420 --> 1:06:23.650 There's something non 0 in between the plates. 1:06:23.650 --> 1:06:26.170 So I'm going to write this μ_0I somehow 1:06:26.172 --> 1:06:28.692 in terms of electric field between the plates and let's see 1:06:28.693 --> 1:06:29.393 how it goes. 1:06:29.389 --> 1:06:35.709 So I'm rewriting the very same term, I is dQ/dt. 1:06:35.710 --> 1:06:38.030 Q is the charge of the capacitor. 1:06:38.030 --> 1:06:43.850 And let me write it as Q/ε_0 d 1:06:43.851 --> 1:06:45.471 by dt. 1:06:45.469 --> 1:06:50.139 Now let me write it as μ_0ε 1:06:50.141 --> 1:06:54.631 _0Ad by dt of Q/A divided by 1:06:54.626 --> 1:06:57.706 ε_0. 1:06:57.710 --> 1:07:00.970 That = μ_0ε 1:07:00.965 --> 1:07:05.845 _0A times d by dt of the 1:07:05.849 --> 1:07:09.489 electric field between the plates. 1:07:09.489 --> 1:07:10.239 Why is that? 1:07:10.239 --> 1:07:14.469 Because the electric field is σ/ 1:07:14.472 --> 1:07:19.872 ε_0 and σ is just 1:07:19.871 --> 1:07:21.461 Q/A. 1:07:21.460 --> 1:07:25.530 Now bring that A in here and you get μ_0 1:07:25.530 --> 1:07:29.270 ε_0 d by dt of electric field 1:07:29.271 --> 1:07:30.521 times A. 1:07:30.518 --> 1:07:33.788 But that we can write as μ_0ε 1:07:33.791 --> 1:07:37.101 _0, the surface integral of the 1:07:37.099 --> 1:07:41.389 electric field over the surface, and that's what we call 1:07:41.387 --> 1:07:45.987 μ_0ε _0 electric flux, 1:07:45.989 --> 1:07:48.329 d by dt. 1:07:48.329 --> 1:07:51.499 Therefore my modified law is going to be 1:07:51.503 --> 1:07:55.743 B⋅dl = μ_0I 1:07:55.735 --> 1:07:59.475 μ_0 ε_0 d by 1:07:59.478 --> 1:08:04.278 dt of Φ _electric. 1:08:04.280 --> 1:08:06.030 That's the bottom line. 1:08:06.030 --> 1:08:09.910 You can fill in these blanks if you have trouble writing and 1:08:09.909 --> 1:08:10.699 listening. 1:08:10.699 --> 1:08:12.839 I'm just saying, go to the region between the 1:08:12.840 --> 1:08:13.280 plates. 1:08:13.280 --> 1:08:14.570 Once you start with dQ/dt, 1:08:14.565 --> 1:08:16.045 turn Q into the σ, 1:08:16.052 --> 1:08:17.542 the charge density on the plates. 1:08:17.538 --> 1:08:20.378 That's simply the electric field, then you'll find, 1:08:20.380 --> 1:08:24.290 this is simply the electric flux, rate of change times 1:08:24.288 --> 1:08:27.828 μ_0ε _0. 1:08:27.828 --> 1:08:38.318 This extra term is called the displacement current. 1:08:38.319 --> 1:08:41.239 I don't know why it's called that name, but you know that you 1:08:41.238 --> 1:08:42.698 have to put that extra term. 1:08:42.698 --> 1:08:45.738 But now look at this equation with that extra term. 1:08:45.739 --> 1:08:49.049 Notice it works everywhere. 1:08:49.050 --> 1:08:51.140 If you took a surface like the first one I took, 1:08:51.140 --> 1:08:54.610 that slices through the wire here, there's no electric field 1:08:54.613 --> 1:08:56.803 anywhere, inside a perfect conductor or 1:08:56.797 --> 1:08:59.187 anything, there's going to be no electric 1:08:59.186 --> 1:08:59.596 field. 1:08:59.600 --> 1:09:03.800 Then μ_0I is going to contribute. 1:09:03.800 --> 1:09:07.050 If I go to the region between the plates, there is no I 1:09:07.051 --> 1:09:10.091 there, but there's the rate of change of electric flux. 1:09:10.090 --> 1:09:13.590 Therefore the last Maxwell equation, 1:09:13.590 --> 1:09:17.170 the Ampere's law, is going to be modified with an 1:09:17.173 --> 1:09:19.053 extra term, μ_0 1:09:19.048 --> 1:09:20.708 ε_0 dΦ 1:09:20.708 --> 1:09:22.498 _electric /dt. 1:09:22.500 --> 1:09:25.740 And it has a nice symmetry now, that the line integral of the 1:09:25.743 --> 1:09:28.773 electric field is the rate of change of magnetic flux. 1:09:28.770 --> 1:09:31.570 The line integral of the magnetic field contains a rate 1:09:31.573 --> 1:09:33.083 of change of electric flux. 1:09:33.078 --> 1:09:37.388 It's got other stuff, but it's got that extra term. 1:09:37.390 --> 1:09:41.920 So I'm going to come next time and start looking at these 1:09:41.918 --> 1:09:42.888 equations. 1:09:42.890 --> 1:09:46.020 And remarkably, these equations imply there are 1:09:46.018 --> 1:09:47.648 electromagnetic waves. 1:09:47.649 --> 1:09:51.989 Without the benefit of any charge, without the benefit of 1:09:51.988 --> 1:09:56.328 any current, there'll be electric and magnetic fields. 1:09:56.329 --> 1:09:58.019 That's the interesting part. 1:09:58.020 --> 1:10:00.840 We know electric fields can be produced by charges, 1:10:00.838 --> 1:10:03.488 and magnetic fields can be produced by currents, 1:10:03.489 --> 1:10:06.309 but I'm saying go light years from everything. 1:10:06.310 --> 1:10:09.690 No ρ, no charge density, no current density. 1:10:09.689 --> 1:10:12.309 There'll be non-zero E and B just moving on 1:10:12.305 --> 1:10:14.965 their own, and you want to understand how that happens. 1:10:14.970 --> 1:10:17.490 First of all, it's amazing it's predicted by 1:10:17.488 --> 1:10:18.248 this stuff. 1:10:18.250 --> 1:10:21.590 So basically, this is a very important day in 1:10:21.588 --> 1:10:26.368 your life, because now you know, all of electromagnetism is that 1:10:26.369 --> 1:10:28.419 equation that equation. 1:10:28.420 --> 1:10:30.240 This is it. 1:10:30.238 --> 1:10:33.128 There is no more stuff any of us knows, at least in classical 1:10:33.126 --> 1:10:33.556 theory. 1:10:33.560 --> 1:10:35.250 In quantum theory, there's new stuff. 1:10:35.250 --> 1:10:39.710 Classical electromagnetism is only that. 1:10:39.710 --> 1:10:42.790 So you don't have to pack your head with all kinds of results. 1:10:42.788 --> 1:10:47.098 You can derive everything from these. 1:10:47.100 --> 1:10:51.090 Okay, so let's do that on Wednesday. 1:10:51.090 --> 1:10:56.000