WEBVTT 00:02.730 --> 00:07.940 Prof: Okay, I left you guys thinking about 00:07.939 --> 00:13.039 inductors last time, so I should start there. 00:13.040 --> 00:17.540 An inductor is very different from the resistor when you start 00:17.539 --> 00:22.359 doing circuit theory, because when you have a 00:22.363 --> 00:28.473 resistor and you connect some voltage to it, 00:28.470 --> 00:35.330 the current is determined by this equation. 00:35.330 --> 00:37.310 This is what I call an algebraic equation. 00:37.310 --> 00:40.070 It's an equation in algebra, one unknown. 00:40.070 --> 00:45.190 You solve for it by dividing and you get I at time 00:45.187 --> 00:50.667 t is V at time t divided by R. 00:50.670 --> 00:53.830 And you can make the network more complicated, 00:53.829 --> 00:57.059 put a few more resistors, put a few more there, 00:57.059 --> 01:00.429 a few more here, resistors within resistors. 01:00.429 --> 01:02.849 It doesn't matter, because we can always in the 01:02.853 --> 01:06.123 end combine the ones in series then lump them with the stuff in 01:06.122 --> 01:08.792 parallel, till the whole thing to the 01:08.786 --> 01:12.206 right of this is one single effective resistor. 01:12.209 --> 01:15.299 Then you find the current coming out of the sources as 01:15.295 --> 01:17.095 V divided by that guy. 01:17.099 --> 01:20.319 Then every time it comes to a branch, we sort of know how to 01:20.323 --> 01:20.983 divide it. 01:20.980 --> 01:23.880 So resistor circuits are very easy. 01:23.879 --> 01:26.739 But now when you bring inductors, things are different, 01:26.735 --> 01:29.745 so I'm just going to tell you a little more about them. 01:29.750 --> 01:31.000 Here is an inductor. 01:31.000 --> 01:33.550 That's the symbol for an inductor. 01:33.550 --> 01:39.470 It's got some inductance L measured in henries. 01:39.470 --> 01:42.460 And the implication of that is the following: 01:42.459 --> 01:46.059 if you have a current going through this inductor, 01:46.060 --> 01:50.750 there is going to be necessarily a voltage with this 01:50.745 --> 01:54.415 polarity if the current is increasing. 01:54.420 --> 01:59.690 And the voltage you need is LdI/dt. 01:59.690 --> 02:00.700 This is the bottom line. 02:00.700 --> 02:05.950 Even if you skip all the other stuff about conservative forces, 02:05.954 --> 02:10.624 this, that, this is something you need to know how to do 02:10.616 --> 02:11.716 problems. 02:11.718 --> 02:15.158 The difference you notice is that the relation between 02:15.155 --> 02:18.265 voltage and current is not an algebraic equation, 02:18.265 --> 02:20.335 but a differential equation. 02:20.340 --> 02:22.960 This is called a differential equation. 02:22.960 --> 02:26.320 Now you don't have to worry, because I will tell you how to 02:26.317 --> 02:29.727 solve it, because I don't know if it's a prerequisite or not 02:29.732 --> 02:30.892 for this course. 02:30.889 --> 02:35.009 I once had a kid in summer school who when I started using 02:35.006 --> 02:38.176 partial derivatives, got very upset and said, 02:38.184 --> 02:40.644 "It's not a prerequisite. 02:40.639 --> 02:43.169 Why are you using partial derivatives?" 02:43.169 --> 02:45.899 So I told him, "When you come to class, 02:45.901 --> 02:49.271 there's always a danger you'll learn something new. 02:49.270 --> 02:50.840 You just have to learn to live with that." 02:50.840 --> 02:53.170 So I'm not going to leave you in the cold. 02:53.169 --> 02:57.579 If I do something you've not done, I will tell you how to do 02:57.578 --> 03:01.088 it, but I cannot avoid using certain notions. 03:01.090 --> 03:02.990 So this one, even though it's called a 03:02.987 --> 03:05.957 differential equation and so on, if V is a constant, 03:05.962 --> 03:07.812 we can easily find the integral. 03:07.810 --> 03:11.830 It will be V over L times t. 03:11.830 --> 03:14.810 Okay, but now we're going to be interested in cases where 03:14.810 --> 03:19.260 V is not simply constant; it could be varying with time 03:19.258 --> 03:21.168 in an arbitrary way. 03:21.169 --> 03:23.909 So you take this guy, the first thing you notice is 03:23.914 --> 03:26.994 that the relation between current and voltage is given by 03:26.991 --> 03:27.871 derivatives. 03:27.870 --> 03:31.220 Second thing is, when a current flows through a 03:31.223 --> 03:35.523 resistor, whatever energy you provide is gone in the form of 03:35.522 --> 03:36.182 heat. 03:36.180 --> 03:37.370 It has dissipated. 03:37.370 --> 03:39.760 The light bulb glows and that's the end. 03:39.758 --> 03:44.008 With an inductor, when you drive a current, 03:44.008 --> 03:47.118 you are building a magnetic field inside the inductor and 03:47.122 --> 03:50.182 there's an energy associated with the magnetic field. 03:50.180 --> 03:55.650 And that stored energy will be given back to you later on. 03:55.650 --> 03:58.010 So it's like a capacitor. 03:58.008 --> 04:00.148 It takes a lot of work to charge a capacitor, 04:00.151 --> 04:02.971 because you've got to take charges from this plate and keep 04:02.973 --> 04:04.583 on piling them in that plate. 04:04.580 --> 04:07.460 It's a lot of work, but then if you connect that to 04:07.461 --> 04:10.291 a bulb and you squeeze the trigger in your camera, 04:10.287 --> 04:12.937 this charge is giving you the energy back. 04:12.938 --> 04:18.188 So the inductor and capacitor are different from the resistor 04:18.187 --> 04:21.887 in two ways: one is, the voltage across the 04:21.891 --> 04:26.741 capacitor is the charge on the capacitor divided by C, 04:26.740 --> 04:29.770 which if youlike is the integral of the current up to 04:29.766 --> 04:31.626 that time divided by C. 04:31.629 --> 04:34.679 Again, calculus comes in. 04:34.680 --> 04:36.640 So these are the two differences, the relation 04:36.644 --> 04:39.274 between current and voltage and voltage is differentiating or 04:39.266 --> 04:41.976 integrating, and these are energy storing 04:41.982 --> 04:44.802 circuit elements compared to this one, 04:44.800 --> 04:47.680 where energy is dissipated. 04:47.680 --> 04:52.340 So let's start with a simple problem where I have a voltage 04:52.336 --> 04:53.056 source. 04:53.060 --> 04:55.470 I'm going to take a fixed voltage, V_0. 04:55.470 --> 04:59.310 There is a switch, there is a resistor R, 04:59.312 --> 05:02.422 and there is this inductor L. 05:02.420 --> 05:06.760 I'm going to assume the current is flowing this way, 05:06.761 --> 05:09.061 in which case--I'm sorry. 05:09.060 --> 05:11.310 You know, this picture's wrong. 05:11.310 --> 05:17.410 What's wrong with this picture? 05:17.410 --> 05:19.260 What is wrong? 05:19.259 --> 05:21.289 Student: The switch is open. 05:21.290 --> 05:23.110 Prof: The switch is open, okay. 05:23.110 --> 05:25.500 That's what's wrong, so nothing is flowing yet. 05:25.500 --> 05:29.610 But when I close the switch, then it's fine. 05:29.610 --> 05:32.340 Now it's flowing. 05:32.339 --> 05:34.329 And the question is, what does it do? 05:34.329 --> 05:36.489 How does it flow? 05:36.490 --> 05:39.720 After all, the inductor is a resistance free wire, 05:39.720 --> 05:43.030 so you may think current = V_0/R will 05:43.029 --> 05:46.319 start flowing immediately, but that's wrong, 05:46.317 --> 05:49.877 because if such a current were flowing, 05:52.670 --> 05:53.530 current in it. 05:53.529 --> 05:56.909 And that energy cannot come from 0 to all of that in no 05:56.911 --> 05:57.351 time. 05:57.350 --> 05:59.360 There's no way to do that. 05:59.360 --> 06:02.430 Or if you like, if the current really grows 06:02.430 --> 06:05.870 rapidly or instantaneously from 0 to some value, 06:05.867 --> 06:09.667 that's a current, dI/dt is infinite here. 06:09.670 --> 06:12.490 dI/dt is infinite, LdI/dt is infinite. 06:12.490 --> 06:15.360 There's an infinite voltage needed somewhere in the circuit 06:15.362 --> 06:17.792 and we don't have it, so that would not happen. 06:17.790 --> 06:22.580 So current will have to be continuous, start from 0 and it 06:22.584 --> 06:25.114 will have to start climbing. 06:25.110 --> 06:28.480 The minute the current starts climbing up it starts flowing 06:28.476 --> 06:29.576 through this guy. 06:29.579 --> 06:34.289 There is the resistance times current voltage drop across 06:34.291 --> 06:34.881 this. 06:34.879 --> 06:38.369 Therefore the amount of voltage available to drive current 06:38.367 --> 06:42.157 through this one or to increase the current through this one is 06:42.160 --> 06:45.460 reduced from V_0 to V_0 - 06:45.464 --> 06:46.264 RI. 06:46.259 --> 06:49.589 That = LdI/dt. 06:49.589 --> 06:52.229 So you can see, as the current increases, 06:52.230 --> 06:54.430 the motivation to increase the current decreases, 06:54.430 --> 06:58.670 because this resistor starts swallowing up more and more of 06:58.672 --> 06:59.772 this voltage. 06:59.769 --> 07:03.009 So the growth in current will be very, very slow, 07:03.009 --> 07:07.129 and we expect that after a very long time, it will settle down 07:07.125 --> 07:08.335 to some value. 07:08.339 --> 07:09.379 What is that value? 07:09.379 --> 07:11.929 We can find that value by saying, "Wait until the 07:11.927 --> 07:13.657 current has stopped growing." 07:13.660 --> 07:16.540 That means dI/dt =0, that current will be 07:16.536 --> 07:20.016 V_0/R and I'm going to give it the 07:20.024 --> 07:22.354 name I_infinity, 07:22.350 --> 07:25.960 because it will turn out that you will reach the value only 07:25.959 --> 07:27.329 after infinite time. 07:27.329 --> 07:30.249 But you can get very, very close to that by waiting 07:30.245 --> 07:33.795 some reasonable amount of time, and part of what we want to do 07:33.802 --> 07:35.612 is, how long should I wait? 07:35.610 --> 07:38.470 Suppose I'm settling for 90 percent of maximum. 07:38.470 --> 07:39.480 When does that happen? 07:39.480 --> 07:42.130 Is it after 1 second or 10 seconds or 10 hours? 07:42.129 --> 07:44.089 What's going to decide that? 07:44.089 --> 07:47.279 Well, once you have the right equations, you can figure 07:47.283 --> 07:48.293 everything out. 07:48.290 --> 07:49.820 So we're going to do that. 07:49.819 --> 07:54.279 So basically we are going to solve the equation 07:54.279 --> 07:55.829 LdI/dt. 07:55.829 --> 07:58.559 So let me show you one more time how it is done, 07:58.560 --> 07:59.490 this equation. 07:59.490 --> 08:01.010 You should know how to write it. 08:01.009 --> 08:04.509 Start anywhere you like, go round this guy. 08:04.509 --> 08:07.689 You gain voltage V_0. 08:07.689 --> 08:11.909 Then go around this guy, you drop by an amount 08:11.913 --> 08:13.043 RI. 08:13.040 --> 08:14.600 You drop by an amount RI. 08:14.600 --> 08:17.450 And here you jump from this end to this end. 08:17.449 --> 08:19.999 Remember, you never go into the inductor. 08:20.000 --> 08:22.000 That's a bad zone. 08:22.000 --> 08:24.780 That's a zone where the potential is not even defined, 08:24.778 --> 08:27.358 but I've told you multiple times that even though there are 08:27.362 --> 08:29.502 non conservative things going on inside this, 08:29.500 --> 08:32.450 once you come outside, you just find a regular 08:32.452 --> 08:35.932 electrostatic potential difference = LdI/dt. 08:35.928 --> 08:38.728 So you drop again, and you come back to where you 08:38.734 --> 08:41.154 start, the whole thing should add up 08:41.153 --> 08:46.053 to 0, which means the equation is 08:46.048 --> 08:51.638 LdI/dt RI = V_0. 08:51.639 --> 08:54.909 That's what we want to solve. 08:54.908 --> 08:57.588 So when you have this thing, as I told you, 08:57.587 --> 09:01.347 solving differential equations is a matter of guess work. 09:01.350 --> 09:02.830 There is no algorithm. 09:02.830 --> 09:05.980 You rely on what you've seen before and you guess the answer. 09:05.980 --> 09:08.900 You put some free parameters, you fiddle with them till 09:08.900 --> 09:09.930 everything works. 09:09.928 --> 09:13.408 And in this kind of problem, we can make everything work. 09:13.409 --> 09:14.859 Some problems we cannot solve. 09:14.860 --> 09:17.820 We can write the equations but we cannot solve them. 09:17.820 --> 09:21.090 This is something you're not used to when you take an 09:21.091 --> 09:23.231 elementary course like this one. 09:23.230 --> 09:26.270 But there are situations, for example, 09:26.269 --> 09:29.419 you know inside the proton there are these quarks, 09:29.418 --> 09:32.928 and the quarks interact with each other with entities called 09:32.932 --> 09:33.472 gluons. 09:33.470 --> 09:36.730 They're called gluons because they glue the quarks and form 09:36.730 --> 09:38.250 the protons and neutrons. 09:38.250 --> 09:43.100 We know the equations governing the behavior of the gluons. 09:43.100 --> 09:45.400 We cannot solve them. 09:45.399 --> 09:46.029 It's really strange. 09:46.029 --> 09:48.039 You have an equation but you cannot solve it. 09:48.038 --> 09:50.628 It's completely possible, because certain 09:50.634 --> 09:54.594 equations--every equation has a solution, but it does not mean 09:54.590 --> 09:57.120 you can write it down analytically. 09:57.120 --> 09:59.570 So quite often, one may be on top of the right 09:59.567 --> 10:03.047 answer, but one may not know how to solve it, so one doesn't even 10:03.048 --> 10:04.788 know if the theory is right. 10:04.788 --> 10:07.278 So suppose this was the equation you got, 10:07.275 --> 10:09.075 but you could not solve it. 10:09.080 --> 10:10.450 It happens to be the right equation. 10:10.450 --> 10:13.960 You don't know what the answer looks like, you can never be 10:13.961 --> 10:16.021 sure this is the right equation. 10:16.019 --> 10:18.699 Similarly when Mr. Schr�dinger invented his 10:18.703 --> 10:21.633 quantum mechanics, he wrote an equation that will 10:21.634 --> 10:24.384 tell you the energy levels of hydrogen. 10:24.379 --> 10:27.049 But luckily, he was able to solve it. 10:27.048 --> 10:29.458 By solving it, he was able to show that his 10:29.457 --> 10:33.127 equation gives the energy levels iand the spectra you expect from 10:33.128 --> 10:34.388 the hydrogen atom. 10:34.389 --> 10:37.229 So you have to remember that not every interesting equation 10:37.227 --> 10:39.827 can be solved, and I gave you the equation for 10:39.831 --> 10:44.091 these gluons, which we can write down but 10:44.087 --> 10:45.847 cannot solve. 10:45.850 --> 10:48.560 There is no function whose derivative we cannot take. 10:48.558 --> 10:50.808 There are lots of functions whose integrals we cannot do. 10:50.808 --> 10:53.868 That's simply asymmetry between those two. 10:53.870 --> 10:58.980 Here's a very simple function: e^(−x2)dx from 0 10:58.977 --> 11:02.177 to 19, or some fixed number, 0 to some 11:02.182 --> 11:04.522 x_max. 11:04.519 --> 11:05.649 It has a definite value. 11:05.649 --> 11:09.139 It's area under the graph, but no one knows how to write a 11:09.144 --> 11:12.274 formula for this in terms of x_max, 11:12.269 --> 11:14.109 where it's in closed form. 11:14.110 --> 11:18.340 But anyway, this guy we can trample to death as follows. 11:18.340 --> 11:21.040 First we say at very, very long times, 11:21.044 --> 11:22.954 the current, I told you, is 11:22.946 --> 11:24.916 V_0/R. 11:24.919 --> 11:27.889 That's when dI/dt is 0. 11:27.889 --> 11:32.749 So we're going to write the actual current in our problem as 11:32.753 --> 11:37.043 the current at infinity some difference I twiddle. 11:37.038 --> 11:39.928 Standard thing, you take out the part you know 11:39.928 --> 11:41.918 and solve for the rest of it. 11:41.918 --> 11:46.268 So let's take that assumed form and put it into the differential 11:46.269 --> 11:47.029 equation. 11:47.029 --> 11:49.739 So when I do LdI/dt, this is a number, 11:49.740 --> 11:51.280 V over R. 11:51.279 --> 11:53.319 It has no d by dt. 11:53.320 --> 11:58.690 So dI/dt becomes dI˜/dt R 11:58.693 --> 12:02.383 times I˜ R times 12:02.383 --> 12:08.183 I_infinity = V_0. 12:08.179 --> 12:09.739 Now you can see why we did that. 12:09.740 --> 12:14.550 We did that because now these two guys cancel and you get 12:14.548 --> 12:17.208 LdI/dt RI = 0. 12:17.210 --> 12:28.330 That means dI˜/dt is −R/L times 12:28.326 --> 12:30.506 I. 12:30.509 --> 12:33.409 So we are saying, get me a function whose time 12:33.410 --> 12:36.510 derivative looks like the function itself up to a 12:36.506 --> 12:38.426 constant, and that's something we know 12:38.433 --> 12:40.943 from high school, is an exponential function. 12:40.940 --> 12:45.320 So I˜ looks like some constant, 12:45.320 --> 12:48.580 I_0e^(-Rt/L). 12:48.580 --> 12:52.210 You can take the derivative of this guy and out will come 12:52.207 --> 12:54.797 -R/L times the function itself. 12:54.798 --> 12:58.938 And I˜_0 is completely arbitrary. 12:58.940 --> 13:01.730 It's not determined by this equation. 13:01.730 --> 13:03.480 There's another useful property to know. 13:03.480 --> 13:05.450 This is a linear equation. 13:05.450 --> 13:09.130 It means if you multiply both sides by 19, you get another 13:09.129 --> 13:12.099 solution which is 19 times the old solution. 13:12.100 --> 13:15.320 So the solution's overall scale is not determined by the 13:15.316 --> 13:15.956 equation. 13:15.960 --> 13:17.700 If I give you one solution, you can multiply it by any 13:17.703 --> 13:19.403 number you like, it's also a solution, 13:19.397 --> 13:21.327 because if you multiply by a number, 13:21.330 --> 13:23.950 say 9, the 9 goes into the derivative, 13:23.950 --> 13:24.970 9 comes here. 13:24.970 --> 13:27.790 You can see 9 times I˜ satisfies the same 13:27.793 --> 13:28.383 equation. 13:28.379 --> 13:30.649 So we don't know this number just from the equation, 13:30.649 --> 13:34.379 but we go back to what we know, which is I of t, 13:34.379 --> 13:40.009 is I_infinity, which is V_0/R 13:40.009 --> 13:44.959 I˜_0 e^(−Rt/L). 13:44.960 --> 13:50.000 But I know that at t = 0, this guy should vanish. 13:50.000 --> 13:52.030 t = 0, this guy should vanish, 13:52.032 --> 13:55.422 because there was no current when I just threw the switch. 13:55.419 --> 13:58.009 So that tells me that equals 0. 13:58.009 --> 14:01.759 That tells me I˜_0 = 14:01.759 --> 14:06.669 -V_0/R and we get our final result, 14:06.668 --> 14:16.558 which is I of t = V_0/R times 1 - 14:16.557 --> 14:19.537 e^(-Rt/L). 14:19.538 --> 14:22.998 So at this level, you are really doing physics 14:22.995 --> 14:25.065 the way we like to do it. 14:25.070 --> 14:28.270 We like to study things, write down some equations and 14:28.270 --> 14:29.600 solve the equations. 14:29.600 --> 14:34.370 Then we get a very precise prediction on what will happen. 14:34.370 --> 14:36.990 Because now we don't have to guess when the current will come 14:36.985 --> 14:38.595 to 90 percent of its maximum value. 14:38.600 --> 14:41.860 We can pick any number you like, because we can now plot 14:41.859 --> 14:42.629 this graph. 14:42.629 --> 14:46.269 And let's plot this graph, I(t) versus t. 14:46.269 --> 14:49.629 At t = 0, e^(-0) is 1. 14:49.629 --> 14:51.889 You get 1 - 1, you get nothing. 14:51.889 --> 14:54.799 At t = infinity, t to the minus infinity 14:54.796 --> 14:55.176 is 0. 14:55.179 --> 14:56.479 Yes? 14:56.480 --> 14:58.840 Student: Over on the right where it says I˜ 14:58.835 --> 15:01.425 sub twiddle = negative V_0 over R, 15:01.429 --> 15:04.209 should that be 0? 15:04.210 --> 15:05.180 Prof: This one? 15:05.179 --> 15:07.319 Yeah, right. Thank you. 15:07.320 --> 15:12.260 Okay, so now this exponential has a full strength of 1 at time 15:12.259 --> 15:13.149 equals 0. 15:13.149 --> 15:15.319 Goes to 0 at time equals infinity. 15:15.320 --> 15:19.490 Therefore it takes away exponentially small stuff at 15:19.486 --> 15:21.036 very large times. 15:21.038 --> 15:24.588 And if you want to know roughly how long should I wait, 15:24.594 --> 15:27.294 the answer is, the time you should wait is 15:27.293 --> 15:29.403 roughly of order L/R. 15:29.399 --> 15:35.589 Because you can write this as e^(-t)/τ, 15:35.591 --> 15:39.131 where τ is L/R. 15:39.129 --> 15:42.069 Whenever we have an exponential, it will always be e 15:42.067 --> 15:45.747 to the minus a physical quantity divided by some number that sets 15:45.754 --> 15:48.064 the scale for the physical quantity. 15:48.059 --> 15:49.799 Exponentials fall exponentially. 15:49.799 --> 15:50.679 We know that. 15:50.679 --> 15:52.049 But how long should I wait? 15:52.048 --> 15:55.478 Well, if you wait t = τ seconds, 15:55.480 --> 15:59.230 then it will be 1/e, which is roughly 1 over third 15:59.225 --> 16:02.895 of the starting value and 1 minus that will be roughly 2 16:02.903 --> 16:04.513 thirds of its value. 16:04.509 --> 16:10.409 If you wait t = 3τ, e^(-3) is 1 over 20, 16:10.409 --> 16:13.929 then you'll be what, 95 percent. 16:13.928 --> 16:21.348 So you can calculate everything you need in this simple problem. 16:21.350 --> 16:23.510 So now let's come to the same problem. 16:23.509 --> 16:28.239 The switch has been closed for a long time and the current has 16:28.240 --> 16:29.250 stabilized. 16:29.250 --> 16:30.530 So here is the switch. 16:30.529 --> 16:32.099 It has been closed. 16:32.100 --> 16:34.650 That's my V_0, that's R. 16:34.649 --> 16:37.419 And the current, you remember, 16:37.417 --> 16:40.087 is V_0/R. 16:40.090 --> 16:44.580 This is L. 16:44.580 --> 16:46.960 So now we want to do something else. 16:46.960 --> 16:54.510 Now I want to open that switch. 16:54.509 --> 17:00.429 So what do you think will happen if you open that switch? 17:00.429 --> 17:01.459 Yes? 17:01.460 --> 17:03.100 Student: The inductor will resist the changing 17:03.100 --> 17:03.410 current. 17:03.408 --> 17:05.008 Prof: But how can it resist? 17:05.009 --> 17:08.789 Student: By inducing the magnetic field. 17:08.788 --> 17:11.548 Prof: It can do what it wants, but how does it drive a 17:11.548 --> 17:13.158 current through the open circuit? 17:13.160 --> 17:15.320 Student: From energy stored in the magnetic field. 17:15.318 --> 17:17.898 Prof: But in what manner will the current flow, 17:17.898 --> 17:19.698 given that I've broken the circuit? 17:19.700 --> 17:22.500 Student: In the direction-- 17:22.500 --> 17:24.260 Prof: No, but how will it manage to go 17:24.259 --> 17:24.619 around? 17:24.618 --> 17:26.978 It's like removing the bridge, right? 17:26.980 --> 17:28.180 How will it go around? 17:28.180 --> 17:30.760 I agree with everything you said, but how will it make the 17:30.757 --> 17:31.207 circuit? 17:31.210 --> 17:32.270 Yes? 17:32.269 --> 17:33.839 Student: There'll be a higher concentration of charges 17:33.842 --> 17:34.592 on one side than the other. 17:34.588 --> 17:37.998 Prof: Yes, but how will you really get rid 17:38.000 --> 17:41.270 of the current in a circuit that seems open? 17:41.269 --> 17:44.609 Have you ever done this? 17:44.609 --> 17:47.429 Okay, that's just fine. 17:47.430 --> 17:50.750 But we know it has to get rid of that current, 17:50.751 --> 17:54.371 because otherwise, what happened to the energy? 17:54.369 --> 17:55.989 It has to go through. 17:55.990 --> 17:58.860 So you know what, you guys don't know what will 17:58.856 --> 18:00.786 happen if I pull that switch. 18:00.789 --> 18:01.569 Yes? 18:01.568 --> 18:04.808 Student: Is the current an electric field? 18:04.808 --> 18:08.518 The change in the magnetic field will create an electric 18:08.518 --> 18:11.618 field which doesn't need the closed circuit. 18:11.618 --> 18:13.388 Prof: Right, no I agree, it will have an 18:13.385 --> 18:14.035 electric field. 18:14.038 --> 18:18.628 But how will the electrons flow when the wire has been 18:18.626 --> 18:21.826 interrupted, is all I'm asking you? 18:21.829 --> 18:23.279 Yes? 18:23.278 --> 18:24.138 Student: Will the current oscillate, 18:24.140 --> 18:24.540 go back and forth? 18:24.539 --> 18:26.669 Prof: No. 18:26.670 --> 18:27.740 Yes? 18:27.740 --> 18:28.970 Student: You will have a static discharge. 18:28.970 --> 18:29.760 Prof: You will have a discharge. 18:29.759 --> 18:31.879 You will have an arc. 18:31.880 --> 18:36.490 You will have a flash, and that's when you'll hear a 18:36.487 --> 18:39.917 zip, and then it will jump this gap. 18:39.920 --> 18:43.170 So whenever you remove the switch, you have to be careful 18:43.166 --> 18:45.716 that there is not stored energy somewhere. 18:45.720 --> 18:47.450 You might think, "Hey, I am pulling the 18:47.454 --> 18:49.074 plug on--" well, I won't say who. 18:49.068 --> 18:53.418 You're pulling the plug, okay, and what can it do to me? 18:53.420 --> 18:56.260 Well, it can zap you, because this is a very 18:56.259 --> 18:58.569 dangerous place to pull the plug. 18:58.569 --> 19:00.819 So you know what people do? 19:00.819 --> 19:02.219 They have another resistor. 19:02.220 --> 19:04.540 Let's call this guy R_1. 19:04.538 --> 19:09.458 This is R_2 and R_2 is a 19:09.457 --> 19:12.647 huge number, 10,000 ohms let's say. 19:12.650 --> 19:15.400 So most of the time, R_2 doesn't do 19:15.402 --> 19:17.932 anything, because when the current comes 19:17.932 --> 19:20.782 here in any situation, it looks at the inductor and 19:20.777 --> 19:22.447 looks at 10,000 ohms and it says, 19:22.450 --> 19:23.570 "I'm going this way." 19:23.568 --> 19:28.348 So primarily, it will all go there, 19:28.354 --> 19:29.344 okay? 19:29.338 --> 19:33.258 In the end, when the switch has been closed for a very long time 19:33.262 --> 19:36.772 in our old experiment, when the current is stabilized, 19:36.772 --> 19:38.982 you remember LdI/dt 0 here. 19:38.980 --> 19:40.950 So there's 0 volts between these two. 19:40.950 --> 19:42.990 That's the same 0 volt across R_2. 19:42.990 --> 19:44.740 No current will flow in R_2. 19:44.740 --> 19:47.310 Everything will flow through L. 19:47.308 --> 19:49.938 That current in fact will be V_0 over 19:49.944 --> 19:52.834 R, because this guy has been cut out of the loop. 19:52.829 --> 19:54.309 This is how it's going. 19:54.308 --> 19:59.138 But now if you throw the switch open, you have given it a path 19:59.141 --> 20:00.331 to discharge. 20:00.328 --> 20:05.238 So the inductor will discharge through the resistor. 20:05.240 --> 20:07.590 The current will continue to flow in this direction. 20:07.588 --> 20:10.308 Of course, it cannot flow forever, because the resistor 20:10.308 --> 20:13.078 will burn up the energy, but you're giving it a path. 20:13.078 --> 20:15.448 You're avoiding--in a way, if you like, 20:15.450 --> 20:18.820 this is also a path with very, very high resistance. 20:18.818 --> 20:21.488 The air is a path with very high resistance. 20:21.490 --> 20:24.190 That means by and large, there are no free carriers in 20:24.190 --> 20:24.700 the air. 20:24.700 --> 20:27.450 But if the two tips get really charged, just like you said, 20:27.450 --> 20:29.680 there'll be an emf that will pile up charges. 20:29.680 --> 20:31.690 They're jumping, they're waiting to jump the 20:31.689 --> 20:31.969 gap. 20:31.970 --> 20:35.240 Eventually they'll polarize air, which is electrically 20:35.240 --> 20:36.970 neutral, into and - parts. 20:36.970 --> 20:39.210 Then the - will go one way, the will go the other way. 20:39.210 --> 20:43.030 You'll recognize that as a discharge. 20:43.029 --> 20:45.849 But you don't have to worry about that now, 20:45.851 --> 20:49.681 because R_2 will take up your current. 20:49.680 --> 20:54.540 So now the equation we write is LdI/dt RI = 0 20:54.542 --> 20:58.992 because if you start anywhere and you go around a loop, 20:58.993 --> 21:01.883 you have no change in anything. 21:01.880 --> 21:05.010 There's no voltage. 21:05.009 --> 21:09.409 And the answer to this one, R_2, 21:09.414 --> 21:13.454 there I = I_0e to the 21:13.452 --> 21:17.952 −R_2/L times t. 21:17.950 --> 21:22.220 So now the current will decay exponentially. 21:22.220 --> 21:25.310 Again, with the time constant, R_2 over 21:25.306 --> 21:27.496 L, which is the 1 over the time 21:27.501 --> 21:29.741 constant, or the time constant is 21:29.740 --> 21:32.310 L over R_2. 21:32.308 --> 21:34.698 So you put L in henries, R in ohms, 21:34.695 --> 21:35.835 you'll get some time. 21:35.838 --> 21:39.058 That will give you an idea of how many times that time you 21:39.059 --> 21:42.449 have to wait before the inductor is completely discharged. 21:42.450 --> 21:44.770 Well, it's never going to be fully discharged. 21:44.769 --> 21:46.649 It's going to take forever, but if you say, 21:46.647 --> 21:49.597 "Look, 1 thousandth of the original current is safe enough. 21:49.598 --> 21:51.498 How long do I have to wait?" 21:51.500 --> 21:55.440 you put .001 here and see what time you get. 21:55.440 --> 22:01.840 So you want e^(−t/τ) to be 22:01.842 --> 22:02.912 .001. 22:02.910 --> 22:08.830 So every big inductor you have in a circuit you will find has 22:08.833 --> 22:15.353 got a resistor in parallel with it to take up the energy it has. 22:15.348 --> 22:18.888 So let's do a little energy calculation. 22:18.890 --> 22:24.010 The energy calculation will be, I had energy in the inductor in 22:26.561 --> 22:29.121 LI_0 ^(2). 22:29.119 --> 22:30.199 What happened to it? 22:30.200 --> 22:31.600 Well, we know what happened. 22:31.598 --> 22:34.808 There's a current in the circuit, and when the current is 22:34.807 --> 22:37.787 flowing in the circuit, it is dissipating heat in the 22:37.785 --> 22:38.525 resistor. 22:38.529 --> 22:42.569 Therefore the power in the resistor will be 22:42.574 --> 22:45.274 I^(2)R_2. 22:45.269 --> 22:49.119 I^(2) is I_0^(2) due to the 22:49.123 --> 22:52.753 minus twice R_2t/L times another 22:52.750 --> 22:54.640 R_2. 22:54.640 --> 22:58.920 Now we integrate this power from 0 to infinity, 22:58.916 --> 23:04.586 you will get I_0 ^(2)R_2. 23:04.588 --> 23:07.928 The integral of e to the minus something is e to 23:07.928 --> 23:10.498 the minus that divided by that infinity to 0. 23:10.500 --> 23:15.320 I can promise you that what you will get will be 23:15.315 --> 23:17.975 L/2R_2. 23:17.980 --> 23:26.610 It will give you then 1 over 2LI_0^(2). 23:26.608 --> 23:31.098 So this is not a surprise, that the stored energy in the 23:31.095 --> 23:33.945 inductor is driving your current. 23:33.950 --> 23:36.810 So if you want, one way to store energy, 23:36.805 --> 23:40.245 it's a capacitor, you charge it and it discharge 23:40.247 --> 23:43.027 when you want the current to flow. 23:43.029 --> 23:46.519 Or you can have an inductor, in which a current has been set 23:46.522 --> 23:46.822 up. 23:46.818 --> 23:50.078 Then when the power shuts off in your house and somebody 23:50.084 --> 23:52.694 throws the switch, that inductor can keep the 23:52.694 --> 23:55.074 light bulb burning, in fact forever. 23:55.068 --> 23:57.148 It will be infinite time before the bulb shuts down, 23:57.150 --> 23:59.810 but you won't be able to see anything after a while, 23:59.808 --> 24:05.228 because this graph is exponentially falling. 24:05.230 --> 24:09.630 This is a very simple problem where we can do the math and we 24:09.630 --> 24:12.050 can understand what it's doing. 24:12.049 --> 24:13.019 Yes? 24:13.019 --> 24:17.269 Student: The graph, what is that triangle thing? 24:17.269 --> 24:19.079 Prof: This one here? 24:19.079 --> 24:20.499 This is called τ. 24:20.500 --> 24:22.340 You mean that symbol? 24:22.338 --> 24:26.328 Tau is a unit of time whose numerical value is 24:26.325 --> 24:31.635 R_2/L and we can write I as I_0 24:31.637 --> 24:34.027 e^(−t/τ). 24:34.029 --> 24:35.909 We always like to write it in this form, 24:35.910 --> 24:40.720 so that if τ = 100 seconds, you will have to make many 24:40.718 --> 24:43.868 multiples of 100 seconds before the current is negligible. 24:43.868 --> 24:46.128 Student: What are the units on the axis? 24:46.130 --> 24:47.850 Prof: Oh, in this graph here? 24:47.848 --> 24:52.228 This is time and this is the current in the loop. 24:52.230 --> 25:01.400 It starts out at some initial value. 25:01.400 --> 25:13.920 Okay, now I'm going to do the next slightly more complicated 25:13.924 --> 25:21.784 circuit, and that is an LC circuit. 25:21.778 --> 25:26.428 If you have a circuit like that, it won't do anything. 25:26.430 --> 25:28.860 You can take these guys, hook them up and wait all day. 25:28.858 --> 25:32.738 Nothing will happen because it's got no energy. 25:32.740 --> 25:33.660 Why should it do anything? 25:33.660 --> 25:35.320 It's just going to sit there. 25:35.318 --> 25:40.688 But if you've charged up your capacitor like this, 25:40.690 --> 25:43.430 and there's an open switch that you then closed, 25:43.430 --> 25:47.700 then those charges are going to leave and find their way around 25:47.703 --> 25:51.563 the other side and neutralize them and the capacitor will 25:51.564 --> 25:52.534 discharge. 25:52.529 --> 25:57.549 But you know that once it discharges, it's not the end of 25:57.545 --> 25:58.615 the story. 25:58.619 --> 25:59.439 So why is that? 25:59.440 --> 26:04.320 Why doesn't it stop when the capacitor is discharged? 26:04.319 --> 26:05.129 Yes? 26:05.130 --> 26:06.990 Student: It's kind of like the inductor has momentum 26:06.986 --> 26:07.706 and it keeps pulling. 26:07.710 --> 26:11.240 Prof: The inductor would have had a current by then, 26:11.236 --> 26:14.576 so the inductor is not going to suddenly stop having its 26:14.579 --> 26:15.309 current. 26:15.308 --> 26:17.958 So it's going to keep driving more current for a while, 26:17.960 --> 26:22.380 until the capacitor's charged up the opposite way to an extent 26:22.378 --> 26:25.238 that this fights that one, that voltage, 26:25.240 --> 26:27.290 and then it will go the other way. 26:27.289 --> 26:31.139 So it will go back and forth. 26:31.140 --> 26:33.700 So let's write down the equation that tells you how it 26:33.698 --> 26:34.758 goes back and forth. 26:34.759 --> 26:36.389 So in other words, you should draw pictures. 26:36.390 --> 26:39.460 Textbooks have nice pictures and I won't even try it. 26:39.460 --> 26:41.990 In the beginning, it may look like--I'm sorry, 26:41.991 --> 26:43.741 in my example, in the beginning, 26:43.737 --> 26:47.167 there's an electric field here, nothing in the capacitor. 26:47.170 --> 26:49.230 After a quarter cycle, there's nothing here, 26:49.230 --> 26:51.290 there's a magnetic field in the inductor. 26:51.288 --> 26:53.568 Another quarter cycle, it's back here, 26:53.566 --> 26:57.316 but with the opposite polarity, and it goes back and forth. 26:57.318 --> 27:05.538 So we can get all that, by writing the equation 27:05.539 --> 27:11.079 LdI/dt Q/C = 0. 27:11.078 --> 27:15.768 So let's write everything in terms of Q. 27:15.769 --> 27:21.519 Then dQ/dt is I, because as the current flows 27:21.523 --> 27:24.703 this way, it builds up charge. 27:24.700 --> 27:26.120 That's my convention. 27:26.118 --> 27:36.488 So I can write it as Ld^(2)Q/dt^(2) (1/C)Q = 27:36.490 --> 27:37.320 0. 27:37.318 --> 27:40.478 Now we have seen exactly this equation before, 27:40.478 --> 27:41.038 right? 27:41.038 --> 27:49.998 We saw the equation for a mass coupled to a spring the equation 27:50.000 --> 27:54.480 md^(2)x/dt^(2) kx = 0. 27:54.480 --> 27:56.140 Now we know what it does. 27:56.140 --> 28:01.290 We know it oscillates back and forth. 28:01.288 --> 28:06.948 And L--this is an SAT question--L is to 28:06.949 --> 28:11.519 m what 1/C is to k. 28:11.519 --> 28:14.879 It's very useful to bear in mind this analogy, 28:14.884 --> 28:18.924 because mathematically, this equation and this equation 28:18.922 --> 28:21.542 have exactly the same solution. 28:21.538 --> 28:23.748 It's no more difficult to solve this one than that one. 28:23.750 --> 28:25.100 This may involve electric charges; 28:25.099 --> 28:26.219 that may involve masses. 28:26.220 --> 28:27.710 You don't care. 28:27.710 --> 28:31.670 So mathematically, here's an equation: 28:31.671 --> 28:38.411 cow times d^(2) dog over dt^(2) let's say elephant 28:38.414 --> 28:42.244 times dog = 0, where dog is a function of 28:42.240 --> 28:44.790 time, has exactly the same solution. 28:44.788 --> 28:47.978 What does it matter what you call the unknown variables, 28:47.976 --> 28:48.436 right? 28:48.440 --> 28:51.600 But you've got to tell me that cow and elephant are time 28:51.597 --> 28:55.157 independent and dog is the only thing that depends on time, 28:55.160 --> 28:57.110 because otherwise the equation is not the same. 28:57.108 --> 29:00.268 So here m and k don't change, L and C don't change. 29:00.269 --> 29:01.969 So what's the answer to this one? 29:01.970 --> 29:04.220 Once again, it's the differential equation. 29:04.220 --> 29:08.750 You write dQ/dt squared is − (1/LC) Q and 29:08.747 --> 29:12.307 you're asking yourself, give me a function which, 29:12.310 --> 29:16.910 when I differentiate twice, looks like itself. 29:16.910 --> 29:18.610 There's always the exponential. 29:18.608 --> 29:21.218 There's also sines and the cosines and you can take any 29:21.218 --> 29:22.328 combination you like. 29:22.328 --> 29:31.888 And the one acceptable solution is some constant A times 29:31.893 --> 29:40.843 cosine square root of 1 over LC t. Is that right? 29:40.839 --> 29:41.299 Yes. 29:41.298 --> 29:43.358 If you take two derivatives of Q, 29:43.358 --> 29:46.178 you'll pull one over root LC each time, 29:46.180 --> 29:49.990 then you'll get a - sine when cosine becomes - sine and one 29:49.990 --> 29:53.210 more derivative will bring it back to - cosine. 29:53.210 --> 29:56.110 So this is called the frequency, which is 29:56.114 --> 29:57.644 1/√LC. 29:57.640 --> 30:02.950 That's the analog of square root of k over m. 30:02.950 --> 30:04.340 Anyway, this will oscillate. 30:04.338 --> 30:08.788 It's a mechanical analog--it's the electrical analog of the 30:08.788 --> 30:10.398 mechanical problem. 30:10.400 --> 30:13.130 So I don't want to do too much of this, because I think you 30:13.133 --> 30:15.213 should be quite familiar with this, at least, 30:15.205 --> 30:16.945 the oscillatory behavior of this. 30:16.950 --> 30:18.590 And the analogy is perfect. 30:18.588 --> 30:22.258 For example, to say that I pulled the mass 30:22.259 --> 30:27.099 by 1 meter is to say that x was given a non-zero 30:27.095 --> 30:30.045 value and dx/dt was 0. 30:30.048 --> 30:32.908 Analogous thing here will be Q was given a non-zero 30:32.910 --> 30:35.370 value and dQ/dt which is I, is 0. 30:35.368 --> 30:38.188 That means the capacitor was charged, the inductor was not 30:38.191 --> 30:39.331 carrying any current. 30:39.329 --> 30:40.209 Then I let it go. 30:40.210 --> 30:41.140 What happens? 30:41.140 --> 30:43.190 The problems are identical. 30:43.190 --> 30:44.480 Yes? 30:44.480 --> 30:48.760 Student: When you solve the differential equation, 30:48.758 --> 30:50.858 why did you omit the sine? 30:50.858 --> 30:52.598 Prof: Yes, I should explain to you. 30:52.599 --> 30:55.009 That's an important point. 30:55.009 --> 30:58.249 Q(t) can be one number, cosine 30:58.250 --> 31:00.950 ω_0t. 31:00.950 --> 31:04.450 ω_0 is this guy another number sine 31:04.448 --> 31:06.288 ω_0t. 31:06.289 --> 31:08.649 And both are acceptable. 31:08.650 --> 31:11.410 And the property of this differential equation is, 31:11.410 --> 31:13.870 if I have one answer, cos ωt, and 31:13.865 --> 31:17.315 another possible answer, sine ωt, then any 31:17.317 --> 31:21.467 constant times cos any constant times sine is also a solution. 31:21.470 --> 31:23.030 You can verify that. 31:23.028 --> 31:25.458 If you have one solution, Q_1, 31:25.460 --> 31:27.440 a second solution, Q_2, 31:27.440 --> 31:29.680 you can add the two equations and you can show 31:29.684 --> 31:32.734 Q_1 Q_2 obeys the same equation. 31:32.730 --> 31:36.480 Not only that, suppose there's one problem you 31:36.480 --> 31:38.630 solve, dQ_1/dt 31:38.630 --> 31:42.060 squared is -ω on R squared 31:42.058 --> 31:43.738 Q_1. 31:43.740 --> 31:48.830 There is some function of time and here's another function of 31:48.827 --> 31:51.707 time, obeying the same equation. 31:51.710 --> 31:55.120 Now you can multiply this by a number A and you can 31:55.116 --> 31:58.216 multiply this by a number B, take the A 31:58.224 --> 32:01.674 inside the derivative, B inside the derivative 32:01.673 --> 32:03.083 and add the two sides. 32:03.078 --> 32:05.668 On the left hand side you will find the second derivative of 32:05.673 --> 32:07.393 AQ_1 BQ_2. 32:07.390 --> 32:08.500 Right hand side, you will find 32:08.502 --> 32:10.542 ω_0^(2) times AQ_1 32:10.537 --> 32:11.417 BQ_2. 32:11.420 --> 32:14.510 That means AQ_1 BQ_2 obeys this 32:14.508 --> 32:16.078 same differential equation. 32:16.078 --> 32:19.718 This is called superposing two solutions. 32:19.720 --> 32:21.580 You've got one solution, another one, 32:21.584 --> 32:23.504 you can multiply one by a constant. 32:23.500 --> 32:26.300 It's very important it's a constant, because that's what 32:26.298 --> 32:28.588 will let you take it inside the derivative. 32:28.588 --> 32:31.388 Therefore the correct answer here is really 32:31.390 --> 32:35.190 Acosωd Bsinωt and you can 32:35.190 --> 32:36.730 ask, why did you write 32:36.732 --> 32:39.012 Acosω _0t? 32:39.009 --> 32:40.869 First of all, that's not a good solution. 32:40.868 --> 32:45.078 I can put a Φ there which is arbitrary. 32:45.078 --> 32:49.038 And it's not hard to show--let me call that C if you 32:49.038 --> 32:50.198 like, C. 32:50.200 --> 32:55.150 This can be written as Ccosω 32:55.154 --> 32:58.384 _0t - Φ. 32:58.380 --> 33:00.730 In other words, it's possible to take either a 33:00.733 --> 33:02.673 solution with a cosine and a sine, 33:02.670 --> 33:06.430 with no extra phases inside, to another one with one 33:06.432 --> 33:08.352 amplitude and one phase. 33:08.348 --> 33:11.378 The phase is the delay in the cosine. 33:11.380 --> 33:15.180 Anybody here has trouble, who does not know the details, 33:15.180 --> 33:18.360 I'll be happy to explain why that's correct. 33:18.358 --> 33:20.458 If you don't know, you should tell me now. 33:20.460 --> 33:25.410 You can check that if you take this thing, 33:25.410 --> 33:30.200 make a right angled triangle with A, 33:30.200 --> 33:34.680 B and C and an angle Φ here, 33:34.680 --> 33:39.630 then A can be written as CcosΦ and 33:39.625 --> 33:44.225 B can be written as CsinΦ, 33:44.230 --> 33:46.360 and that just happens to be cosωt - 33:46.363 --> 33:50.053 Φ, this one, some trigonometry. 33:50.048 --> 33:53.868 So the differential equation, with the second order in time, 33:53.865 --> 33:56.965 will always have two unknown parameters in it. 33:56.970 --> 34:00.060 You can choose them to be these numbers A and B, 34:00.055 --> 34:02.225 or this number C and Φ. 34:02.230 --> 34:04.710 And the equation won't tell you what they are, 34:04.710 --> 34:07.960 because once you tell me there's a mass coupled to a 34:07.955 --> 34:10.375 spring and I say, "Okay, this is the spring 34:10.382 --> 34:11.652 constant this is the mass?" 34:11.650 --> 34:13.780 and I ask you, "Where is the mass right 34:13.775 --> 34:14.365 now?" 34:14.369 --> 34:15.069 I don't know. 34:15.070 --> 34:18.130 It depends on when you started it and how you released it. 34:18.130 --> 34:20.930 So I need to know what's called initial conditions, 34:20.929 --> 34:24.339 which is the value of Q or the value of x at the 34:24.340 --> 34:27.640 initial time and the value of the velocity or the current at 34:27.641 --> 34:28.761 an initial time. 34:28.760 --> 34:32.550 With those two pieces of information, I can solve for 34:32.550 --> 34:34.300 A and B. 34:34.300 --> 34:38.470 This I think you should know, this kind of stuff you should 34:38.469 --> 34:41.559 know, so I won't say too much about that. 34:41.559 --> 34:46.479 All right, so now we do another problem. 34:46.480 --> 34:52.880 We put an alternating voltage on these guys. 34:52.880 --> 34:57.990 So this is V_0 cosωt and this 34:57.987 --> 35:01.067 is C and this is L. 35:01.070 --> 35:05.080 Now we ask, what's the current? 35:05.079 --> 35:10.519 This omega is not the natural frequency of oscillation. 35:10.518 --> 35:14.578 It is some externally given omega, like 60 hertz from your 35:14.579 --> 35:16.929 power supply, from your socket. 35:16.929 --> 35:20.049 So that's driving the circuit and you can ask, 35:20.050 --> 35:21.370 what happens now? 35:21.369 --> 35:25.349 The answer will be, you write the same equation. 35:25.349 --> 35:34.889 You can write Ld^(2)Q/dt^(2) (1/C)Q = 35:34.894 --> 35:41.794 V_0cos ωt. 35:41.789 --> 35:43.549 Now what are you going to do? 35:43.550 --> 35:46.840 You have to again guess the solution. 35:46.840 --> 35:50.170 So I want a function, Q of t, 35:50.170 --> 35:53.730 so that when I take two derivatives and add it to some 35:53.731 --> 35:56.291 multiple of itself, I get something, 35:56.289 --> 35:58.179 something times the cosine. 35:58.179 --> 36:01.809 So what should that function look like? 36:01.809 --> 36:02.929 Yes? 36:02.929 --> 36:07.739 Student: > 36:07.739 --> 36:09.939 Prof: No, no, what function Q 36:09.940 --> 36:12.770 of t do you think will satisfy this equation? 36:12.768 --> 36:15.798 I'm asking for a function of time which, when I put into 36:15.797 --> 36:18.437 this, has some chance of obeying the equation. 36:18.440 --> 36:23.480 What's the functional form? 36:23.480 --> 36:24.540 Yes? 36:24.539 --> 36:26.029 Student: Cosine times some constant. 36:26.030 --> 36:26.760 Prof: Right. 36:26.760 --> 36:30.100 In other words, I can take it to be some 36:30.101 --> 36:33.791 constant, C times cosωt, 36:33.786 --> 36:34.726 period. 36:34.730 --> 36:37.830 Not even times Φ. 36:37.829 --> 36:41.609 Take this one, see what happens. 36:41.610 --> 36:46.230 Now when you take one derivative, you get 36:46.226 --> 36:50.726 -ωLCcos ωt. 36:50.730 --> 36:53.370 When you take another derivative, you get 36:53.365 --> 36:55.735 −ω ^(2)L ^(2)-- 36:55.739 --> 36:59.959 I'm sorry, this becomes sine times cosωt, 36:59.960 --> 37:04.030 and this one is just oh my god--this is my nightmare. 37:04.030 --> 37:06.560 You can see the nightmare now? 37:06.559 --> 37:08.969 How many people see the nightmare? 37:08.969 --> 37:09.879 Yes? 37:09.880 --> 37:10.770 Student: Constant C. 37:10.768 --> 37:13.758 Prof: Yes, so I picked that C to be 37:13.757 --> 37:15.497 the same as this C. 37:15.500 --> 37:22.480 So we'll put a hat on this guy so we can tell them about. 37:22.480 --> 37:27.730 So C˜/C times C˜ is 37:27.728 --> 37:32.738 V_0 cosωt. 37:32.739 --> 37:35.879 And what this tells you is, if you take two 37:35.880 --> 37:39.170 derivatives--I'm sorry, this is so sloppy. 37:39.170 --> 37:47.470 If you take two derivatives here, you will get 37:47.467 --> 37:51.337 ω^(2). 37:51.340 --> 37:59.300 cosω be also be with the - sign. 37:59.300 --> 38:01.330 Q/C. 38:01.329 --> 38:02.799 Okay. 38:02.800 --> 38:03.820 You guys buy that? 38:03.820 --> 38:07.930 There is no L^(2), just an L. 38:07.929 --> 38:14.239 That = V_0 cosωt. 38:14.239 --> 38:14.569 Right? 38:14.570 --> 38:18.610 Every derivative brings an ω and there's a net - 38:18.614 --> 38:19.154 sign. 38:19.150 --> 38:22.910 So you find out that your guess, C˜c 38:22.914 --> 38:27.624 osωt is going to work, provided this equation is 38:27.621 --> 38:28.721 satisfied. 38:28.719 --> 38:30.529 And the cosωt can be canceled. 38:30.530 --> 38:33.220 That's the whole point of doing this thing. 38:33.219 --> 38:40.349 So then you get C˜ = V_0 - 38:40.351 --> 38:47.481 V_0 divided by ω^(2)L − 38:47.481 --> 38:49.161 1/C. 38:49.159 --> 38:53.649 Or if you like, -V_0/L divided 38:53.648 --> 38:58.528 by ω^(2) - ω_0^(2). 38:58.530 --> 39:01.780 Don't worry about the numerator; look at the denominator. 39:01.780 --> 39:05.950 The denominator says that if your driving frequency is equal 39:05.951 --> 39:09.771 to the resonant frequency, C˜ blows up. 39:09.768 --> 39:12.148 That means when a system has got a resonance--yes? 39:12.150 --> 39:15.870 Student: Where did the ω^(2) come from? 39:15.869 --> 39:19.069 Prof: I took two derivatives--oh, 39:19.065 --> 39:19.635 here. 39:19.639 --> 39:20.999 Oh, I'm sorry. 39:21.000 --> 39:23.420 Yes, thank you. 39:23.420 --> 39:26.810 You know, it's good to do these things slowly and not rely on 39:26.813 --> 39:28.343 what you heard somewhere. 39:28.340 --> 39:29.740 So let me do it for you again. 39:29.739 --> 39:30.739 Sorry about that. 39:30.739 --> 39:32.599 So this time I take two derivatives. 39:32.599 --> 39:34.529 Each one brings an ω. 39:34.530 --> 39:37.400 There is an L, and the - sine comes because 39:37.398 --> 39:39.308 the cosine becomes - sine. 39:39.309 --> 39:41.809 Differentiate one more, you get - the cosine. 39:41.809 --> 39:43.139 This guy is nothing. 39:43.139 --> 39:46.999 It just says "divide my by C" 39:47.001 --> 39:49.321 and that's equal to that. 39:49.320 --> 39:52.800 And then I wrote 1/LC is ω_0^(2). 39:52.800 --> 39:53.890 Thank you very much. 39:53.889 --> 39:55.379 Okay? 39:55.380 --> 39:57.400 I was really hung up on the final result, 39:57.400 --> 39:59.590 which tells you, you'd better not drive this at 39:59.585 --> 40:03.495 the resonant frequency, because then current amplitude 40:03.503 --> 40:06.013 will build up indefinitely. 40:06.010 --> 40:07.900 That's also true in a swing. 40:07.900 --> 40:11.900 If you've got a swing and the kid's coming back and forth, 40:11.900 --> 40:14.300 and you're reading a newspaper just pushing the kid, 40:14.300 --> 40:17.800 you've got to push at the right time to get the best result. 40:17.800 --> 40:20.620 And that kid's not going to fly off, because there's one more 40:20.621 --> 40:23.161 term in the equation for the kid, which is friction. 40:23.159 --> 40:25.749 But if you have a frictionless swing and you're doing this, 40:25.750 --> 40:27.880 you've got to watch out, because soon there'll be nobody 40:27.882 --> 40:31.142 around, because the amplitude will keep 40:31.137 --> 40:31.967 growing. 40:31.969 --> 40:34.449 Well, these are unrealistic problems, but I want you to 40:34.447 --> 40:35.317 notice one thing. 40:35.320 --> 40:37.050 Here's what I want you to notice. 40:37.050 --> 40:41.120 The voltage you were given looked like V_0 40:41.117 --> 40:45.257 cosωt and the current that you got, 40:45.260 --> 40:50.570 I can obtain by taking the derivative of this charge, 40:50.570 --> 40:53.700 which is like dQ/dt. 40:53.699 --> 40:56.639 That's going to be C˜ω 40:56.637 --> 41:00.737 sinωt, and C˜, we can 41:00.737 --> 41:04.987 write as V_0/L divided by ω^(2) - 41:04.987 --> 41:08.137 ω_0 ^(2)sinωt. 41:08.139 --> 41:10.559 This is I. 41:10.559 --> 41:14.019 Here's what I want you to notice. 41:14.018 --> 41:17.548 It looks a lot like Ohm's law, because the current looks like 41:17.552 --> 41:20.622 voltage divided by some number, but that's a very big 41:20.615 --> 41:21.495 difference. 41:21.500 --> 41:27.000 The voltage is a cosine and the current is a sine. 41:27.000 --> 41:29.060 That's something I want you to think about. 41:29.059 --> 41:33.529 The current is not in step with the voltage, whereas in a 41:33.529 --> 41:36.719 resistor circuit, the current follows the 41:36.722 --> 41:37.682 voltage. 41:37.679 --> 41:40.549 It's the same profile as the voltage, except you divide by 41:40.545 --> 41:41.095 R. 41:41.099 --> 41:43.449 Here one is a cosine, one is a sine. 41:43.449 --> 41:47.589 That means when one guy is at maximum, other is at minimum. 41:47.590 --> 41:50.110 It's called out of phase, in fact out of phase by 90 41:50.114 --> 41:50.614 degrees. 41:50.610 --> 41:51.420 Yes? 41:51.420 --> 41:54.840 Student: ________ by the C that you got, 41:54.844 --> 41:56.434 where did the omega go? 41:56.429 --> 41:57.299 Prof: Here? 41:57.300 --> 41:58.610 Student: On the other side. 41:58.610 --> 42:02.990 When you multiply that through by the C-- 42:02.989 --> 42:04.759 Prof: Oh, ω did not go, 42:04.755 --> 42:05.255 it's back. 42:05.260 --> 42:06.650 Right here. You're right. 42:06.650 --> 42:07.700 There's the ω. 42:07.699 --> 42:08.289 That came from... 42:08.289 --> 42:21.889 42:21.889 --> 42:25.969 Okay, these problems, the only thing I'm focusing on 42:25.969 --> 42:28.239 right now, I would say of all the things I 42:28.235 --> 42:30.795 wrote there, to be the most important thing 42:30.797 --> 42:33.257 is number one, this is what I want you to bear 42:33.257 --> 42:33.657 in mind. 42:33.659 --> 42:35.969 That's why sometimes I'm not paying attention to some 42:35.967 --> 42:36.497 constants. 42:36.500 --> 42:41.050 What is important to notice is that this is an equation you can 42:41.045 --> 42:42.655 solve by inspection. 42:42.659 --> 42:45.209 Why was it easier to solve it by inspection? 42:45.210 --> 42:48.410 Because you are trying to get in the end a cosine to balance 42:48.411 --> 42:51.251 the right hand side, and you're trying to find a 42:51.253 --> 42:55.263 function who is itself a cosine, or whose second derivative is a 42:55.259 --> 42:57.219 cosine, and we know the answer to that 42:57.217 --> 43:00.037 is a cosine, so we can guess the answer. 43:00.039 --> 43:02.649 Once you guess the answer, you put it in and you analyze 43:02.652 --> 43:04.412 the solution, you notice also that the 43:04.409 --> 43:06.499 current and the voltage are not in step. 43:06.500 --> 43:07.730 One is a cosine. 43:07.730 --> 43:10.120 You know what a cosine does, it does that. 43:10.119 --> 43:13.039 Other is a sine, so they are out of step. 43:13.039 --> 43:15.649 When one is at maximum, the other is at minimum and so 43:15.652 --> 43:15.902 on. 43:15.900 --> 43:20.230 That means a current as a function of time is not equal to 43:20.233 --> 43:24.493 the voltage as a function of time divided by anything. 43:24.489 --> 43:29.049 There is nothing you can divide a cosine by to turn it into a 43:29.047 --> 43:29.577 sine. 43:29.579 --> 43:34.469 Whereas with resistors, you just divide by R, 43:34.472 --> 43:36.922 you get the current. 43:36.920 --> 43:43.860 Now for the more realistic problem, the realistic problem 43:43.860 --> 43:50.060 has got a capacitor, an inductor and a resistor and 43:50.056 --> 43:53.276 no other power supply. 43:53.280 --> 44:03.000 The equation obeyed by this one will be Ld^(2)Q/dt^(2) 44:03.003 --> 44:11.513 (that stands for LdI/dt) RdQ/dt 44:11.510 --> 44:14.810 Q/C = 0. 44:14.809 --> 44:19.639 This is analogous to md^(2)x/dt^(2) (I don't 44:19.639 --> 44:24.469 know how you guys wrote this thing last semester. 44:24.469 --> 44:35.779 It doesn't matter) γdx/dt kx = 44:35.784 --> 44:36.694 0. 44:36.690 --> 44:40.030 Notice that it's the same form. 44:40.030 --> 44:43.820 And I'm going to give this to you as a homework problem to 44:43.817 --> 44:46.007 analyze the answer to this one. 44:46.010 --> 44:50.930 So this describes a problem where you have a mass and a 44:50.927 --> 44:54.567 spring and some friction on the table. 44:54.570 --> 44:57.950 This means if you pull it and let it go, the oscillations will 44:57.952 --> 45:00.452 eventually get damped and it will die down. 45:00.449 --> 45:06.419 And the general solution for the case when it's oscillating 45:06.423 --> 45:11.993 will look like some number A e to the minus some 45:11.985 --> 45:16.925 number αt times cosine times some frequency 45:16.929 --> 45:19.299 ω't. 45:19.300 --> 45:20.530 Let me see. 45:20.530 --> 45:24.340 You can always add a phase Φ, 45:24.335 --> 45:27.355 but I'm not going to do that. 45:27.360 --> 45:29.210 If you want, you can put an extra Φ, 45:29.213 --> 45:32.163 but I'll choose my origin of time so that I don't have that. 45:32.159 --> 45:34.759 This is what the answer's going to look like, 45:34.760 --> 45:37.840 where ω' and α are going to be 45:37.838 --> 45:41.218 controlled by L, R and C. 45:41.219 --> 45:43.399 That's the thing I don't want to do in class. 45:43.400 --> 45:45.670 Have you seen this before? 45:45.670 --> 45:48.230 Professor Harris tells me you've seen this last time. 45:48.230 --> 45:52.110 And the hints in the homework will guide you on how to do 45:52.114 --> 45:52.604 this. 45:52.599 --> 45:56.809 You just assume the solution x at t looks like 45:56.811 --> 46:00.241 some A e to the - (I don't want to call it 46:00.237 --> 46:04.017 α now) β times t. 46:04.018 --> 46:07.478 You put it in the equation and solve for β and 46:07.483 --> 46:10.523 you'll find β will have a real part and 46:10.521 --> 46:11.861 an imaginary part. 46:11.860 --> 46:17.010 And you have to combine the two to get this answer. 46:17.010 --> 46:21.140 Now that brings me to another thing, so I don't know how 46:21.137 --> 46:25.117 prepared you guys are for what's about to happen next, 46:25.117 --> 46:28.117 which is the use of complex numbers. 46:28.119 --> 46:30.339 So everybody familiar with complex numbers? 46:30.340 --> 46:34.190 Who doesn't know complex numbers? 46:34.190 --> 46:36.440 How do you do your taxes? 46:36.440 --> 46:39.430 You don't know imaginary numbers? 46:39.429 --> 46:42.809 So I will tell you. 46:42.809 --> 46:44.099 I'll give you a lightning review. 46:44.099 --> 46:47.199 I'm assuming everybody has seen them in high school. 46:47.199 --> 46:50.579 I will only tell you the part you need, but I'm going to 46:50.581 --> 46:52.921 assume that I can use them fluently. 46:52.920 --> 46:56.840 I don't want to stop every time and worry about you guys. 46:56.840 --> 46:59.400 So I really need a show of hands. 46:59.400 --> 47:04.370 Anybody never seen x iy and x - iy? 47:04.369 --> 47:05.959 And how about e^(iθ)? 47:05.960 --> 47:07.930 You know that guy? 47:07.929 --> 47:09.039 Okay, that's all you need. 47:09.039 --> 47:10.959 So I'm going to tell you what the deal is. 47:10.960 --> 47:12.900 So everyone knows what a complex number is. 47:12.900 --> 47:15.600 We know i is square root of -1. 47:15.599 --> 47:19.929 Then we know that we can write a complex number as x iy 47:19.934 --> 47:24.274 and visualize it in a complex plane where you measure x 47:24.268 --> 47:27.678 this way and iy is measured that way. 47:27.679 --> 47:34.309 That's usually called a generic complex number z. 47:34.309 --> 47:39.119 But now let us find the length of that--the complex number is a 47:39.123 --> 47:41.923 single point in the complex plane. 47:41.920 --> 47:47.340 The length of that is square root of x^(2) y^(2). 47:47.340 --> 47:52.480 So I can also write z as x divided by square root 47:52.483 --> 47:57.633 of x^(2) y^(2) i times y divided by square 47:57.626 --> 48:02.516 root of x^(2) y^(2) times the square root of x^(2) 48:02.521 --> 48:03.851 y^(2). 48:03.849 --> 48:06.339 Just done nothing, just rearranged stuff. 48:06.340 --> 48:11.480 This is going to be called the modulus of z, 48:11.483 --> 48:13.443 and what is this? 48:13.440 --> 48:16.510 This angle is θ here. 48:16.510 --> 48:19.420 This is really cosθ 48:19.416 --> 48:24.476 isinθ times modulus of z. 48:24.480 --> 48:28.670 And that, thanks to this great identity by Euler, 48:28.670 --> 48:34.570 is e^(iθ). Now I don't know how much you know 48:34.568 --> 48:40.668 about this great formula that relates e^(iθ) to 48:40.673 --> 48:42.333 cosine sine. 48:42.329 --> 48:44.539 How many people know where it comes from? 48:44.539 --> 48:49.029 What does it mean to raise e to a complex power? 48:49.030 --> 48:51.040 You know where it really comes from? 48:51.039 --> 48:51.809 Yes? 48:51.809 --> 48:52.739 Student: > 48:52.739 --> 48:54.339 Prof: It comes from power series. 48:54.340 --> 48:57.510 It turns out you can write a power series for cosine. 48:57.510 --> 48:59.420 You can write a power series for sine, 48:59.420 --> 49:03.750 then, for example, cosine of θ = 1 49:03.751 --> 49:08.181 − θ^(2) over 2 factorial, 49:08.179 --> 49:10.329 θ to the 4 factorial, 49:10.329 --> 49:14.369 or 6 factorial, etc. 49:14.369 --> 49:16.959 And you can write a power series for e^(θ) if 49:16.963 --> 49:19.463 you like, which is 1 θ 49:19.458 --> 49:21.838 θ^(2)/2. 49:21.840 --> 49:25.510 This defines cosθ in the sense that if you put 49:25.514 --> 49:29.194 θ = Π/2 in this infinite series, 49:29.190 --> 49:31.770 you will get 0. 49:31.768 --> 49:33.188 And if you put θ = Π, you will get 49:33.188 --> 49:33.678 Π -1. 49:33.679 --> 49:36.479 In other words, in spite of this funny looking 49:36.481 --> 49:38.601 form, it really is the same guy. 49:38.599 --> 49:40.189 It will oscillate, it will have zeros, 49:40.186 --> 49:41.126 it will be periodic. 49:41.130 --> 49:43.680 None of it is obvious, but this power series is 49:43.681 --> 49:47.181 numerically equal to this one if you keep the infinite number of 49:47.175 --> 49:47.725 turns. 49:47.730 --> 49:51.000 Likewise, e^(θ) is defined by this series. 49:51.000 --> 49:52.570 And similarly, there's a formula for 49:52.570 --> 49:53.470 sinθ. 49:53.469 --> 49:57.889 Now once a power series is defined, you can put e raised to 49:57.894 --> 49:59.884 anything you want there. 49:59.880 --> 50:03.410 So you know what I'm going to put there. 50:03.409 --> 50:07.739 e raised to dog is 1 dog dog squared. 50:07.739 --> 50:09.119 This is not a joke. 50:09.119 --> 50:10.339 This is really true. 50:10.340 --> 50:11.270 This is the definition. 50:11.268 --> 50:13.398 If someone says, "How do I raise e 50:13.398 --> 50:14.528 to the power dog?" 50:14.530 --> 50:15.830 you do this. 50:15.829 --> 50:17.939 This will have all the properties of the exponential. 50:17.940 --> 50:19.290 It doesn't matter what's in the exponent. 50:19.289 --> 50:20.969 That's the key. 50:20.969 --> 50:24.109 People originally put real numbers, then they put complex 50:24.108 --> 50:24.668 numbers. 50:24.670 --> 50:27.500 Now they put matrices, operators, anything you want. 50:27.500 --> 50:30.760 e raised to anything, you formally define to be this 50:30.762 --> 50:33.242 infinite series, provided the infinite series 50:33.239 --> 50:35.939 converges and gives you a meaningful answer. 50:35.940 --> 50:38.330 In that sense, if you put e^(iθ) 50:38.331 --> 50:40.531 here, and compare the result to 50:40.530 --> 50:42.840 cosθ isinθ, 50:42.840 --> 50:44.440 it matches, that's all. 50:44.440 --> 50:48.140 I'm just going to keep using that result. 50:48.139 --> 50:51.819 Then you should also know that for every complex number, 50:51.822 --> 50:55.442 there's a complex conjugate, which is x - iy. 50:55.440 --> 50:58.120 That means i goes to -i. 50:58.119 --> 51:02.059 That means z, the angle θ 51:02.061 --> 51:05.711 will change to -θ. 51:05.710 --> 51:09.980 And given any complex number, the real part of z, 51:09.980 --> 51:14.170 which is x, is z z* over 2 and the 51:14.172 --> 51:20.022 imaginary part of z, which is y, 51:20.019 --> 51:26.889 is z − z* over 2i. 51:26.889 --> 51:29.329 This is what you need to know. 51:29.329 --> 51:32.219 Every complex number has a real part and an imaginary part. 51:32.219 --> 51:34.919 In fact, z* looks like this. 51:34.920 --> 51:37.820 And if you add z z*, the vertical parts cancel 51:37.820 --> 51:40.110 and you get double the horizontal part. 51:40.110 --> 51:41.840 That's why you divide by 2. 51:41.840 --> 51:45.650 You subtract and divide by 2i, you get y. 51:45.650 --> 51:49.180 Basically, that's all I want you to know, but I want you to 51:49.181 --> 51:51.861 be able to manipulate them rapidly enough. 51:51.860 --> 51:56.490 Here's something very useful about complex numbers. 51:56.489 --> 52:02.719 Let's take a complex number z_1 which is 52:02.717 --> 52:07.387 mod z_1 e^(iθ1), 52:07.389 --> 52:09.909 and another complex number, z_2, 52:09.909 --> 52:15.419 which is mod z_2 e^(iθ2). 52:15.420 --> 52:16.840 So what do they look like? 52:16.840 --> 52:18.900 Well, z_1 looks like this, 52:18.900 --> 52:20.760 with some angle θ_1, 52:20.760 --> 52:22.850 and z_2_ may look like that. 52:22.849 --> 52:25.019 It's got some angle θ_2. 52:25.018 --> 52:28.118 So every complex number has a length and an angle. 52:28.119 --> 52:29.239 You can think of it two ways. 52:29.239 --> 52:31.779 It's got an x part and a y part, 52:31.784 --> 52:34.894 and a real and an imaginary, or it's got a modulus and a 52:34.893 --> 52:35.463 phase. 52:35.460 --> 52:38.570 That's the Cartesian version of the number and the polar number 52:38.567 --> 52:39.367 of the number. 52:39.369 --> 52:42.839 And the connection between them comes from Euler's formula. 52:42.840 --> 52:45.910 But now look what happens when I take the product 52:45.911 --> 52:48.281 z_1z _2. 52:48.280 --> 52:51.260 I get mod z_1 mod z_2 52:51.260 --> 52:54.540 e^(iθ)_1^( θ)_2, 52:54.539 --> 52:59.559 because exponentials add when you multiply them. 52:59.559 --> 53:02.069 Well, this allows you to immediately guess what the 53:02.065 --> 53:03.665 product is going to look like. 53:03.670 --> 53:06.050 The product is going to have a length equal to the length of 53:06.048 --> 53:08.468 z_1 times length of z_2. 53:08.469 --> 53:11.419 It's going to have an angle, I've not done a good picture 53:11.418 --> 53:13.718 here, which is the sum of the two 53:13.715 --> 53:14.995 angles, because it's 53:15.000 --> 53:17.640 θ_1 θ_2. 53:17.639 --> 53:21.859 So listen to this statement very carefully. 53:21.860 --> 53:24.420 When you take a complex number, say number 1, 53:24.420 --> 53:27.100 and you multiply it by a second complex number, 53:27.099 --> 53:29.369 you do two things to the first guy. 53:29.369 --> 53:33.619 You rescale it and you rotate it. 53:33.619 --> 53:36.789 So two operations are done in one shot when you use complex 53:36.791 --> 53:37.341 numbers. 53:37.340 --> 53:38.930 The mod z_2 rescale is the mod 53:38.931 --> 53:40.831 z_1 and θ_2 adds 53:40.833 --> 53:42.013 to θ_1. 53:42.010 --> 53:43.660 Likewise, if you divide, if you take 53:43.655 --> 53:46.285 z_1 over z_2, it is mod 53:46.286 --> 53:48.866 z_1 over mod z_2 times 53:48.871 --> 53:51.881 e^(iθ)_1 ^(- θ)_2, 53:51.880 --> 53:55.630 because the exponential θ_2 is 53:55.632 --> 53:56.572 downstairs. 53:56.570 --> 53:59.620 So multiplication by real numbers is very easy. 53:59.619 --> 54:02.579 You take a real number 4, you multiply by 8, 54:02.577 --> 54:06.357 you'll get something in the real axis 8 times longer. 54:06.360 --> 54:10.180 You multiply by -8, you'll get something back here. 54:10.179 --> 54:12.089 With complex numbers, you take a number, 54:12.088 --> 54:14.928 you multiply another number, you rescale and you rotate. 54:14.929 --> 54:15.979 That's what I want you to know. 54:15.980 --> 54:19.920 That's going to be very important for what I'm saying. 54:19.920 --> 54:22.890 Now I come to the problem I really want to solve, 54:22.889 --> 54:31.639 which is an LCR circuit driven by an alternating source, 54:31.639 --> 54:34.449 V_0 cosωt. 54:34.449 --> 54:37.599 This is R, this is C and this is 54:37.599 --> 54:38.369 L. 54:38.369 --> 54:45.589 And the equation we have to solve is L dI/dt 54:45.592 --> 54:52.672 RI 1/C integral I dt = 54:52.672 --> 54:59.032 V_0 cosωt. 54:59.030 --> 55:03.510 Because Q is the integral up to time t of the 55:03.512 --> 55:06.072 current, that's just Q/C. It's 55:06.065 --> 55:09.075 the same equation, but now, unlike in this 55:09.077 --> 55:12.507 problem, where I had no driving voltage, 55:12.510 --> 55:14.990 I have a driving voltage, and I've also written it in 55:14.987 --> 55:17.367 terms of current rather than in terms of charge. 55:17.369 --> 55:19.509 Because in electrical engineering, you don't really 55:19.514 --> 55:21.064 watch the charge in the capacitor. 55:21.059 --> 55:24.809 You look at the current flowing through the circuit. 55:24.809 --> 55:28.189 So you have to solve this equation. 55:28.190 --> 55:31.220 The question is, can you guess the answer? 55:31.219 --> 55:32.149 That's the only way we know. 55:32.150 --> 55:34.450 You've got to guess. 55:34.449 --> 55:38.459 You're trying to find a function, I of t, 55:38.460 --> 55:41.740 so that when you differentiate it, you get a cosine 55:41.735 --> 55:44.235 ωt, when you integrate it, 55:44.237 --> 55:46.177 you get a cosine ωt. 55:46.179 --> 55:47.659 So far we can do it. 55:47.659 --> 55:49.349 Sine ωt will do it. 55:49.349 --> 55:53.589 But when you leave it alone, then also you get a cosine 55:53.588 --> 55:56.648 ωt. You cannot do that, 55:56.650 --> 55:59.790 because sine will become a cosine and sine will become a 55:59.789 --> 56:00.189 sine. 56:00.190 --> 56:03.840 In the LC circuit, when you didn't have this guy, 56:03.840 --> 56:06.360 we were okay, because if you pick a sine, 56:06.360 --> 56:10.080 this became a cosine and that also became a cosine and you can 56:10.079 --> 56:11.969 combine them to get a cosine. 56:11.969 --> 56:14.479 But you cannot have dI/dt, 56:14.478 --> 56:19.098 I and integral of I all in one equation and all satisfied by 56:19.103 --> 56:21.223 trigonometric function. 56:21.219 --> 56:24.569 The only time we can guess the answer is if it looks like 56:24.567 --> 56:26.717 V_0 e^(αt). 56:26.719 --> 56:29.789 Suppose that was the voltage. 56:29.789 --> 56:33.579 Then can you guess the answer, what form the answer will have? 56:33.579 --> 56:36.539 It will also be e^(αt) times some number, 56:36.539 --> 56:38.629 because then this will look e^(αt), that will 56:38.630 --> 56:40.580 look like e^(αt), that will look like 56:40.576 --> 56:41.386 e^(αt). 56:41.389 --> 56:43.759 e^(αt) has a great property that whether you 56:43.764 --> 56:45.394 multiply it, whether you integrate it, 56:45.394 --> 56:46.874 differentiate it or leave it alone, 56:46.869 --> 56:48.629 it looks the same. 56:48.630 --> 56:50.580 Unfortunately, no one's interested in this 56:50.577 --> 56:52.857 voltage, because it's growing exponentially fast, 56:52.860 --> 56:55.380 or if you put a - sign, it's dying exponentially. 56:55.380 --> 56:58.390 What we really want is this. 56:58.389 --> 57:02.379 So the question is, how do we solve this problem 57:02.380 --> 57:04.590 with cosωt? 57:04.590 --> 57:08.130 So we're going to use a certain trick. 57:08.130 --> 57:12.710 The trick I'm going to use is the following: 57:12.708 --> 57:18.878 let me take a general case where this is some function V 57:18.884 --> 57:23.444 of t, where I'm not even assuming 57:23.436 --> 57:24.976 V is real. 57:24.980 --> 57:28.410 I'm not assuming I is real, but L and R 57:28.413 --> 57:29.813 and C are real. 57:29.809 --> 57:33.119 Everything else can be complex numbers. 57:33.119 --> 57:36.099 Now once you have an equation and you've found a solution, 57:36.101 --> 57:38.561 if you take the complex conjugate of both sides, 57:38.559 --> 57:39.919 they will still match. 57:39.920 --> 57:41.480 Do you understand that? 57:41.480 --> 57:44.750 If two things are equal, their complex conjugates are 57:44.753 --> 57:47.323 equal, because this thing is the real 57:47.318 --> 57:51.448 imaginary that balances the real imaginary on the other side. 57:51.449 --> 57:54.339 Complex conjugate just reverses the imaginary terms, 57:54.335 --> 57:57.275 so they will still match when you flip both sides. 57:57.280 --> 58:05.220 So it means if you take L dI* of dt R times 58:05.221 --> 58:13.871 I* 1/C integral I* of t' dt' t = V* of 58:13.871 --> 58:15.291 t. 58:15.289 --> 58:18.129 So star is the rule by which if V had a real and 58:18.125 --> 58:20.535 imaginary part, you flip the imaginary part. 58:20.539 --> 58:25.499 So that solution for V of t implies a second 58:25.501 --> 58:26.461 equation. 58:26.460 --> 58:29.090 In other words, if the voltage V drives 58:29.090 --> 58:31.840 a current I, the conjugate of the voltage 58:31.840 --> 58:34.180 drives the conjugate of the current. 58:34.179 --> 58:35.929 That simply follows from the equation. 58:35.929 --> 58:38.799 It comes from the fact that when I conjugated it, 58:38.804 --> 58:40.964 L, R and C are real. 58:40.960 --> 58:42.530 Do you understand that? 58:42.530 --> 58:45.680 Suppose L were imaginary or had imaginary part, 58:45.682 --> 58:48.482 then I must also put a conjugate on L. 58:48.480 --> 58:50.570 Then I* obeys a different equation, 58:50.574 --> 58:53.644 because this had L in it, this had L* in it. 58:53.639 --> 58:57.339 But if everything is real, I* obeys the same 58:57.338 --> 59:00.888 equation when you have a V* driving it. 59:00.889 --> 59:02.509 I'm almost done now. 59:02.510 --> 59:06.070 I want you to add the left hand side to the left hand side and 59:06.067 --> 59:09.797 the right hand side to the right hand side, and what do I get? 59:09.800 --> 59:13.870 Well, if I write it here, you may not be able to see it. 59:13.869 --> 59:15.469 Let me put it here then. 59:15.469 --> 59:21.959 If you add these two, you'd get L d/dt of 59:21.956 --> 59:29.546 I I* R times I I* 1/C 59:29.547 --> 59:37.407 integral of I I* dt = V V*. This again comes 59:37.414 --> 59:44.044 because the equation is a linear equation. 59:44.039 --> 59:46.959 You can always add the left hand side and the right hand 59:46.956 --> 59:49.866 side to get another problem where the driving voltage is 59:49.871 --> 59:52.631 V V* and the driving current is I I*, 59:52.630 --> 59:58.760 etc. 59:58.760 --> 1:00:05.800 But I I* is 2 times the real part of I and V 1:00:05.795 --> 1:00:11.185 V* is 2 times the real part of V. 1:00:11.190 --> 1:00:17.640 Therefore this implies that L d/dt of the real part 1:00:17.644 --> 1:00:24.554 of I R plus the real part of I 1/C 1:00:24.552 --> 1:00:31.462 times the integral of the real part of I = real part of 1:00:31.460 --> 1:00:33.160 V. 1:00:33.159 --> 1:00:36.289 So let me say in words what the equation means. 1:00:36.289 --> 1:00:40.659 If you by luck solve an equation with a complex 1:00:40.655 --> 1:00:43.455 potential V, and you get a current, 1:00:43.461 --> 1:00:47.301 then the answer to the problem where the potential is only the 1:00:47.297 --> 1:00:50.437 real part of the actual potential you applied, 1:00:50.440 --> 1:00:53.630 the current will be the real part of the answer you got. 1:00:53.630 --> 1:00:55.270 That's all I want you to know. 1:00:55.268 --> 1:00:58.568 If you solve a complex problem with a driving voltage which is 1:00:58.565 --> 1:01:00.285 complex-- which is just a mathematical 1:01:00.293 --> 1:01:02.263 fiction, in real life you don't have 1:01:02.259 --> 1:01:05.609 that--then the current that you get will also be complex. 1:01:05.610 --> 1:01:09.000 But its real part will be due to the real part of your 1:01:09.000 --> 1:01:09.640 voltage. 1:01:09.639 --> 1:01:12.119 And you can also show the imaginary part of the current 1:01:12.123 --> 1:01:14.473 will be due to the imaginary part of the voltage. 1:01:14.469 --> 1:01:17.709 This is just the principle of superposition. 1:01:17.710 --> 1:01:19.260 So here is the trick we are going to use. 1:01:19.260 --> 1:01:22.480 We are going to go back to original equation, 1:01:22.480 --> 1:01:29.470 LdI/dt RI 1/C integral I dt = V_0cos 1:01:29.469 --> 1:01:33.849 ωt, and we're going to solve a new 1:01:33.849 --> 1:01:34.489 problem. 1:01:34.489 --> 1:01:39.029 We're going to solve a problem where this is V_0 1:01:39.027 --> 1:01:40.537 e^(iωt). 1:01:40.539 --> 1:01:42.829 And the current for that, I'm going to call some 1:01:42.831 --> 1:01:43.711 I˜. 1:01:43.710 --> 1:01:49.800 It's a complex current, because the driving voltage is 1:01:49.800 --> 1:01:52.100 a complex voltage. 1:01:52.099 --> 1:01:54.599 So think of Ve^(iωt) as the 1:01:54.597 --> 1:01:56.837 V in the previous problem. 1:01:56.840 --> 1:02:00.280 The actual problem I was given had V_0cos 1:02:00.282 --> 1:02:02.552 ωt, but you know that 1:02:02.554 --> 1:02:04.604 V_0 e^(iωt) is 1:02:04.597 --> 1:02:07.257 V_0 cosωt I 1:02:07.264 --> 1:02:10.164 times V_0 sinωt. 1:02:10.159 --> 1:02:13.559 Consequently, this potential has a real part 1:02:13.559 --> 1:02:16.539 which is a cosine, an imaginary part which is a 1:02:16.536 --> 1:02:19.306 sine, and the current will have a real and imaginary part. 1:02:19.309 --> 1:02:22.739 The answer to the original question is the real part of the 1:02:22.740 --> 1:02:24.280 answer to this question. 1:02:24.280 --> 1:02:25.760 This is what you should think about. 1:02:25.760 --> 1:02:28.850 This is what you should understand. 1:02:28.849 --> 1:02:33.589 So I've manufactured a problem with an unphysical complex 1:02:33.588 --> 1:02:35.868 voltage, and all I know is that at the 1:02:35.869 --> 1:02:38.529 end of the day, if I take the real part of the 1:02:38.534 --> 1:02:40.564 current, I'll get the answer to the real 1:02:40.556 --> 1:02:42.926 part of the potential, which happens to be the actual 1:02:42.931 --> 1:02:44.571 potential, V_0 1:02:44.574 --> 1:02:46.264 cosωt. 1:02:46.260 --> 1:02:48.290 So why would I do all this? 1:02:48.289 --> 1:02:50.709 Why would I take a problem that's bad enough with the 1:02:50.713 --> 1:02:52.953 cosine and turn it into a complex exponential? 1:02:52.949 --> 1:02:56.679 I think you can sort of guess what the reason is. 1:02:56.679 --> 1:03:00.139 The reason is that it may be complex, but it's still an 1:03:00.141 --> 1:03:01.041 exponential. 1:03:01.039 --> 1:03:04.049 That means its derivative is going to look just the same. 1:03:04.050 --> 1:03:07.920 That means I can make a guess that the current I˜ 1:03:07.920 --> 1:03:11.590 is proportional to e^(iωt). So you can 1:03:11.586 --> 1:03:15.246 make the assumption that I˜ of t is 1:03:15.253 --> 1:03:19.063 some constant I˜ times e^(iωt). 1:03:19.056 --> 1:03:23.266 Make that assumption and put it into this equation and see what 1:03:23.266 --> 1:03:24.486 you get. 1:03:24.489 --> 1:03:30.719 You will find you will get IωL times 1:03:30.722 --> 1:03:37.812 I˜ R times I˜ 1/IωC 1:03:37.809 --> 1:03:42.699 times I˜ times e^(Iωt) is 1:03:42.697 --> 1:03:46.727 V_0e^(iωt). 1:03:46.730 --> 1:03:51.430 This is because if you assume the current is an exponential, 1:03:51.429 --> 1:03:55.259 every derivative = an iω and every integral 1:03:55.264 --> 1:03:59.384 = 1/iω, and multiplying is just multiplying. 1:03:59.380 --> 1:04:02.170 So d/dt has got to be replaced by ω, 1:04:02.170 --> 1:04:05.730 and you get this equation for I˜. That means 1:04:05.731 --> 1:04:10.891 you will satisfy the equation, provided I˜ times 1:04:10.894 --> 1:04:13.454 all of this = V . 1:04:13.449 --> 1:04:16.359 And I'm going to write this as Z times I˜ 1:04:16.362 --> 1:04:18.972 = V where Z is the name for all the guys 1:04:18.965 --> 1:04:20.625 multiplying I˜. 1:04:20.630 --> 1:04:21.780 So I'll start with that here. 1:04:21.780 --> 1:04:39.420 1:04:39.420 --> 1:04:43.470 Okay. 1:04:43.469 --> 1:04:50.499 So I wrote iωL R 1 over iωC 1:04:50.501 --> 1:04:56.261 I˜ e^(iωt) is V_0 1:04:56.255 --> 1:04:59.065 e^(iωt). 1:04:59.070 --> 1:05:02.960 Everybody get that? 1:05:02.960 --> 1:05:07.360 That's because differentiating on an exponential is just 1:05:07.360 --> 1:05:09.760 multiplying by the exponent. 1:05:09.760 --> 1:05:12.220 So this is a complex number. 1:05:12.219 --> 1:05:17.199 R iωL − i/ωC. 1:05:17.199 --> 1:05:20.969 I want you to know that i in the bottom is like -i 1:05:20.965 --> 1:05:22.045 in the top. 1:05:22.050 --> 1:05:26.650 And this is called the impedance Z, 1:05:26.653 --> 1:05:29.353 it's a complex number. 1:05:29.349 --> 1:05:32.069 You can visualize the complex number as follows: 1:05:32.074 --> 1:05:34.514 it's got a real part, which is R. 1:05:34.510 --> 1:05:38.270 It's got an imaginary part, which is ωL − 1:05:38.268 --> 1:05:41.578 1/ωC, and the impedance Z is a 1:05:41.577 --> 1:05:44.857 complex number with some modulus and some phase. 1:05:44.860 --> 1:05:47.430 If you know the real and imaginary parts, 1:05:47.427 --> 1:05:50.957 you can construct the modulus in the phase like that. 1:05:50.960 --> 1:05:53.860 If you want, it's the modulus of Z. 1:05:53.860 --> 1:06:00.980 Therefore canceling this, I find I˜ 1:06:00.983 --> 1:06:04.963 = V_0/Z. 1:06:04.960 --> 1:06:10.060 That becomes V_0 over the modulus of Z times 1:06:10.063 --> 1:06:14.693 e^(i)^(Φ), where Φ is this angle 1:06:14.688 --> 1:06:19.108 defined by tanΦ = ωL − 1:06:19.112 --> 1:06:22.342 1/ωC divided by R. 1:06:22.340 --> 1:06:24.660 So don't worry too much about writing all of this down, 1:06:24.655 --> 1:06:26.795 because this you're going to find in every book. 1:06:26.800 --> 1:06:31.110 There's nothing novel or different. 1:06:31.110 --> 1:06:34.300 So this is the formula for I twiddle. 1:06:34.300 --> 1:06:38.540 Now I itself, you remember, 1:06:38.539 --> 1:06:41.569 = I˜ of 0. 1:06:41.570 --> 1:06:44.070 I'm sorry, this is I˜ of 0. 1:06:44.070 --> 1:06:48.460 Did I define it that way? 1:06:48.460 --> 1:06:49.850 Yeah, I should write it--I'm sorry. 1:06:49.849 --> 1:06:52.819 I should write it as I˜ of 0. 1:06:52.820 --> 1:06:59.000 There's a subscript 0 there, because the full I twiddle has 1:06:58.996 --> 1:07:02.826 got an e^(iωt) in it. 1:07:02.829 --> 1:07:10.329 That looks like V_0 over mod 1:07:10.331 --> 1:07:19.881 Z times e^(iωt - )^(Φ). 1:07:19.880 --> 1:07:23.330 But this is not the physical current, because it's the result 1:07:23.327 --> 1:07:24.647 of a complex voltage. 1:07:24.650 --> 1:07:26.770 The physical current is a real part of this, 1:07:26.768 --> 1:07:30.428 without the twiddle, that we write as real part of 1:07:30.427 --> 1:07:34.007 I˜. Where do I get the real part? 1:07:34.010 --> 1:07:36.530 V_0 and absolute value of Z are 1:07:36.530 --> 1:07:37.090 both real. 1:07:37.090 --> 1:07:41.300 Real part of e to the i something is cosine of 1:07:41.302 --> 1:07:43.732 something and that's our answer. 1:07:43.730 --> 1:07:51.210 That's our answer, the final answer you get. 1:07:51.210 --> 1:07:54.300 This tells you that the current has a magnitude which is 1:07:54.304 --> 1:07:57.684 V_0 over the absolute value of Z, 1:07:57.679 --> 1:08:00.749 and it's got a phase Φ by which it is 1:08:00.753 --> 1:08:02.653 behind the driving voltage. 1:08:02.650 --> 1:08:05.070 The magnitude of Z for as in any complex number, 1:08:05.070 --> 1:08:07.090 if someone says, "What's the magnitude of 1:08:07.088 --> 1:08:07.938 Z?" 1:08:07.940 --> 1:08:18.990 it is simply square root of real part squared imaginary part 1:08:18.987 --> 1:08:20.857 squared. 1:08:20.859 --> 1:08:26.369 So I want you to think about the magic of complex numbers. 1:08:26.368 --> 1:08:30.608 Your final answer has a current which is a cosωt - 1:08:30.613 --> 1:08:31.643 Φ. 1:08:31.640 --> 1:08:34.720 Your driving voltage is a cosωt. 1:08:34.720 --> 1:08:38.340 There is no way you can divide the voltage by anything to get 1:08:38.336 --> 1:08:39.236 that current. 1:08:39.238 --> 1:08:44.618 But in the complex language, the complex current is the 1:08:44.623 --> 1:08:49.713 complex voltage divided by the complex impedance. 1:08:49.710 --> 1:08:50.710 How does the happen? 1:08:50.710 --> 1:08:54.550 That happens because impendence Z has got a magnitude and 1:08:54.548 --> 1:08:56.768 a phase, therefore if you had a voltage 1:08:56.765 --> 1:08:59.235 V_0 e^(iωt) and you divide 1:08:59.240 --> 1:09:01.820 by mod Z e^(i) ^(Φ) , that 1:09:01.819 --> 1:09:04.499 becomes e^(-i) ^(Φ) upstairs. 1:09:04.500 --> 1:09:08.770 You're able to change the magnitude and you're able to 1:09:08.770 --> 1:09:11.270 change the phase by dividing. 1:09:11.270 --> 1:09:14.380 So if you want to take the voltage and you want to rescale 1:09:14.380 --> 1:09:16.710 it and shift its phase, all in one shot, 1:09:16.711 --> 1:09:19.561 you can do it if you divide by a complex number, 1:09:19.560 --> 1:09:23.970 because a complex number will rescale it and also shift its 1:09:23.970 --> 1:09:24.580 phase. 1:09:24.579 --> 1:09:28.079 And then you take the real part. 1:09:28.078 --> 1:09:30.198 In other words, cosωt divided by 1:09:30.203 --> 1:09:31.993 nothing will give you cosωt - 1:09:31.988 --> 1:09:35.098 Φ, but e^(iωt), 1:09:35.100 --> 1:09:37.010 when divided by e^(i)^(Φ), 1:09:37.010 --> 1:09:38.670 will in fact give you e^(iωt - 1:09:38.670 --> 1:09:39.720 i)^(Φ). 1:09:39.720 --> 1:09:43.060 So that phase shift we can produce only by dividing with a 1:09:43.061 --> 1:09:44.061 complex number. 1:09:44.060 --> 1:09:47.630 So I'll come back and explain to you a little bit more on how 1:09:47.627 --> 1:09:50.897 you do circuit problems using the complex impedance, 1:09:50.899 --> 1:09:53.889 but I think it will be helpful for those of you who don't know 1:09:53.887 --> 1:09:56.237 complex numbers to get use to this part of it. 1:09:56.239 --> 1:10:01.999