1 00:00:00 --> 00:00:00,229 2 00:00:00,229 --> 00:00:03,517 Class average on the exam was fifty-five. 3 00:00:03,517 --> 00:00:06,805 Three homework problems were on the exam. 4 00:00:06,805 --> 00:00:10,751 I'm a little puzzled by the fringe field problem. 5 00:00:10,751 --> 00:00:14,779 That was the one homework problem that was graded. 6 00:00:14,779 --> 00:00:19,218 It was graded on your homework, and I went back to your 7 00:00:19,218 --> 00:00:22,835 homework scores and they were ninety percent. 8 00:00:22,835 --> 00:00:26,288 You all did extremely well on that problem. 9 00:00:26,288 --> 00:00:29,74 But on the exam, it was only forty percent. 10 00:00:29,74 --> 00:00:34,368 And so perhaps this is telling you something. 11 00:00:34,368 --> 00:00:36,889 And maybe it's telling me something too. 12 00:00:36,889 --> 00:00:39,862 If you turn in a correct solution for homework, 13 00:00:39,862 --> 00:00:43,675 it's not very useful if you do not understand what you wrote 14 00:00:43,675 --> 00:00:45,162 down. The understanding, 15 00:00:45,162 --> 00:00:48,264 of course, is what matters, not that you somehow, 16 00:00:48,264 --> 00:00:50,978 one way or another, get a correct solution. 17 00:00:50,978 --> 00:00:54,791 If I had to judge you only on the basis of exam one and two, 18 00:00:54,791 --> 00:00:57,829 forgetting the quizzes, forgetting the homework, 19 00:00:57,829 --> 00:01:01,254 then those with eighty or lower would fail the course, 20 00:01:01,254 --> 00:01:03,71 as of now. But of course those who have 21 00:01:03,71 --> 00:01:07,683 between eighty and ninety are by no 22 00:01:07,683 --> 00:01:11,102 means home free. They are still in the danger 23 00:01:11,102 --> 00:01:13,822 zone. And anyone who has ninety-five 24 00:01:13,822 --> 00:01:17,629 or even a hundred, I cannot guarantee you that you 25 00:01:17,629 --> 00:01:20,194 will pass the course. The depends, 26 00:01:20,194 --> 00:01:23,535 of course, on how you will do in the future, 27 00:01:23,535 --> 00:01:25,944 the third exam and on the final. 28 00:01:25,944 --> 00:01:29,363 Today I want to discuss with you RC circuits. 29 00:01:29,363 --> 00:01:31,85 We already discussed RL circuits. 30 00:01:31,85 --> 00:01:36,745 Now we get RC circuits. I have here a battery 31 00:01:36,745 --> 00:01:42,469 with an EMF V zero. And I have here a switch 32 00:01:42,469 --> 00:01:48,592 connecting to a capacitor. And here a resistor. 33 00:01:48,592 --> 00:01:55,38 And I close the loop. When I have the switch in this 34 00:01:55,38 --> 00:02:01,369 position, the capacitor is going to charge up. 35 00:02:01,369 --> 00:02:08,157 I get a current I going in this direction. 36 00:02:08,157 --> 00:02:15,496 If I call this point A here and this point P and this point S to 37 00:02:15,496 --> 00:02:20,622 make sure that we are on the same wavelength, 38 00:02:20,622 --> 00:02:25,748 I will call the potential over the capacitor, 39 00:02:25,748 --> 00:02:32,97 I will call that V A minus V P. The potential over the resistor 40 00:02:32,97 --> 00:02:40,891 is of course always I times R, is then V of P minus V of S. 41 00:02:40,891 --> 00:02:45,292 The question now is what is the potential over the capacitor 42 00:02:45,292 --> 00:02:49,842 doing as a function of time and what is the current doing as a 43 00:02:49,842 --> 00:02:53,274 function of time as I charge up this capacitor. 44 00:02:53,274 --> 00:02:57,451 And your intuition will help you a great deal without any 45 00:02:57,451 --> 00:03:01,702 fancy differential equations. It is clear that at T equals 46 00:03:01,702 --> 00:03:04,238 zero, the capacitor is not charged. 47 00:03:04,238 --> 00:03:06,849 There is no charge on the capacitor. 48 00:03:06,849 --> 00:03:10,206 And it will take time to charge the capacitor. 49 00:03:10,206 --> 00:03:15,214 So at T equals zero, you expect that the potential 50 00:03:15,214 --> 00:03:19,556 over the capacitor is zero. If you make T a little larger 51 00:03:19,556 --> 00:03:23,588 than zero, you are going to charge up this capacitor, 52 00:03:23,588 --> 00:03:27,465 and so the potential over the capacitor will go up, 53 00:03:27,465 --> 00:03:30,411 and therefore the current will go down. 54 00:03:30,411 --> 00:03:35,141 And if you wait long enough -- we call that infinitely long -- 55 00:03:35,141 --> 00:03:38,243 then the capacitor will be fully charged. 56 00:03:38,243 --> 00:03:42,12 It will have the potential V zero of 57 00:03:42,12 --> 00:03:46,22 the battery, and then the current has become 58 00:03:46,22 --> 00:03:49,462 zero. No current is flowing anymore 59 00:03:49,462 --> 00:03:54,326 if the battery -- if the capacitor is fully charged. 60 00:03:54,326 --> 00:03:58,427 And so this capacitor is going to charge up, 61 00:03:58,427 --> 00:04:03,004 this becomes positive, and this becomes negative. 62 00:04:03,004 --> 00:04:08,249 And so you can construct a, a plot whereby you plot as a 63 00:04:08,249 --> 00:04:13,589 function of time here the potential over the capacitor -- 64 00:04:13,589 --> 00:04:19,532 we haven't used any differential equations yet. 65 00:04:19,532 --> 00:04:26,002 You know that if you wait long enough you will reach that value 66 00:04:26,002 --> 00:04:29,968 V zero. And it's going to build up like 67 00:04:29,968 --> 00:04:33,829 this, asymptotically reach that value. 68 00:04:33,829 --> 00:04:39,255 And if C is very large, then the current will be more 69 00:04:39,255 --> 00:04:44,89 like this, and if C is very small, then it will go much 70 00:04:44,89 --> 00:04:47,604 faster, of course, small C. 71 00:04:47,604 --> 00:04:52,613 The current as a function of time. 72 00:04:52,613 --> 00:04:56,403 In the beginning, the current will be high, 73 00:04:56,403 --> 00:05:01,998 but ultimately the current will die down, when the capacitor is 74 00:05:01,998 --> 00:05:05,788 fully charged. So you expect something like 75 00:05:05,788 --> 00:05:08,947 this. So this you can do without any 76 00:05:08,947 --> 00:05:13,459 differential equations. Let's now do it the correct 77 00:05:13,459 --> 00:05:16,437 way. The closed loop integral of E 78 00:05:16,437 --> 00:05:20,498 dot DL happens to be zero, which will make Mr. 79 00:05:20,498 --> 00:05:23,503 Kirchoff very happy. 80 00:05:23,503 --> 00:05:28,288 The closed loop integral of E dot DL in this circuit is zero. 81 00:05:28,288 --> 00:05:30,442 Mr. Faraday is happy and Mr. 82 00:05:30,442 --> 00:05:33,871 Kirchoff is happy. There is no magnetic flux 83 00:05:33,871 --> 00:05:37,301 change here; we don't have self-inductances. 84 00:05:37,301 --> 00:05:42,086 The electric field inside the capacitor is in this direction, 85 00:05:42,086 --> 00:05:45,596 from plus to minus. The electric field in the 86 00:05:45,596 --> 00:05:50,381 resistor is in this direction, the current is flowing in this 87 00:05:50,381 --> 00:05:53,891 direction. And this battery, 88 00:05:53,891 --> 00:05:58,817 which has this as the positive side and this as the negative 89 00:05:58,817 --> 00:06:03,242 side, inside the battery the electric field is in this 90 00:06:03,242 --> 00:06:06,582 direction. So if I start at point A and I 91 00:06:06,582 --> 00:06:10,841 go around the circuit, then E and DL are in the same 92 00:06:10,841 --> 00:06:15,433 direction if I go from A to P, and so I get plus V of C, 93 00:06:15,433 --> 00:06:18,022 that's the E dot DL from A to P. 94 00:06:18,022 --> 00:06:24,367 Then I go through the resistor. Again, E and DL are in the same 95 00:06:24,367 --> 00:06:27,309 direction, so I get plus I times R. 96 00:06:27,309 --> 00:06:29,991 Then I come through the battery. 97 00:06:29,991 --> 00:06:34,836 Now the electric field is opposing the direction in which 98 00:06:34,836 --> 00:06:38,73 I go, so I get minus V zero. And that is zero. 99 00:06:38,73 --> 00:06:41,671 So that's my differential equation. 100 00:06:41,671 --> 00:06:46,257 And I can write it differently, because I know that I, 101 00:06:46,257 --> 00:06:49,977 the current, is dQ dT, Q being the charge on 102 00:06:49,977 --> 00:06:55,081 the capacitor. And it is only if that number 103 00:06:55,081 --> 00:06:59,797 changes, if the capacitor is either charging up or 104 00:06:59,797 --> 00:07:03,55 discharging, is there a current flowing. 105 00:07:03,55 --> 00:07:08,554 And in addition I know that V of C is Q divided by C. 106 00:07:08,554 --> 00:07:12,019 That's the definition of capacitance. 107 00:07:12,019 --> 00:07:17,6 And so I can write down for this V of C, I can write down Q 108 00:07:17,6 --> 00:07:21,642 divided by C. For I I can write down dQ dT. 109 00:07:21,642 --> 00:07:26,165 So I get R times dQ dT minus V zero equals zero. 110 00:07:26,165 --> 00:07:31,503 And this is a differential equation in Q. 111 00:07:31,503 --> 00:07:36,179 We have seen an identical differential equation. 112 00:07:36,179 --> 00:07:38,964 It was not in Q, it was in I, 113 00:07:38,964 --> 00:07:43,938 but it had of course a completely similar solution. 114 00:07:43,938 --> 00:07:48,714 And the solution to this differential equation is 115 00:07:48,714 --> 00:07:53,688 actually quite simple. I will put it on this board. 116 00:07:53,688 --> 00:08:00,254 Q -- so this is going to be Q as a function of time -- 117 00:08:00,254 --> 00:08:06,569 is V zero C times one minus E to the minus T over R C. 118 00:08:06,569 --> 00:08:12,288 If I take the derivative of this Q then I have I, 119 00:08:12,288 --> 00:08:17,531 because I is dQ dT. So I, which is dQ dT then 120 00:08:17,531 --> 00:08:22,536 becomes V zero times C, I get a minus sign, 121 00:08:22,536 --> 00:08:28,375 I get another minus sign, then I get one over R C, 122 00:08:28,375 --> 00:08:34,711 and then I get E to the minus T divided by R C. 123 00:08:34,711 --> 00:08:40,212 So the two minus signs eat each other up and I lose one C here. 124 00:08:40,212 --> 00:08:45,537 And so I get that the current as a function of time is V zero 125 00:08:45,537 --> 00:08:49,974 divided by R times E to the minus T divided by R C. 126 00:08:49,974 --> 00:08:52,902 So that's the curve at the bottom. 127 00:08:52,902 --> 00:08:57,783 And the potential over the capacitor is now very simple, 128 00:08:57,783 --> 00:09:04,083 because the potential over the capacitor is Q divided by C. 129 00:09:04,083 --> 00:09:09,914 And I already have Q here, so I simply have to divide this 130 00:09:09,914 --> 00:09:13,698 C out. So I get V zero times one minus 131 00:09:13,698 --> 00:09:19,529 E to the minus T over R C. And so that is the upper curve. 132 00:09:19,529 --> 00:09:25,87 And so we can now make a small table and we can look at various 133 00:09:25,87 --> 00:09:30,269 values for T. Maybe I should do that here on 134 00:09:30,269 --> 00:09:35,996 the blackboard, because I don't want to erase 135 00:09:35,996 --> 00:09:39,143 anything yet. So we have T here, 136 00:09:39,143 --> 00:09:44,117 we have I, and we have V of C. And when T is zero, 137 00:09:44,117 --> 00:09:48,583 you go to your equation of I, so this is one. 138 00:09:48,583 --> 00:09:52,744 So you have a current V zero divided by R. 139 00:09:52,744 --> 00:09:55,891 And you V C -- V C of C is zero. 140 00:09:55,891 --> 00:09:58,936 You can see that. If T is zero, 141 00:09:58,936 --> 00:10:04,214 you get one minus one -- oh sorry, I have to go here. 142 00:10:04,214 --> 00:10:10,031 You get one minus one, so you see indeed that the 143 00:10:10,031 --> 00:10:13,52 potential over the capacitance is still zero. 144 00:10:13,52 --> 00:10:17,564 If you wait long enough, then the current must go to 145 00:10:17,564 --> 00:10:20,339 zero. That exponential function goes 146 00:10:20,339 --> 00:10:25,017 to zero if you wait long enough. And your potential over the 147 00:10:25,017 --> 00:10:29,696 capacitance then reaches V zero, which is exactly consistent 148 00:10:29,696 --> 00:10:32,867 with our solution. If you wait a time RC, 149 00:10:32,867 --> 00:10:36,594 that's called the time constant of this circuit. 150 00:10:36,594 --> 00:10:42,63 In the case of the current, it's also called the decay time 151 00:10:42,63 --> 00:10:46,442 of the circuit. Then of course your current is 152 00:10:46,442 --> 00:10:49,492 one over E times V zero divided by R. 153 00:10:49,492 --> 00:10:53,304 And one over E is a roughly point three seven, 154 00:10:53,304 --> 00:10:57,286 oh point three seven. So your current is down to 155 00:10:57,286 --> 00:11:01,692 thirty-seven percent of what it was at the beginning. 156 00:11:01,692 --> 00:11:05,335 And after a time R C, the potential over the 157 00:11:05,335 --> 00:11:10,503 capacitor is one minus one over E times V zero, 158 00:11:10,503 --> 00:11:16,667 and that is then about oh point seven three -- uh, 159 00:11:16,667 --> 00:11:22,957 six three, oh point six three, sixty-three percent. 160 00:11:22,957 --> 00:11:28,366 In other words, if I go back here to my plot 161 00:11:28,366 --> 00:11:35,411 and if I draw a line at time R C, then this value here is 162 00:11:35,411 --> 00:11:42,078 thirty-seven percent of the maximum and this 163 00:11:42,078 --> 00:11:46,713 value here is then sixty-three percent of the maximum. 164 00:11:46,713 --> 00:11:50,56 So it's in -- the solution is rather obvious, 165 00:11:50,56 --> 00:11:54,408 very intuitive. And the R C times can vary an 166 00:11:54,408 --> 00:11:57,468 enormous amount, as you can imagine, 167 00:11:57,468 --> 00:12:00,704 depending upon the values for R and C. 168 00:12:00,704 --> 00:12:05,95 If we have here an R and a C and we want to know what the R C 169 00:12:05,95 --> 00:12:11,197 time is, convince yourself that the product of R and C indeed 170 00:12:11,197 --> 00:12:14,254 has units of seconds. 171 00:12:14,254 --> 00:12:19,052 Remember we had L over R before, which had also units of 172 00:12:19,052 --> 00:12:22,368 seconds. R C also has units of seconds. 173 00:12:22,368 --> 00:12:26,904 If you have R equals one ohm and C is one microfarad, 174 00:12:26,904 --> 00:12:30,307 then the R C time is only a microsecond. 175 00:12:30,307 --> 00:12:34,931 But if you have R equals hundred megaohms and you have 176 00:12:34,931 --> 00:12:39,119 this one millifarad, then this is ten to the five 177 00:12:39,119 --> 00:12:42,172 seconds, which is longer than a day. 178 00:12:42,172 --> 00:12:46,447 And that would mean that it would 179 00:12:46,447 --> 00:12:51,901 take you even three days to reach ninety-five percent of V 180 00:12:51,901 --> 00:12:54,006 zero. After three days, 181 00:12:54,006 --> 00:12:58,885 you would still have only ninety-five percent of the 182 00:12:58,885 --> 00:13:04,338 potential difference of the capacitor of the maximum value 183 00:13:04,338 --> 00:13:08,931 that you can get. Now what I want to do is I make 184 00:13:08,931 --> 00:13:13,14 a change here. I have here a conducting wire. 185 00:13:13,14 --> 00:13:18,115 I call this position one of the switch. 186 00:13:18,115 --> 00:13:21,149 And I'm going to put the switch in this position, 187 00:13:21,149 --> 00:13:23,804 position two. And I can do that without any 188 00:13:23,804 --> 00:13:26,079 danger. Remember, I waited until this 189 00:13:26,079 --> 00:13:29,682 capacitor was fully charged. There was no current running. 190 00:13:29,682 --> 00:13:33,285 And so when no current was running, I can quietly take the 191 00:13:33,285 --> 00:13:35,434 switch and put it in this position. 192 00:13:35,434 --> 00:13:38,973 And what is going to happen now is of course this side is 193 00:13:38,973 --> 00:13:42,07 positively charged and this is negatively charged, 194 00:13:42,07 --> 00:13:45,673 so now you're going to get a current which is going to run 195 00:13:45,673 --> 00:13:48,138 counterclockwise, in opposite direction. 196 00:13:48,138 --> 00:13:51,615 And what is going to happen is the 197 00:13:51,615 --> 00:13:56,341 capacitor is discharging now. And the resistor will dissipate 198 00:13:56,341 --> 00:13:59,098 the energy that is in the capacitor. 199 00:13:59,098 --> 00:14:04,06 The one half C V squared energy stored in the capacitor is going 200 00:14:04,06 --> 00:14:08,471 to be dissipated in the resistor in terms of I squared R, 201 00:14:08,471 --> 00:14:12,016 in terms of heat. And if you wait long enough, 202 00:14:12,016 --> 00:14:16,664 the current will become zero. So it should be obvious what's 203 00:14:16,664 --> 00:14:20,366 going to happen. If I return to this curve here, 204 00:14:20,366 --> 00:14:24,226 if I redefine my time equals zero, 205 00:14:24,226 --> 00:14:28,704 and if this is the moment that I put the switch in position 206 00:14:28,704 --> 00:14:32,719 two, then I expect that the capacitor will discharge. 207 00:14:32,719 --> 00:14:37,042 You get a curve like this. And I expect that the current, 208 00:14:37,042 --> 00:14:41,752 which now becomes negative -- it reverses direction and I call 209 00:14:41,752 --> 00:14:44,995 that negative. And so the current will come 210 00:14:44,995 --> 00:14:47,929 like this. And if you wait long enough, 211 00:14:47,929 --> 00:14:51,79 of course, the current will aga- again become zero. 212 00:14:51,79 --> 00:14:54,338 You have discharged the capacitor. 213 00:14:54,338 --> 00:14:58,352 So if you want the formal solution, 214 00:14:58,352 --> 00:15:01,693 you have to go back to the differential equation. 215 00:15:01,693 --> 00:15:05,172 And you take this term out, because it's not there. 216 00:15:05,172 --> 00:15:09,208 And now you have to solve this differential equation again, 217 00:15:09,208 --> 00:15:13,244 which is now utterly trivial. And I would like you to solve 218 00:15:13,244 --> 00:15:17,002 that differential equation. You couldn't have an easier 219 00:15:17,002 --> 00:15:19,438 one. I will give you the solution to 220 00:15:19,438 --> 00:15:23,195 I as a function of time, and you then will come up with 221 00:15:23,195 --> 00:15:25,701 this part. I as a function of time is 222 00:15:25,701 --> 00:15:29,946 exactly the same as this except with a minus 223 00:15:29,946 --> 00:15:34,273 sign, provided that I call this T equals zero. 224 00:15:34,273 --> 00:15:39,851 So I redefine the zero time. And so I as a function of time 225 00:15:39,851 --> 00:15:45,717 is the equation that you have here, but now with a minus sign. 226 00:15:45,717 --> 00:15:49,372 So you get an exponential change again, 227 00:15:49,372 --> 00:15:52,449 but the current has flipped over. 228 00:15:52,449 --> 00:15:57,931 I can demonstrate this to you. I have a electronic switch, 229 00:15:57,931 --> 00:16:05,572 so I go between one and two -- every four milliseconds I throw 230 00:16:05,572 --> 00:16:09,303 the switch. And so what I have is, 231 00:16:09,303 --> 00:16:14,843 as a function of time, what we call a square wave. 232 00:16:14,843 --> 00:16:19,479 So this is my battery. And this time here, 233 00:16:19,479 --> 00:16:24,002 from here to here, is eight milliseconds. 234 00:16:24,002 --> 00:16:28,525 This is time. And so this is that value V 235 00:16:28,525 --> 00:16:32,03 zero. And the values that I have 236 00:16:32,03 --> 00:16:35,395 chosen for R and C, 237 00:16:35,395 --> 00:16:40,179 I will give them to you. The value for V zero here is 238 00:16:40,179 --> 00:16:43,675 one volt, and here of course it's zero. 239 00:16:43,675 --> 00:16:48,275 The value that I have chosen for R is six kiloohms. 240 00:16:48,275 --> 00:16:52,783 And the value I have for C I think is oh point one 241 00:16:52,783 --> 00:16:55,911 microfarad. Yes, that's what it is. 242 00:16:55,911 --> 00:17:00,419 And there's a reason why I've chosen these values. 243 00:17:00,419 --> 00:17:04,742 This is oh point one microfarads. 244 00:17:04,742 --> 00:17:11,246 And so the R C time is six times ten to the minus four 245 00:17:11,246 --> 00:17:14,682 seconds. So this is point six 246 00:17:14,682 --> 00:17:19,959 milliseconds, which is substantially smaller 247 00:17:19,959 --> 00:17:24,868 than four. And so you can expect that the 248 00:17:24,868 --> 00:17:31,126 capacitor becomes almost fully charged in these four 249 00:17:31,126 --> 00:17:36,597 milliseconds. That's why I chose the R C time 250 00:17:36,597 --> 00:17:40,181 substantially smaller than four milliseconds. 251 00:17:40,181 --> 00:17:43,766 And my setup is such that I can show you the, 252 00:17:43,766 --> 00:17:47,024 um, the input, the driving voltage of the 253 00:17:47,024 --> 00:17:49,305 battery. I can show you this. 254 00:17:49,305 --> 00:17:54,111 I can show you then how the capacitor is charged and how the 255 00:17:54,111 --> 00:17:58,184 capacitor is discharged with the time constant R C. 256 00:17:58,184 --> 00:18:02,095 And also how the current goes through the system. 257 00:18:02,095 --> 00:18:04,783 So I can show you also this curve. 258 00:18:04,783 --> 00:18:09,101 And then I can change the capacitor 259 00:18:09,101 --> 00:18:14,498 so that the R C time changes. I can also change the 260 00:18:14,498 --> 00:18:18,492 resistance. So let me change first the 261 00:18:18,492 --> 00:18:23,997 lights, so that you get high quality for your money. 262 00:18:23,997 --> 00:18:29,826 You already see there the switching voltage between one 263 00:18:29,826 --> 00:18:34,468 volt and zero. And now I'm going to show you 264 00:18:34,468 --> 00:18:40,944 the voltage over the capacitor, and I will take the other one 265 00:18:40,944 --> 00:18:44,778 out. And here you see indeed exactly 266 00:18:44,778 --> 00:18:48,067 the image that we discussed. So you see early on the 267 00:18:48,067 --> 00:18:51,677 charging of the capacitor, and it reaches effectively the 268 00:18:51,677 --> 00:18:54,321 maximum value. You know, you don't have to 269 00:18:54,321 --> 00:18:57,158 wait infinitely long. That's always -- we say 270 00:18:57,158 --> 00:18:59,995 infinitely long, but that's -- clearly if you 271 00:18:59,995 --> 00:19:02,896 wait three or four, five times R C then you're 272 00:19:02,896 --> 00:19:05,41 almost there. And then here I'm going to 273 00:19:05,41 --> 00:19:09,279 switch it back to position two, and you see it's discharging. 274 00:19:09,279 --> 00:19:13,722 And then it's charging up again, and discharging. 275 00:19:13,722 --> 00:19:16,555 And now I can also show you the current. 276 00:19:16,555 --> 00:19:20,986 You see it on the same plot as the battery -- as the capacitor 277 00:19:20,986 --> 00:19:23,892 is charging, you see the current is high. 278 00:19:23,892 --> 00:19:27,596 But by the time that the capacitor is fully charged, 279 00:19:27,596 --> 00:19:31,301 the current is zero. But the moment that I throw the 280 00:19:31,301 --> 00:19:35,005 switch to position two, the current flips direction, 281 00:19:35,005 --> 00:19:38,056 becomes negative, and then as the capacitor 282 00:19:38,056 --> 00:19:42,851 discharges and heat is dissipated in the resistor, 283 00:19:42,851 --> 00:19:46,397 ultimately the current again will become zero. 284 00:19:46,397 --> 00:19:50,338 What I can do now is to increase my capacitance by, 285 00:19:50,338 --> 00:19:52,702 for instance, a factor of five. 286 00:19:52,702 --> 00:19:57,037 But that would change my R C time to three milliseconds. 287 00:19:57,037 --> 00:20:01,608 So now there is no way that the battery -- not the battery, 288 00:20:01,608 --> 00:20:05,864 no, there is no way that the capacitor can become fully 289 00:20:05,864 --> 00:20:09,332 charged in the four milliseconds that it has. 290 00:20:09,332 --> 00:20:13,115 And you will see that indeed if I 291 00:20:13,115 --> 00:20:16,226 make this five, point five microfarads, 292 00:20:16,226 --> 00:20:20,894 oh point five microfarads, then you see that the capacitor 293 00:20:20,894 --> 00:20:25,972 has not enough time to get fully charged, and so here you begin 294 00:20:25,972 --> 00:20:29,329 to discharge. Notice also that the current 295 00:20:29,329 --> 00:20:33,424 doesn't reach to zero, for the same reason that the 296 00:20:33,424 --> 00:20:36,29 capacitor doesn't get fully charged. 297 00:20:36,29 --> 00:20:41,122 So it never reaches the point that the current becomes zero. 298 00:20:41,122 --> 00:20:45,872 I'm already switching here to position two, 299 00:20:45,872 --> 00:20:50,741 and the current then flips over but never becomes zero. 300 00:20:50,741 --> 00:20:54,888 All right, so let's go back to point one value. 301 00:20:54,888 --> 00:20:58,766 There it is. This is easier than LR circuits 302 00:20:58,766 --> 00:21:04,176 for the reason that we don't have to deal with Faraday's Law. 303 00:21:04,176 --> 00:21:07,602 We don't have a non-conservative field. 304 00:21:07,602 --> 00:21:12,832 And so it's easier to imagine things, because we're dealing 305 00:21:12,832 --> 00:21:18,423 with Kirchoff's Law. So potential differences are 306 00:21:18,423 --> 00:21:23,013 uniquely defined and don't depend on the path, 307 00:21:23,013 --> 00:21:28,42 whereas with Faraday's Law they do depend on the path. 308 00:21:28,42 --> 00:21:33,113 So now I want to change to a different subject, 309 00:21:33,113 --> 00:21:37,806 which is transformers. Transformers play a very 310 00:21:37,806 --> 00:21:43,519 important role in our lives. Transformers are key in many 311 00:21:43,519 --> 00:21:48,314 instruments that you use at home but 312 00:21:48,314 --> 00:21:54,275 also key in getting the energy all the way from the power 313 00:21:54,275 --> 00:21:57,469 station to us, as you will see. 314 00:21:57,469 --> 00:22:02,579 A full understanding of transformers is not easy. 315 00:22:02,579 --> 00:22:08,009 It's extremely complicated. It's really more like an 316 00:22:08,009 --> 00:22:12,48 engineering problem than a physics problem. 317 00:22:12,48 --> 00:22:18,761 I will give you a simplified version, in which I leave a lot 318 00:22:18,761 --> 00:22:23,126 of details out, but the basic idea will be 319 00:22:23,126 --> 00:22:25,787 there. Here I have a coil, 320 00:22:25,787 --> 00:22:31,111 and I call this coil the primary. 321 00:22:31,111 --> 00:22:36,631 This coil has N1 windings and has a self-inductance L1. 322 00:22:36,631 --> 00:22:41,028 And I put here a voltmeter, which I call V1. 323 00:22:41,028 --> 00:22:46,651 It's always in that circuit. And a current I1 is running 324 00:22:46,651 --> 00:22:52,478 through the coil and returning. And there is a teeny weeny 325 00:22:52,478 --> 00:22:57,897 little current I1 -- I put a little I there -- running 326 00:22:57,897 --> 00:23:03,724 through this voltmeter. Insignificantly small, 327 00:23:03,724 --> 00:23:10,357 so I just call this also I1, because this I1 is so small. 328 00:23:10,357 --> 00:23:15,805 But it gives me, it allows me to always monitor 329 00:23:15,805 --> 00:23:18,766 V1. So that's the primary. 330 00:23:18,766 --> 00:23:24,925 Now we have a secondary, which is wound in such a way 331 00:23:24,925 --> 00:23:31,321 that there is a magnetic flux coupling between the two. 332 00:23:31,321 --> 00:23:37,964 This is the secondary. It has N1 -- N2 windings. 333 00:23:37,964 --> 00:23:43,651 Self-inductance is L2. And I put here a voltmeter V2. 334 00:23:43,651 --> 00:23:49,556 The current I2 is flowing here. This is I2, here is the 335 00:23:49,556 --> 00:23:53,603 consumer, this is where you are maybe. 336 00:23:53,603 --> 00:23:57,868 And I2 is flowing back through the coil. 337 00:23:57,868 --> 00:24:03,992 But here is this teeny weeny little current I2 flowing so 338 00:24:03,992 --> 00:24:09,133 that I can monitor the value of V2. 339 00:24:09,133 --> 00:24:14,001 What I can do now is I can write down the closed loop 340 00:24:14,001 --> 00:24:18,588 integral of E dot DL for this closed circuit here. 341 00:24:18,588 --> 00:24:22,802 And I have to apply, of course, Faraday's Law, 342 00:24:22,802 --> 00:24:26,827 because I now have a changing magnetic flux. 343 00:24:26,827 --> 00:24:31,04 They're always AC. Transformers always require 344 00:24:31,04 --> 00:24:35,815 alternating current. And so if I start here and I go 345 00:24:35,815 --> 00:24:41,918 through the self-inductor, I know there is no electric 346 00:24:41,918 --> 00:24:47,001 field in the self-inductor, so that contribution of E dot 347 00:24:47,001 --> 00:24:50,36 DL is zero. Then I arrive here at this 348 00:24:50,36 --> 00:24:55,171 current, which is opposing me. I go in this direction, 349 00:24:55,171 --> 00:24:59,983 but the E field is opposing me, and so I get minus V1. 350 00:24:59,983 --> 00:25:03,614 And that now, according to Faraday's Law, 351 00:25:03,614 --> 00:25:08,607 since I went in the direction of the current through the 352 00:25:08,607 --> 00:25:13,327 self-inductor, equals minus L1 DI1 DT. 353 00:25:13,327 --> 00:25:17,547 And I'll tel- tell you what the simplification is that I made, 354 00:25:17,547 --> 00:25:21,56 because I want to be honest with you about where I am rigid 355 00:25:21,56 --> 00:25:24,811 and where I am not. here is really a second term 356 00:25:24,811 --> 00:25:28,339 here, which is the mutual inductance between the two 357 00:25:28,339 --> 00:25:30,691 coils. And there should really be a 358 00:25:30,691 --> 00:25:34,357 term here in terms of that mutual inductance capital M 359 00:25:34,357 --> 00:25:36,917 times DI2 DT. And I leave it out here. 360 00:25:36,917 --> 00:25:40,722 I leave it out because the final result in most cases is 361 00:25:40,722 --> 00:25:43,904 not affected by it. But I want you to know that 362 00:25:43,904 --> 00:25:48,411 strictly speaking, this equation is 363 00:25:48,411 --> 00:25:52,065 simplified. I know that this is the E 364 00:25:52,065 --> 00:25:56,937 induced in side one. And now I do the same closed 365 00:25:56,937 --> 00:26:03,027 loop integral through the coil, through the voltmeter back to 366 00:26:03,027 --> 00:26:05,26 the coil. I start here. 367 00:26:05,26 --> 00:26:11,35 The current I2 is in the same direction as the direction that 368 00:26:11,35 --> 00:26:13,887 I move. So I have plus V2. 369 00:26:13,887 --> 00:26:18,962 Then I go through the self-inductor. 370 00:26:18,962 --> 00:26:24,254 There's no electric field in the self-inductor, 371 00:26:24,254 --> 00:26:30,465 because there's no electric field in a wire that has no 372 00:26:30,465 --> 00:26:33,226 resistance. And that now, 373 00:26:33,226 --> 00:26:38,287 since I went in the direction of the current, 374 00:26:38,287 --> 00:26:43,118 is minus L2 DI2 DT. And again, I bury the M 375 00:26:43,118 --> 00:26:47,719 contribution. And this now equals the EMF 376 00:26:47,719 --> 00:26:53,7 induced in the secondary. But clearly this is also N1 377 00:26:53,7 --> 00:26:58,747 times D phi B DT, with a minus sign. 378 00:26:58,747 --> 00:27:03,69 This is the magnetic flux change through one loop in the 379 00:27:03,69 --> 00:27:07,105 primary. The surface of one loop in the 380 00:27:07,105 --> 00:27:11,868 primary sees a magnetic flux going through it which is 381 00:27:11,868 --> 00:27:16,092 changing with time, and that value D phi B DT is 382 00:27:16,092 --> 00:27:19,418 through one loop. But I have N1 loops, 383 00:27:19,418 --> 00:27:23,552 so I have an N1 there. And so in the secondary, 384 00:27:23,552 --> 00:27:28,046 I have N2 times the same D fee by DT 385 00:27:28,046 --> 00:27:34,475 if I have perfect magnetic coupling between the two loops, 386 00:27:34,475 --> 00:27:41,131 which is not always the case. And so now you see that I have 387 00:27:41,131 --> 00:27:45,643 that V2 over V1, if I take the magnitude, 388 00:27:45,643 --> 00:27:51,396 is now simply N2 over N1. That is an amazing result. 389 00:27:51,396 --> 00:27:59,067 It tells you that you can make V2 on the secondary side way 390 00:27:59,067 --> 00:28:02,239 larger than V1. We call that transforming up, 391 00:28:02,239 --> 00:28:06,421 by making N2 larger than N1. But you can also make it lower 392 00:28:06,421 --> 00:28:10,315 -- we call that a step-down transformer -- by making N2 393 00:28:10,315 --> 00:28:13,416 smaller than N1. When I write down capital V 394 00:28:13,416 --> 00:28:17,598 here you should think of that. It's an alternating current, 395 00:28:17,598 --> 00:28:21,492 so you have cosine omega Ts in there, or sines omega T. 396 00:28:21,492 --> 00:28:25,241 You should think of this perhaps as the maximum value 397 00:28:25,241 --> 00:28:27,621 possible. But of course that's not 398 00:28:27,621 --> 00:28:30,618 important in the concept. 399 00:28:30,618 --> 00:28:34,733 But there are always cosine omega Ts all over the place. 400 00:28:34,733 --> 00:28:38,549 The power station puts the electricity, so to speak, 401 00:28:38,549 --> 00:28:41,766 on the line at three hundred thousand volts. 402 00:28:41,766 --> 00:28:44,983 We discussed that earlier, why they do that, 403 00:28:44,983 --> 00:28:49,173 that's such high voltage. So they have a transformer from 404 00:28:49,173 --> 00:28:53,587 their generator to transform the potential up to a very high 405 00:28:53,587 --> 00:28:56,43 value, to three hundred thousand volts. 406 00:28:56,43 --> 00:29:01,667 When it arrives at Boston, it's stepped down to twelve 407 00:29:01,667 --> 00:29:04,203 kilovolts. And so there are transformers, 408 00:29:04,203 --> 00:29:06,232 which you see, huge transformers, 409 00:29:06,232 --> 00:29:09,021 which have a ratio N1 over N2 of twenty-five. 410 00:29:09,021 --> 00:29:11,62 So N1 is twenty-five times larger than N2. 411 00:29:11,62 --> 00:29:13,966 So I step it down to twelve kilovolts. 412 00:29:13,966 --> 00:29:17,009 But you don't want twelve kilovolts at your home. 413 00:29:17,009 --> 00:29:20,179 You want hundred ten, hundred twenty volts at home. 414 00:29:20,179 --> 00:29:23,982 So when you look at these power poles outside people's homes, 415 00:29:23,982 --> 00:29:26,962 you see again transformers in these power poles, 416 00:29:26,962 --> 00:29:30,575 which bring it down from twelve kilovolts to about hundred 417 00:29:30,575 --> 00:29:36,427 twenty volts. So there you have a ratio N1 418 00:29:36,427 --> 00:29:44,215 over N2 of about one hundred. To calculate the ratios of the 419 00:29:44,215 --> 00:29:49,099 current, I1 on I2, is way more tricky. 420 00:29:49,099 --> 00:29:56,228 There are some big ifs. And one big if is that R in the 421 00:29:56,228 --> 00:30:01,64 primary -- in the secondary side, is much, 422 00:30:01,64 --> 00:30:08,505 much smaller than omega L. That is one if. 423 00:30:08,505 --> 00:30:12,48 The second if is that no energy is lost in terms of eddy 424 00:30:12,48 --> 00:30:15,299 currents. Very often these two coils are 425 00:30:15,299 --> 00:30:18,624 coupled through an iron core, and then you get, 426 00:30:18,624 --> 00:30:22,672 get eddy currents in the iron core, then you lose energy. 427 00:30:22,672 --> 00:30:26,141 Also, do you not always have ideal flux coupling. 428 00:30:26,141 --> 00:30:30,334 So it is not always true that the magnetic flux through one 429 00:30:30,334 --> 00:30:34,743 coil in the primary is the same as the one through one coil in 430 00:30:34,743 --> 00:30:37,706 the secondary. But if all of that were the 431 00:30:37,706 --> 00:30:40,536 case, so if R is much, 432 00:30:40,536 --> 00:30:45,186 much less than omega L and if there is no energy lost in eddy 433 00:30:45,186 --> 00:30:48,751 currents and if there is perfect flux coupling, 434 00:30:48,751 --> 00:30:53,091 then you can show that whatever power is delivered on the 435 00:30:53,091 --> 00:30:57,586 primary side is going to be consumed on the secondary side. 436 00:30:57,586 --> 00:31:01,384 And if that is true, if the power delivered on the 437 00:31:01,384 --> 00:31:03,554 primary side, which is I1 V1, 438 00:31:03,554 --> 00:31:07,971 if that is the same as the power consumed on the secondary 439 00:31:07,971 --> 00:31:11,743 side, then that must be I2 V2. 440 00:31:11,743 --> 00:31:15,764 But if this is true, which is only under those 441 00:31:15,764 --> 00:31:21,125 conditions, and this is also correct, then of course you will 442 00:31:21,125 --> 00:31:25,592 find now that I2 divided by I1 is N1 divided by N2, 443 00:31:25,592 --> 00:31:30,417 and let's put here magnitudes again, so that we are not 444 00:31:30,417 --> 00:31:33,455 worried about possible minus signs. 445 00:31:33,455 --> 00:31:39,621 And then you look at this, you see that you can generate, 446 00:31:39,621 --> 00:31:44,04 if you want to, in the secondary an enormously 447 00:31:44,04 --> 00:31:49,245 high current by simply having a large ratio, N1 to N2. 448 00:31:49,245 --> 00:31:54,843 And I will demonstrate that today, that indeed if you meet 449 00:31:54,843 --> 00:31:59,164 those conditions that you can indeed do that. 450 00:31:59,164 --> 00:32:03,878 I2 over I1, that's fine, V2 over V1, that's fine. 451 00:32:03,878 --> 00:32:06,923 So I will do two demonstrations. 452 00:32:06,923 --> 00:32:13,895 The first thing I want to demonstrate is that if I make -- 453 00:32:13,895 --> 00:32:18,979 oh, that's beautiful. I have to erase something here 454 00:32:18,979 --> 00:32:23,664 -- oh, actually, I could work on this board here 455 00:32:23,664 --> 00:32:27,551 for a while. I have here a demonstration 456 00:32:27,551 --> 00:32:31,239 whereby you see here the primary coil. 457 00:32:31,239 --> 00:32:34,428 You see it right in front of you. 458 00:32:34,428 --> 00:32:38,216 It has two hundred and twenty windings. 459 00:32:38,216 --> 00:32:41,306 So N1 is two hundred and twenty. 460 00:32:41,306 --> 00:32:45,791 And V1 is hundred and ten volts, and it is AC, 461 00:32:45,791 --> 00:32:50,904 has to be. The frequency is sixty hertz. 462 00:32:50,904 --> 00:32:54,41 It's just what we get out of the plug here, 463 00:32:54,41 --> 00:32:57,165 like you get it in your dormitory. 464 00:32:57,165 --> 00:33:01,924 That means omega is about three hundred and seventy-seven. 465 00:33:01,924 --> 00:33:05,598 It's two pi times F. If I give you really the 466 00:33:05,598 --> 00:33:09,772 correct value for V1, it's really the maximum value 467 00:33:09,772 --> 00:33:12,611 for V1 times the cosine of omega T. 468 00:33:12,611 --> 00:33:16,952 And this value is really hundred ten times the square 469 00:33:16,952 --> 00:33:21,627 root of two. I've mentioned that earlier 470 00:33:21,627 --> 00:33:24,231 in my lectures. The voltmeter, 471 00:33:24,231 --> 00:33:29,439 however, are cir- are designed in such a way that they only 472 00:33:29,439 --> 00:33:34,467 show you hundred and ten. And so we call that hundred and 473 00:33:34,467 --> 00:33:37,878 ten volts. I can now make the secondary 474 00:33:37,878 --> 00:33:41,919 just one winding. I have a wire that I re- ro- 475 00:33:41,919 --> 00:33:45,869 wrap around once. And if indeed that equation 476 00:33:45,869 --> 00:33:50,268 holds, which it will, I expect now that V2 will be 477 00:33:50,268 --> 00:33:56,699 about oh point five volts, namely the hundred ten volts 478 00:33:56,699 --> 00:34:01,904 divided by two twenty, which is the ratio primary over 479 00:34:01,904 --> 00:34:05,243 secondary. And then I will put four 480 00:34:05,243 --> 00:34:09,663 windings around, and then you expect about two 481 00:34:09,663 --> 00:34:13,296 volts. And you will see it there right 482 00:34:13,296 --> 00:34:17,716 in front of you. This is where you're supposed 483 00:34:17,716 --> 00:34:21,055 to see the reading. I can't see it. 484 00:34:21,055 --> 00:34:27,635 So I have to trust you when you give me those values. 485 00:34:27,635 --> 00:34:32,991 OK, so here is my primary coil, and here is my secondary. 486 00:34:32,991 --> 00:34:37,677 This is my secondary coil. And the way I have this 487 00:34:37,677 --> 00:34:41,981 demonstration is that this is completely open. 488 00:34:41,981 --> 00:34:46,381 So I only show you immediately the value of V2. 489 00:34:46,381 --> 00:34:51,068 I'll show you this value. It has a huge value of a 490 00:34:51,068 --> 00:34:57,094 resistor, several megaohms. So for sure I don't meet 491 00:34:57,094 --> 00:35:00,555 this condition. But that condition is not 492 00:35:00,555 --> 00:35:05,573 necessary for this to hold. That condition is necessary for 493 00:35:05,573 --> 00:35:08,515 this to hold, not for this to hold. 494 00:35:08,515 --> 00:35:13,014 So I have no fears that I am going to be embarrassed. 495 00:35:13,014 --> 00:35:17,254 So I'm going to put one -- this is very long wire. 496 00:35:17,254 --> 00:35:20,888 And I'm going to put one loop around there. 497 00:35:20,888 --> 00:35:26,166 I'm now powering this solenoid. There's an iron core in there. 498 00:35:26,166 --> 00:35:30,492 Sixty hertz, hundred and ten volts. 499 00:35:30,492 --> 00:35:34,242 And there we go, one loop around it. 500 00:35:34,242 --> 00:35:37,992 I measure oh point four eight volts. 501 00:35:37,992 --> 00:35:41,1 That's not bad. Now I put two, 502 00:35:41,1 --> 00:35:45,385 three, four around it. What do I see now? 503 00:35:45,385 --> 00:35:51,171 I see one point nine nine six. You may not see the six. 504 00:35:51,171 --> 00:35:55,135 Very close to two. What I will do now, 505 00:35:55,135 --> 00:36:02,849 I will move it up a little, and when I move it up a little, 506 00:36:02,849 --> 00:36:06,1 the secondary, the flux coupling is not ideal 507 00:36:06,1 --> 00:36:09,571 anymore, and so you expect that V2 will go down. 508 00:36:09,571 --> 00:36:13,855 Just move it up a little along the primary, and you see the 509 00:36:13,855 --> 00:36:17,179 value goes down. Because now the flux coupling 510 00:36:17,179 --> 00:36:20,356 is not ideal. And if I bring it further out, 511 00:36:20,356 --> 00:36:22,867 the flux coupling becomes hopeless. 512 00:36:22,867 --> 00:36:26,856 And when I bring it here, the flux coupling is so poor, 513 00:36:26,856 --> 00:36:30,401 but I still get oh point three volts, by the way. 514 00:36:30,401 --> 00:36:36,011 So that's the first part, to show you that this is indeed 515 00:36:36,011 --> 00:36:39,732 quite accurate. Now I want to do a second part, 516 00:36:39,732 --> 00:36:44,342 and that is more interesting. I'm going to try to convince 517 00:36:44,342 --> 00:36:48,872 you that I can get a pr- secondary current which is huge, 518 00:36:48,872 --> 00:36:51,379 maybe even one thousand amperes. 519 00:36:51,379 --> 00:36:54,777 But now I have to take special precautions. 520 00:36:54,777 --> 00:36:58,902 I have to do the experiment in a very different way. 521 00:36:58,902 --> 00:37:03,431 And the way I'm going to do this experiment now is with a 522 00:37:03,431 --> 00:37:09,356 secondary which is specially designed to 523 00:37:09,356 --> 00:37:17,16 have an enormously low value of resistance, because I want to 524 00:37:17,16 --> 00:37:24,444 approach this situation so that this approximately holds. 525 00:37:24,444 --> 00:37:31,858 We have a very thick copper secondary, almost half an inch 526 00:37:31,858 --> 00:37:35,501 thick. It's like so. 527 00:37:35,501 --> 00:37:39,665 This is copper. And I put here a nail, 528 00:37:39,665 --> 00:37:44,618 an iron nail. And the resistance of this iron 529 00:37:44,618 --> 00:37:50,358 nail is about four times ten to the minus four ohms. 530 00:37:50,358 --> 00:37:55,874 The self-inductance of this ring, purely geometry, 531 00:37:55,874 --> 00:38:01,502 has nothing to do with the kind of material I have, 532 00:38:01,502 --> 00:38:06,679 is easy to estimate. I made a calculation and I 533 00:38:06,679 --> 00:38:11,073 found about five times ten to the 534 00:38:11,073 --> 00:38:15,282 minus seven henry. So omega times L2 -- I have to 535 00:38:15,282 --> 00:38:20,368 multiply this by three hundred and seventy-seven -- becomes 536 00:38:20,368 --> 00:38:24,05 about two times ten to the minus four ohms. 537 00:38:24,05 --> 00:38:28,785 Even though R2 is not much, much smaller than omega L2, 538 00:38:28,785 --> 00:38:32,205 they are now beginning to be comparable. 539 00:38:32,205 --> 00:38:36,238 My I1, the current that I drive in the primary, 540 00:38:36,238 --> 00:38:39,911 is going to be twenty amperes. 541 00:38:39,911 --> 00:38:44,811 I do not expect that I2 will be two hundred and twenty times 542 00:38:44,811 --> 00:38:48,381 larger than I1, because my -- I use the same 543 00:38:48,381 --> 00:38:52,035 two hundred and twenty windings for the coil, 544 00:38:52,035 --> 00:38:56,685 and this is just one winding. So I have the same ratio N1 545 00:38:56,685 --> 00:39:00,339 over N2 that I had before in that experiment, 546 00:39:00,339 --> 00:39:04,657 when I did N2 equals one. I don't expect that this is 547 00:39:04,657 --> 00:39:09,141 going to be two hundred and twenty times 548 00:39:09,141 --> 00:39:12,338 twenty amperes, which would be about four 549 00:39:12,338 --> 00:39:17,134 thousand four hundred amperes, because I am really not in the 550 00:39:17,134 --> 00:39:20,89 domain that R is much much smaller than omega L. 551 00:39:20,89 --> 00:39:25,126 But I probably get five hundred to a thousand amperes. 552 00:39:25,126 --> 00:39:29,442 And if my current I2 becomes five hundred to a thousand 553 00:39:29,442 --> 00:39:33,118 amperes, you can calculate what I square R2 is, 554 00:39:33,118 --> 00:39:36,635 because you know R2. That's all in that nail. 555 00:39:36,635 --> 00:39:42,634 And that comes out to be about hundred to four hundred watts. 556 00:39:42,634 --> 00:39:46,798 You better believe that that that nail is going to be 557 00:39:46,798 --> 00:39:50,321 red-hot, is going to glow, and probably melt. 558 00:39:50,321 --> 00:39:54,565 And this is the idea behind so-called induction ovens. 559 00:39:54,565 --> 00:39:58,809 This is purposely done to get a very high temperature. 560 00:39:58,809 --> 00:40:02,893 It's also done in welding to get very high currents. 561 00:40:02,893 --> 00:40:07,297 And so this second part, the demonstration with the same 562 00:40:07,297 --> 00:40:13,463 primary, I want to make it dark so that you can see the glowing 563 00:40:13,463 --> 00:40:16,864 of that nail. And I will also show you, 564 00:40:16,864 --> 00:40:22,056 of course, first the -- so here is the coil, and here is my 565 00:40:22,056 --> 00:40:25,726 secondary. This is that very thick copper. 566 00:40:25,726 --> 00:40:31,186 And then you see the nail here. Make sure there is no current. 567 00:40:31,186 --> 00:40:34,498 My God, I almost shoved it over there. 568 00:40:34,498 --> 00:40:38,168 [laughter]. Thank goodness that I realized 569 00:40:38,168 --> 00:40:44,255 that there was current. So the current is now not going 570 00:40:44,255 --> 00:40:47,361 through the solenoid, and so here is the, 571 00:40:47,361 --> 00:40:51,4 the, the secondary now. Put some pieces of wood under 572 00:40:51,4 --> 00:40:54,118 there so that you can see it better. 573 00:40:54,118 --> 00:40:58,001 And so now if I power the primary, I can't tell you 574 00:40:58,001 --> 00:41:01,807 exactly what the current is through the secondary, 575 00:41:01,807 --> 00:41:06,078 but it's probably somewhere in the vicinity between five 576 00:41:06,078 --> 00:41:08,564 hundred and one thousand amperes. 577 00:41:08,564 --> 00:41:11,981 It will glow. It may take one minute to melt. 578 00:41:11,981 --> 00:41:17,697 It may not melt at all. It may take three seconds. 579 00:41:17,697 --> 00:41:22,014 Unpredictable, because we don't know exactly 580 00:41:22,014 --> 00:41:27,635 the resistance of that nail. So I will tell you when I do 581 00:41:27,635 --> 00:41:29,743 it. I will count down. 582 00:41:29,743 --> 00:41:31,852 Three, two, one, zero. 583 00:41:31,852 --> 00:41:35,064 It's already gone. That was fast. 584 00:41:35,064 --> 00:41:40,183 So you saw it glowing, and you can actually see that 585 00:41:40,183 --> 00:41:44,701 the nail melted. So you've seen an example now 586 00:41:44,701 --> 00:41:49,92 where the current is immensely high. 587 00:41:49,92 --> 00:41:55,847 So now I want to discuss with you another practical 588 00:41:55,847 --> 00:42:01,181 application, which has immediate consequences, 589 00:42:01,181 --> 00:42:06,16 or applications I should say, in your cars. 590 00:42:06,16 --> 00:42:09,478 Spark plugs. Cars have coils, 591 00:42:09,478 --> 00:42:15,05 and coils are run by a car battery, which is DC, 592 00:42:15,05 --> 00:42:20,146 it's not AC. But you got to get high voltage 593 00:42:20,146 --> 00:42:24,76 to get a spark going in spark 594 00:42:24,76 --> 00:42:28,031 plug. And that's done in a very 595 00:42:28,031 --> 00:42:31,957 clever way. So here is my twelve volt 596 00:42:31,957 --> 00:42:35,01 battery. There is always some 597 00:42:35,01 --> 00:42:39,481 resistance, of course, in, in the circuit. 598 00:42:39,481 --> 00:42:44,716 If you have a coil, you always have some final -- 599 00:42:44,716 --> 00:42:49,296 finite resistance. And then here is a coil, 600 00:42:49,296 --> 00:42:55,947 and here is a switch. And this has N1 windings, 601 00:42:55,947 --> 00:43:03,096 has a certain self-inductance, the same situation that we had 602 00:43:03,096 --> 00:43:06,552 before. And then I have here a 603 00:43:06,552 --> 00:43:12,39 secondary, where I have N2 windings, which is way, 604 00:43:12,39 --> 00:43:16,918 way larger than N1. I close the switch. 605 00:43:16,918 --> 00:43:24,305 A current is going to build up. And the time constant is L over 606 00:43:24,305 --> 00:43:30,143 R seconds -- we discussed that before, 607 00:43:30,143 --> 00:43:34,14 it's going to build up. And now I'm going to do 608 00:43:34,14 --> 00:43:39,093 something extremely cruel. I'm going to open this circuit, 609 00:43:39,093 --> 00:43:42,916 open the switch. Imagine what is -- what I am 610 00:43:42,916 --> 00:43:46,131 doing. I'm cutting instantaneously the 611 00:43:46,131 --> 00:43:49,346 current. The current is going happily, 612 00:43:49,346 --> 00:43:54,038 reaches a maximum value after several L over R seconds, 613 00:43:54,038 --> 00:43:59,077 and now I *wssht* open it up. So I get a huge value for DI1 614 00:43:59,077 --> 00:44:02,552 DT. I1 is now the current 615 00:44:02,552 --> 00:44:04,864 in the primary. The current, 616 00:44:04,864 --> 00:44:10,087 of course, will die down on a time scale L over R which is the 617 00:44:10,087 --> 00:44:14,026 new value for R. And the new value for R is the 618 00:44:14,026 --> 00:44:17,793 fact that I open it up. I make the resistance 619 00:44:17,793 --> 00:44:21,731 infinitely large here. And if the resistance is 620 00:44:21,731 --> 00:44:26,184 infinitely large here, my L over R time goes to zero. 621 00:44:26,184 --> 00:44:32,519 So that makes you see indeed why the DI1 DT becomes enormous. 622 00:44:32,519 --> 00:44:37,061 With a twelve volt battery, you can easily get several 623 00:44:37,061 --> 00:44:41,174 hundred volts EMF now, because the DI1 DT will of 624 00:44:41,174 --> 00:44:45,629 course create an EMF, an induced EMF in that circuit, 625 00:44:45,629 --> 00:44:50,427 because you get a D phi DT. You get an enormous change in 626 00:44:50,427 --> 00:44:54,797 the magnetic flux here, which is directly coupled to 627 00:44:54,797 --> 00:44:58,31 the current. And that induced EMF could be 628 00:44:58,31 --> 00:45:03,965 several hundred volts. But now look at the secondary. 629 00:45:03,965 --> 00:45:07,909 The secondary has an N2 which is way larger than N1. 630 00:45:07,909 --> 00:45:12,008 So the induced EMF in the secondary is very roughly N2 631 00:45:12,008 --> 00:45:15,411 over N1 times the induced EMF in the primary. 632 00:45:15,411 --> 00:45:19,587 That was the ratio that we had earlier, the V2 over V1, 633 00:45:19,587 --> 00:45:23,686 remember was N2 over N1. So now I get in the secondary 634 00:45:23,686 --> 00:45:27,166 an absolutely horrendous potential difference. 635 00:45:27,166 --> 00:45:30,646 I could get up to a million volts if I wanted. 636 00:45:30,646 --> 00:45:34,59 In your cars, that is not necessary. 637 00:45:34,59 --> 00:45:39,601 It is enough in your cars that you get here something like ten 638 00:45:39,601 --> 00:45:42,887 kilovolts. And that will comfortably give 639 00:45:42,887 --> 00:45:45,516 you a spark over your spark plug. 640 00:45:45,516 --> 00:45:50,363 And so what is happening with your car without you realizing 641 00:45:50,363 --> 00:45:55,128 it, there is a circuit that closes and opens and closes and 642 00:45:55,128 --> 00:45:59,811 opens and closes and opens, and it's only when you open it 643 00:45:59,811 --> 00:46:04,083 that you get this pathetic high voltage 644 00:46:04,083 --> 00:46:06,767 here and that the spark flies over. 645 00:46:06,767 --> 00:46:11,502 And if you run your engine at three thousand RPM and you have 646 00:46:11,502 --> 00:46:15,133 a four cylinder, that happens two hundred times 647 00:46:15,133 --> 00:46:18,132 per second. But it is the breaking that 648 00:46:18,132 --> 00:46:21,052 does it. And I can demonstrate this to 649 00:46:21,052 --> 00:46:23,578 you. We have a very special spark 650 00:46:23,578 --> 00:46:26,261 plug. Maybe it should not be called 651 00:46:26,261 --> 00:46:29,024 spark plug. It's a beautiful device. 652 00:46:29,024 --> 00:46:31,865 You'll see it all the way over there. 653 00:46:31,865 --> 00:46:38,517 And we have there a situation whereby we don't even know what 654 00:46:38,517 --> 00:46:42,397 N2 over N1 is. This is ancient work of art, 655 00:46:42,397 --> 00:46:46,831 probably built in the nineteenth century or maybe 656 00:46:46,831 --> 00:46:51,542 early twenty century. We think that N2 over N1 is at 657 00:46:51,542 --> 00:46:56,438 least several thousand, but let's say it is ten to the 658 00:46:56,438 --> 00:46:59,856 third. But it could be way higher than 659 00:46:59,856 --> 00:47:03,92 that, we don't know. And we run it with a car 660 00:47:03,92 --> 00:47:07,348 battery, just like your car, 661 00:47:07,348 --> 00:47:10,507 twelve volts. We let the current build up in 662 00:47:10,507 --> 00:47:14,915 the primary, just like we have here, and then in a very cruel 663 00:47:14,915 --> 00:47:18,955 way we open the primary and we create, in the secondary, 664 00:47:18,955 --> 00:47:21,82 up to three, four, five hundred thousand 665 00:47:21,82 --> 00:47:24,244 volts. And then what you will see, 666 00:47:24,244 --> 00:47:28,064 you will see a spark fly over here, at three hundred, 667 00:47:28,064 --> 00:47:30,488 four, five hundred thousand volts. 668 00:47:30,488 --> 00:47:34,455 You can draw a spark over a distance of ten centimeters 669 00:47:34,455 --> 00:47:38,789 easily, because all you have to do is get 670 00:47:38,789 --> 00:47:43,105 above three million volts per meter, which is the breakdown 671 00:47:43,105 --> 00:47:46,527 electric field. And this is what I want to show 672 00:47:46,527 --> 00:47:49,132 you now. And for that I also have to 673 00:47:49,132 --> 00:47:52,182 make it a little dark, because if you see, 674 00:47:52,182 --> 00:47:55,828 want to see sparks of course, you want to keep it, 675 00:47:55,828 --> 00:47:59,771 make it a little dark. So this is the way we are going 676 00:47:59,771 --> 00:48:02,598 to do it. Maybe you want to take a look 677 00:48:02,598 --> 00:48:05,575 at that beautiful coil first. It's there. 678 00:48:05,575 --> 00:48:08,179 So this is the, uh, the whole setup. 679 00:48:08,179 --> 00:48:12,197 Here you see the car battery there. 680 00:48:12,197 --> 00:48:16,949 It's just twelve volts. And here you see the, 681 00:48:16,949 --> 00:48:21,377 the secondary. This is the open end of the 682 00:48:21,377 --> 00:48:26,885 secondary, those two little balls that I have there. 683 00:48:26,885 --> 00:48:32,933 And all of this is enclosed. We cannot measure very much. 684 00:48:32,933 --> 00:48:37,577 And so now I will change the lights for you. 685 00:48:37,577 --> 00:48:44,166 OK, I'm now running a current through the primary. 686 00:48:44,166 --> 00:48:48,708 And that current is about -- oh, we don't even know what the 687 00:48:48,708 --> 00:48:50,863 current is. I can't tell you. 688 00:48:50,863 --> 00:48:55,329 All I can tell you is now I will open up the current in the 689 00:48:55,329 --> 00:48:59,024 primary, so -- boy, I didn't mean to kill myself, 690 00:48:59,024 --> 00:49:02,873 but I did in the process. OK, I open up the switch, 691 00:49:02,873 --> 00:49:07,185 and every time that I open it you see there this enormous 692 00:49:07,185 --> 00:49:11,342 voltage at the secondary. This is the way that your car 693 00:49:11,342 --> 00:49:15,192 coil works, and that you get sparks 694 00:49:15,192 --> 00:49:21,881 in your spark plug. That is an enormous potential 695 00:49:21,881 --> 00:49:28,431 difference there. We can also set this in such a 696 00:49:28,431 --> 00:49:35,956 way that it opens and closes without my having to do it 697 00:49:35,956 --> 00:49:39,579 manually. And then you see, 698 00:49:39,579 --> 00:49:45,432 you get an idea of how your spark 699 00:49:45,432 --> 00:49:48,157 plug works. But that's not the reason why I 700 00:49:48,157 --> 00:49:51,205 showed it to you. This is such a wonderful piece 701 00:49:51,205 --> 00:49:54,124 of engineering. And a lot of research was done 702 00:49:54,124 --> 00:49:58,016 in the early part of the twenty century with instruments like 703 00:49:58,016 --> 00:50:00,222 this. It was named after the person 704 00:50:00,222 --> 00:50:02,622 who invented this, which is Ruhmkorff, 705 00:50:02,622 --> 00:50:05,865 and when I was a student we always referred to this 706 00:50:05,865 --> 00:50:09,368 instrument as a Ruhmkorff. Well, I'll let it go so when 707 00:50:09,368 --> 00:50:12,741 you walk out you can come close. But be very careful. 708 00:50:12,741 --> 00:50:16,114 You're dealing there with several 709 00:50:16,114 --> 50:21 hundred thousand volts. OK, have a good weekend.