1 00:00:00,469 --> 00:00:02,401 - So we have something interesting going on. 2 00:00:02,401 --> 00:00:05,821 I have this ring of conductor right here, this square ring, 3 00:00:05,821 --> 00:00:09,733 it has a resistance of two ohms, we see that it is 4 00:00:09,733 --> 00:00:12,682 two meters by two meters so the area 5 00:00:12,682 --> 00:00:15,723 of this ring would be four square meters 6 00:00:15,723 --> 00:00:18,324 and we see a magnetic field going through 7 00:00:18,324 --> 00:00:21,040 the surface defined by the ring and it's constant, 8 00:00:21,040 --> 00:00:23,455 it's a constant magnetic field of five teslas 9 00:00:23,455 --> 00:00:26,183 and it's going exactly perpendicularly to, 10 00:00:26,183 --> 00:00:30,362 perpendicularly to the surface of the ring. 11 00:00:30,362 --> 00:00:32,962 Now what we're going to happen, what we're 12 00:00:32,962 --> 00:00:35,761 going to see happen is over the next four seconds, 13 00:00:35,761 --> 00:00:38,744 and this is going to happen at a linear rate, 14 00:00:38,744 --> 00:00:40,659 it's going to happen at a constant rate, 15 00:00:40,659 --> 00:00:43,120 we're going to see the magnetic field over four seconds 16 00:00:43,120 --> 00:00:46,226 go from five teslas to 10 teslas so it's going to 17 00:00:46,226 --> 00:00:48,704 double over those four seconds and by doing so 18 00:00:48,704 --> 00:00:50,573 we're going to have a change in flux. 19 00:00:50,573 --> 00:00:51,955 Let's think about what the change 20 00:00:51,955 --> 00:00:54,660 in flux is over this four seconds. 21 00:00:54,660 --> 00:00:57,875 So our initial flux, let me write it over here, 22 00:00:57,875 --> 00:01:01,915 so flux, let me use a different color 23 00:01:01,915 --> 00:01:03,703 and at any time if you are so inspired 24 00:01:03,703 --> 00:01:04,956 I encourage you to pause the video 25 00:01:04,956 --> 00:01:07,679 and figure out what our change in flux is. 26 00:01:07,679 --> 00:01:11,785 So our flux, flux initial 27 00:01:11,785 --> 00:01:14,725 is going to be, well it's the 28 00:01:14,725 --> 00:01:16,937 it's going to be the constant magnetic field, 29 00:01:16,937 --> 00:01:18,493 you could say the average magnetic field 30 00:01:18,493 --> 00:01:19,898 over the surface but since it's constant 31 00:01:19,898 --> 00:01:23,980 that's just going to be five teslas, so five teslas 32 00:01:23,980 --> 00:01:25,818 and it helps for, it helps us in this problem 33 00:01:25,818 --> 00:01:28,046 that the magnetic field vectors are exactly 34 00:01:28,046 --> 00:01:30,555 perpendicular to the surface, to the surface 35 00:01:30,555 --> 00:01:32,436 defined by the ring, if they weren't we would 36 00:01:32,436 --> 00:01:35,013 have to find the component that is perpendicular 37 00:01:35,013 --> 00:01:36,499 but we have that right over there so we have 38 00:01:36,499 --> 00:01:38,739 the five teslas, that's the average magnetic field 39 00:01:38,739 --> 00:01:41,558 or the average component of the magnetic field 40 00:01:41,558 --> 00:01:43,990 that is perpendicular to the surface, 41 00:01:43,990 --> 00:01:46,680 so five teslas times the area of the surface. 42 00:01:46,680 --> 00:01:50,969 So times, well two meters times two meters is four 43 00:01:50,969 --> 00:01:55,509 square meters so that is going to be equal to, 44 00:01:55,509 --> 00:01:58,470 that is equal to 20 45 00:01:58,470 --> 00:02:01,180 tesla meter squared, 46 00:02:01,180 --> 00:02:05,051 tesla meter squared, fair enough. 47 00:02:05,051 --> 00:02:08,858 Now what's the final, what's the final flux? 48 00:02:08,858 --> 00:02:13,641 The final flux, flux final is going to be equal to, 49 00:02:13,641 --> 00:02:16,695 well now the average magnetic field or 50 00:02:16,695 --> 00:02:18,332 the average components of the magnetic field 51 00:02:18,332 --> 00:02:19,460 that are perpendicular in the way 52 00:02:19,460 --> 00:02:20,486 I've defined this magnetic field, 53 00:02:20,486 --> 00:02:22,388 the vectors are already perpendicular 54 00:02:22,388 --> 00:02:26,078 is 10 teslas, so 10 teslas. 55 00:02:26,078 --> 00:02:29,048 The area of our ring hasn't changed, 56 00:02:29,048 --> 00:02:31,741 it's still four square meters, 57 00:02:31,741 --> 00:02:34,552 so times four square meters, 58 00:02:34,552 --> 00:02:36,653 and so what is this going to be? 59 00:02:36,653 --> 00:02:40,870 So our final flux is going to be 60 00:02:40,870 --> 00:02:44,422 final flux is going to be 40, 61 00:02:44,422 --> 00:02:48,590 40 tesla meters squared. 62 00:02:48,590 --> 00:02:52,003 So what is our, what is our change in flux? 63 00:02:52,003 --> 00:02:56,471 Let me write this over here, our change in flux, change in 64 00:02:56,471 --> 00:03:00,381 flux, which is going to be our final flux 65 00:03:01,161 --> 00:03:05,667 minus our initial flux is going to be 66 00:03:05,667 --> 00:03:09,149 40 tesla meters squared minus 20 tesla meters squared, 67 00:03:09,149 --> 00:03:13,863 which is just going to be 20 tesla meters squared. 68 00:03:14,223 --> 00:03:16,846 So we figured out the change in flux, we actually know 69 00:03:16,846 --> 00:03:19,319 the change in time is going to be four seconds 70 00:03:19,319 --> 00:03:22,001 and actually using that we can now figure out 71 00:03:22,001 --> 00:03:24,857 what the voltage induced is going to be, 72 00:03:24,857 --> 00:03:27,306 the voltage induced, or the voltage that's going 73 00:03:27,306 --> 00:03:29,814 to now induce, induce a current. 74 00:03:29,814 --> 00:03:32,809 And if you were to look up Faraday's law on the internet, 75 00:03:32,809 --> 00:03:34,991 you were to look up for a formula for Faraday's law 76 00:03:34,991 --> 00:03:36,693 you would see something that looks like this, 77 00:03:36,693 --> 00:03:39,963 you would see voltage generated is equal to 78 00:03:39,963 --> 00:03:42,797 negative and, at least if you're not using the 79 00:03:42,797 --> 00:03:47,797 calculus version of it, negative N times our change in flux, 80 00:03:47,799 --> 00:03:51,897 change in, let me write change in flux not just flux, 81 00:03:51,897 --> 00:03:54,889 change in flux, delta flux 82 00:03:54,889 --> 00:03:57,599 over change in time. 83 00:03:57,599 --> 00:04:00,112 So one way to think about this, and to do this problem 84 00:04:00,112 --> 00:04:02,357 right we're assuming we have a constant 85 00:04:02,357 --> 00:04:05,267 or the rate of change is constant in our flux 86 00:04:05,267 --> 00:04:09,040 so you have your average rate of change of your flux 87 00:04:09,040 --> 00:04:11,617 and then you're going to multiply it times N. 88 00:04:11,617 --> 00:04:14,086 N is actually the number of loops you have, or you can 89 00:04:14,086 --> 00:04:16,562 think of it as the number of surfaces defined by it. 90 00:04:16,562 --> 00:04:19,197 In this exact example, in this exact example 91 00:04:19,197 --> 00:04:21,380 N is just going to be one, we just have one loop 92 00:04:21,380 --> 00:04:23,563 so that simplifies it right over there 93 00:04:23,563 --> 00:04:25,803 and then, so this is going to be, and you might say 94 00:04:25,803 --> 00:04:27,359 what is this negative because it's a bit of 95 00:04:27,359 --> 00:04:29,210 a strange thing because you know, how are we 96 00:04:29,210 --> 00:04:30,935 defining direction, you know what's in the-- 97 00:04:30,935 --> 00:04:33,530 and all of that and that's why I'm a little bit, 98 00:04:33,530 --> 00:04:35,625 I'm not a huge fan of this negative sign. 99 00:04:35,625 --> 00:04:38,063 This is, you know if you look it up in a textbook 100 00:04:38,063 --> 00:04:39,920 they'll often say, and you're not using calculus, 101 00:04:39,920 --> 00:04:42,579 they'll say, oh this reminder to use Lenz's law, 102 00:04:42,579 --> 00:04:45,424 they'll write literally Lenz's law 103 00:04:45,424 --> 00:04:47,013 and I would say if they want a reminder to use 104 00:04:47,013 --> 00:04:48,337 Lenz's law why don't they just remind you 105 00:04:48,337 --> 00:04:50,159 to use Lenz's law instead of putting 106 00:04:50,159 --> 00:04:51,982 a kind of bizarre negative sign there. 107 00:04:51,982 --> 00:04:53,619 And the negative sign actually does make sense 108 00:04:53,619 --> 00:04:56,574 if you were, if you were doing kind of the 109 00:04:56,574 --> 00:04:58,947 using the vectors here and taking the, 110 00:04:58,947 --> 00:05:00,607 and using a little bit of the, 111 00:05:00,607 --> 00:05:03,405 well, doing more sophisticated mathematics 112 00:05:03,405 --> 00:05:05,831 but this is just saying that the voltage induced 113 00:05:05,831 --> 00:05:08,002 is going to be in a direction so to 114 00:05:08,002 --> 00:05:11,004 induce a current whose, whose induced 115 00:05:11,004 --> 00:05:13,275 magnetic field will go in the direction, 116 00:05:13,275 --> 00:05:15,746 will counteract the change in flux, 117 00:05:15,746 --> 00:05:17,510 so that's just Lenz's law there. 118 00:05:17,510 --> 00:05:19,903 So the real key here, at least for this example 119 00:05:19,903 --> 00:05:22,142 is to find our change in flux over change in time 120 00:05:22,142 --> 00:05:25,068 or our average, our average rate of change in flux 121 00:05:25,068 --> 00:05:26,600 and what is this going to be? 122 00:05:26,600 --> 00:05:29,769 Well this is going to be 20 tesla meters squared, 123 00:05:29,769 --> 00:05:34,312 20 tesla meters squared, that was our change in flux 124 00:05:34,762 --> 00:05:37,362 right over there divided by our change in time, 125 00:05:37,362 --> 00:05:41,008 which is four seconds, over four seconds, 126 00:05:41,008 --> 00:05:42,829 which is going to be equal to, and I'll, 127 00:05:42,829 --> 00:05:45,295 I could throw that negative there if we want to, 128 00:05:45,295 --> 00:05:49,475 that negative 20 divided by four is five, 129 00:05:50,445 --> 00:05:55,228 five tesla meters squared or square meters 130 00:05:55,228 --> 00:05:59,904 per second and this actually turns out to be a volt, 131 00:05:59,904 --> 00:06:02,576 so we could say this is negative five volts, 132 00:06:02,576 --> 00:06:05,991 negative, negative five volts, 133 00:06:07,511 --> 00:06:09,462 negative five volts. 134 00:06:09,462 --> 00:06:13,223 So if you have a voltage of, well let's just say five volts, 135 00:06:13,223 --> 00:06:15,266 we can think about the negative later, 136 00:06:15,266 --> 00:06:18,545 if you have a voltage of five volts across a, 137 00:06:18,545 --> 00:06:22,162 across a circuit that has a resistance of two ohms 138 00:06:22,162 --> 00:06:25,529 what is the current, what is the current going to be? 139 00:06:25,529 --> 00:06:27,665 Well we just have to remind ourselves 140 00:06:27,665 --> 00:06:30,288 V is equal to I-R 141 00:06:30,288 --> 00:06:32,842 or voltage is equal to the current divided by 142 00:06:32,842 --> 00:06:34,584 the resistance, or voltage is equal to 143 00:06:34,584 --> 00:06:37,444 the current times the resistance or you could say 144 00:06:37,444 --> 00:06:41,151 that the current, the current is equal to 145 00:06:41,151 --> 00:06:43,461 the voltage divided by the resistance. 146 00:06:43,461 --> 00:06:46,351 So in this case the current, the current induced 147 00:06:46,351 --> 00:06:47,570 is going to be the voltage and I'm just going to 148 00:06:47,570 --> 00:06:49,172 focus on its absolute value now, we can 149 00:06:49,172 --> 00:06:51,053 think about its direction in a second. 150 00:06:51,053 --> 00:06:53,132 It's going to be its voltage, five volts 151 00:06:53,132 --> 00:06:57,032 divided by the resistance, so two ohms, 152 00:06:57,032 --> 00:07:01,757 two ohms, which is going to be equal to, 153 00:07:01,757 --> 00:07:06,418 this is going to be equal to 2.5, 2.5 154 00:07:06,418 --> 00:07:10,974 amperes, 2.5 amperes. 155 00:07:11,914 --> 00:07:13,865 So we now know the magnitude of the current 156 00:07:13,865 --> 00:07:15,316 that's going to be induced while we 157 00:07:15,316 --> 00:07:16,744 have this change in flux, remember this 158 00:07:16,744 --> 00:07:18,961 is going to happen while, over the course 159 00:07:18,961 --> 00:07:21,724 of those four seconds, as we have this rate of change 160 00:07:21,724 --> 00:07:24,139 of flux, this average rate of change of flux, 161 00:07:24,139 --> 00:07:26,716 which we'll assume is the actual rate of change of flux, 162 00:07:26,716 --> 00:07:30,257 we're assuming that it's changing at a constant rate 163 00:07:30,257 --> 00:07:32,777 and so while it is changing we were just able to figure out 164 00:07:32,777 --> 00:07:36,398 that it would induce a current of 2.5 amperes. 165 00:07:36,398 --> 00:07:37,954 Now the next question we should ask ourselves 166 00:07:37,954 --> 00:07:40,020 and this is where this little negative comes in, 167 00:07:40,020 --> 00:07:43,015 is a reminder for us to use Lenz's law is, well 168 00:07:43,015 --> 00:07:45,256 which direction is that current going to go in? 169 00:07:45,256 --> 00:07:48,135 Is it going to go in, let me pick two orientations, 170 00:07:48,135 --> 00:07:52,279 is it going to go in a, is it going to go in a, 171 00:07:52,279 --> 00:07:56,853 in a clockwise direction, is it going to go that way 172 00:07:56,853 --> 00:07:59,451 over the course of this change in flux 173 00:07:59,451 --> 00:08:01,450 or is it going to go in a counterclockwise direction, 174 00:08:01,450 --> 00:08:03,609 is it going to go that way? 175 00:08:03,609 --> 00:08:05,316 And to think about that we just have to use 176 00:08:05,316 --> 00:08:07,615 the right hand rule, take our right hand, 177 00:08:07,615 --> 00:08:08,729 point our thumb in the direction 178 00:08:08,729 --> 00:08:12,850 of the proposed direction of the current 179 00:08:12,850 --> 00:08:15,636 and so if we went with this one, our right hand, 180 00:08:15,636 --> 00:08:17,169 our right hand would look like this, 181 00:08:17,169 --> 00:08:19,758 I'm literally taking my left hand out and-- 182 00:08:19,758 --> 00:08:22,753 I mean my right hand out and I'm drawing it 183 00:08:22,753 --> 00:08:25,411 and I'm looking at it to think about what would happen, 184 00:08:25,411 --> 00:08:27,954 so that's my right hand so if I use the right hand 185 00:08:27,954 --> 00:08:29,718 if the current went in this direction then it would 186 00:08:29,718 --> 00:08:33,539 induce a magnetic field that went, 187 00:08:33,539 --> 00:08:36,022 that went like this and so if the current went 188 00:08:36,022 --> 00:08:38,448 in this direction the magnetic field it induces 189 00:08:38,448 --> 00:08:42,510 inside the surface would only reinforce the change in flux 190 00:08:42,510 --> 00:08:45,541 so it would only add to the flux so, and it's going 191 00:08:45,541 --> 00:08:47,445 in the same direction as the change in flux, 192 00:08:47,445 --> 00:08:49,629 which would just keep us, you know as we 193 00:08:49,629 --> 00:08:51,985 talked about in the Lenz's law video, 194 00:08:51,985 --> 00:08:54,921 that would turn into just this source of energy 195 00:08:54,921 --> 00:08:56,991 that comes out of nowhere and defies 196 00:08:56,991 --> 00:08:59,391 the law of conservation of energy so this 197 00:08:59,391 --> 00:09:01,632 absolutely not, is not going to be the direction 198 00:09:01,632 --> 00:09:03,129 and so we know that the direction 199 00:09:03,129 --> 00:09:06,391 is going to be in a clockwise one. 200 00:09:06,391 --> 00:09:10,049 So the current, the 2.5 ampere current 201 00:09:10,049 --> 00:09:14,936 is going to flow, is going to flow like that, 202 00:09:14,936 --> 00:09:15,982 and we're done! 203 00:09:15,982 --> 00:09:17,768 By thinking about our change in flux 204 00:09:17,768 --> 00:09:20,811 and how long it's taking us, we were able to figure out 205 00:09:20,811 --> 00:09:22,873 not only the magnitude of the current, we were able 206 00:09:22,873 --> 00:09:24,826 to figure out the orientation of the direction 207 00:09:24,826 --> 00:00:00,000 that it's actually going to flow in.