1 00:00:00,626 --> 00:00:01,896 - [Instructor] So if you've ever run current 2 00:00:01,896 --> 00:00:03,938 through a resistor, you might've noticed that 3 00:00:03,938 --> 00:00:05,117 that resistor warms up. 4 00:00:05,117 --> 00:00:08,135 And in fact if you run too much current through the resistor 5 00:00:08,135 --> 00:00:09,877 that resistor can get so hot it can burn you 6 00:00:09,877 --> 00:00:11,552 when you touch it, so you have to be careful. 7 00:00:11,552 --> 00:00:13,058 So what I'm saying is when you have current 8 00:00:13,058 --> 00:00:15,745 flowing through a resistor it warms up, 9 00:00:15,745 --> 00:00:17,121 and we want to explain why. 10 00:00:17,121 --> 00:00:20,511 Conceptually, why does current moving through a resistor 11 00:00:20,511 --> 00:00:23,141 heat it up, and is there way a to calculate 12 00:00:23,141 --> 00:00:25,915 exactly how much that current would heat up that resistor 13 00:00:25,915 --> 00:00:27,771 over a given amount of time? 14 00:00:27,771 --> 00:00:29,014 There is a way to calculate it. 15 00:00:29,014 --> 00:00:30,551 We'll derive that in this video. 16 00:00:30,551 --> 00:00:32,716 But first we should just explain conceptually 17 00:00:32,716 --> 00:00:35,506 why is it that current moving through a resistor 18 00:00:35,506 --> 00:00:36,742 heats up the resistor. 19 00:00:36,742 --> 00:00:38,907 And so we'll explain that with this current. 20 00:00:38,907 --> 00:00:40,495 Now the way I've got it drawn here, 21 00:00:40,495 --> 00:00:42,692 notice that I've got these positive charges. 22 00:00:42,692 --> 00:00:45,489 Positive charges don't actually move through a wire, 23 00:00:45,489 --> 00:00:47,700 but physicists always pretend like they do 24 00:00:47,700 --> 00:00:50,099 because positives moving one way through a wire 25 00:00:50,099 --> 00:00:52,310 is equivalent to negatives moving the other way 26 00:00:52,310 --> 00:00:55,219 through the wire, and if you use the positive description, 27 00:00:55,219 --> 00:00:56,961 even though it's technically incorrect, 28 00:00:56,961 --> 00:00:59,502 you don't have to deal with all those negative signs. 29 00:00:59,502 --> 00:01:01,440 It's easier to deal with, and it's equivalent, 30 00:01:01,440 --> 00:01:02,928 so we may as well use it. 31 00:01:02,928 --> 00:01:04,400 But you can go through this whole description 32 00:01:04,400 --> 00:01:06,614 with negative charges moving the other way, 33 00:01:06,614 --> 00:01:07,903 and it works just as well. 34 00:01:07,903 --> 00:01:09,654 You just have to be very careful with the negatives, 35 00:01:09,654 --> 00:01:12,094 and it kind of obscures the conceptual meaning 36 00:01:12,094 --> 00:01:14,303 'cause it's hiding behind a bunch of negative signs. 37 00:01:14,303 --> 00:01:16,040 So we'll just use these positive charges, 38 00:01:16,040 --> 00:01:18,300 but know that it's really negatives going the other way. 39 00:01:18,300 --> 00:01:20,680 So why do positive charges flowing through a resistor 40 00:01:20,680 --> 00:01:22,114 cause that resistor to heat up? 41 00:01:22,114 --> 00:01:22,947 Well here's why. 42 00:01:22,947 --> 00:01:25,370 So we know that when current flows through a resistor, 43 00:01:25,370 --> 00:01:27,838 there's a voltage across that resistor. 44 00:01:27,838 --> 00:01:31,263 In other words, between this point here and this point here, 45 00:01:31,263 --> 00:01:33,479 there's a difference in electrical potential. 46 00:01:33,479 --> 00:01:35,947 So there's a voltage across that resistor. 47 00:01:35,947 --> 00:01:37,886 Technically I'm gonna call it delta V 48 00:01:37,886 --> 00:01:40,179 'cause it's a difference in electric potential. 49 00:01:40,179 --> 00:01:43,070 In other words, V on this side, the electric potential 50 00:01:43,070 --> 00:01:44,823 on this first side of the resistor 51 00:01:44,823 --> 00:01:47,029 is gonna have a different value from the electric potential 52 00:01:47,029 --> 00:01:49,152 on this second side of the resistor. 53 00:01:49,152 --> 00:01:50,673 So why does this matter? 54 00:01:50,673 --> 00:01:53,867 It matters 'cause this final side where the charges end up, 55 00:01:53,867 --> 00:01:56,332 is gonna have a lower electrical potential 56 00:01:56,332 --> 00:01:57,431 than the beginning. 57 00:01:57,431 --> 00:01:59,891 So these positive charges are gonna be moving 58 00:01:59,891 --> 00:02:03,112 from a high potential region to a low potential region. 59 00:02:03,112 --> 00:02:04,622 And that means they're gonna be changing 60 00:02:04,622 --> 00:02:06,615 their potential energy, so these charges 61 00:02:06,615 --> 00:02:08,193 have electric potential energy, 62 00:02:08,193 --> 00:02:11,007 and if they go from a region of higher electric potential 63 00:02:11,008 --> 00:02:13,029 to a region of lower electric potential, 64 00:02:13,029 --> 00:02:15,599 they've started with more potential energy, 65 00:02:15,599 --> 00:02:18,450 electrical potential energy than they end with. 66 00:02:18,450 --> 00:02:21,118 So they're decreasing their electric potential energy. 67 00:02:21,118 --> 00:02:24,203 And in case that's confusing, remember that the definition 68 00:02:24,203 --> 00:02:25,990 of electric potential is the amount 69 00:02:25,990 --> 00:02:29,235 of electric potential energy per charge. 70 00:02:29,235 --> 00:02:31,557 So to just put a number in here so it's not so abstract, 71 00:02:31,557 --> 00:02:35,174 let's say V two was two joules per coulomb. 72 00:02:35,174 --> 00:02:37,362 That would mean for every coulomb of charge 73 00:02:37,362 --> 00:02:39,974 at that point V two, there'd be two joules 74 00:02:39,974 --> 00:02:41,676 of electric potential energy. 75 00:02:41,676 --> 00:02:44,276 And if V one is at a higher electric potential, 76 00:02:44,276 --> 00:02:46,411 maybe this is at six joules per coulomb, 77 00:02:46,411 --> 00:02:48,821 that would mean over here at this position V one, 78 00:02:48,821 --> 00:02:51,881 for every coulomb of charge there'd be six joules 79 00:02:51,881 --> 00:02:53,412 of electric potential energy. 80 00:02:53,412 --> 00:02:55,832 So as these charges move through the resistor, 81 00:02:55,832 --> 00:02:58,916 they're gonna be decreasing their electric potential energy, 82 00:02:58,916 --> 00:03:00,542 and so the obvious question is, 83 00:03:00,542 --> 00:03:02,055 well where does that energy go? 84 00:03:02,055 --> 00:03:03,906 If these charges are decreasing 85 00:03:03,906 --> 00:03:05,411 their electric potential energy, 86 00:03:05,411 --> 00:03:07,101 where's that potential energy going? 87 00:03:07,101 --> 00:03:10,113 My first guess is that they'd increase their kinetic energy 88 00:03:10,113 --> 00:03:13,265 because I'd remember then on Earth if you drop a ball 89 00:03:13,265 --> 00:03:15,884 and it decreases its potential energy, 90 00:03:15,884 --> 00:03:18,084 its gravitational potential energy, 91 00:03:18,084 --> 00:03:19,690 we know that when it decreases 92 00:03:19,690 --> 00:03:22,169 its gravitational potential energy and falls down, 93 00:03:22,169 --> 00:03:25,135 it increases its kinetic energy, it just speeds up. 94 00:03:25,135 --> 00:03:28,765 So the decrease in gravitational potential energy 95 00:03:28,765 --> 00:03:30,649 just corresponds to an increase 96 00:03:30,649 --> 00:03:32,121 in kinetic energy of that object. 97 00:03:32,121 --> 00:03:34,066 And so maybe that's happening over here. 98 00:03:34,066 --> 00:03:36,688 Maybe as these charges lose potential energy, 99 00:03:36,688 --> 00:03:38,633 they speed up, but that can't happen. 100 00:03:38,633 --> 00:03:41,589 Remember the current on one side of a resistor 101 00:03:41,589 --> 00:03:43,995 has to be the same as the current on the other side. 102 00:03:43,995 --> 00:03:45,844 These charges don't speed up. 103 00:03:45,844 --> 00:03:48,421 They're losing potential energy, but they don't speed up. 104 00:03:48,421 --> 00:03:50,102 This is a little counterintuitive. 105 00:03:50,102 --> 00:03:51,585 We're used to things speeding up 106 00:03:51,585 --> 00:03:52,955 when they lose potential energy, 107 00:03:52,955 --> 00:03:55,051 but these charges aren't going to speed up. 108 00:03:55,051 --> 00:03:58,305 What they do is they just heat up the resistor. 109 00:03:58,305 --> 00:04:00,871 So as these charges fly through this resistor, 110 00:04:00,871 --> 00:04:02,673 they strike the atoms and molecules 111 00:04:02,673 --> 00:04:05,550 in this lattice structure of this solid. 112 00:04:05,550 --> 00:04:08,431 So this resistor's made out of atoms and molecules, 113 00:04:08,431 --> 00:04:10,347 and as these charges flow through here, 114 00:04:10,347 --> 00:04:12,599 and again it's really electrons flowing the other way, 115 00:04:12,599 --> 00:04:14,939 but as the charges flow through, same idea. 116 00:04:14,939 --> 00:04:16,748 They strike the atoms and molecules, 117 00:04:16,748 --> 00:04:18,712 they transfer energy into them. 118 00:04:18,712 --> 00:04:21,404 And as they pass through in their wake, 119 00:04:21,404 --> 00:04:23,727 they leave a resistor that's hotter, 120 00:04:23,727 --> 00:04:24,673 at a higher temperature. 121 00:04:24,673 --> 00:04:27,297 Which means these atoms are jiggling around 122 00:04:27,297 --> 00:04:28,979 more than they were before. 123 00:04:28,979 --> 00:04:31,204 And since they're oscillating more than they were before, 124 00:04:31,204 --> 00:04:32,951 they're jiggling, they've got more energy, 125 00:04:32,951 --> 00:04:35,348 the temperature of this resistor increases. 126 00:04:35,348 --> 00:04:37,479 So these charges, rather than keeping all the energy 127 00:04:37,479 --> 00:04:39,871 for themselves, they actually just spread it out 128 00:04:39,871 --> 00:04:42,135 over that resistor as they pass through, 129 00:04:42,135 --> 00:04:44,085 and they spread it out in the form of heat, 130 00:04:44,085 --> 00:04:45,194 or thermal energy. 131 00:04:45,194 --> 00:04:47,359 And they emerge with basically the same kinetic energy 132 00:04:47,359 --> 00:04:50,105 that they started with, so this change in potential energy, 133 00:04:50,105 --> 00:04:53,001 electrical potential energy, corresponds to an increase 134 00:04:53,001 --> 00:04:54,980 in thermal energy of this resistor. 135 00:04:54,980 --> 00:04:56,866 And that's why the resistors heat up. 136 00:04:56,866 --> 00:04:59,126 But is there a way to calculate exactly how much 137 00:04:59,126 --> 00:05:00,554 this resistor will heat up? 138 00:05:00,554 --> 00:05:03,335 How much energy it's gonna gain per time? 139 00:05:03,335 --> 00:05:06,184 There is, we just have to use the definition of power. 140 00:05:06,184 --> 00:05:09,151 So we know the definition of power is the work per time, 141 00:05:09,151 --> 00:05:11,948 or since work is the change in energy, 142 00:05:11,948 --> 00:05:14,352 or the energy transferred, we can just write this 143 00:05:14,352 --> 00:05:18,312 as the amount of energy this resistor's gaining per time. 144 00:05:18,312 --> 00:05:21,156 What we want is a formula that tells us how much energy 145 00:05:21,156 --> 00:05:24,819 are these charges depositing in the resistor per time. 146 00:05:24,819 --> 00:05:27,011 Well this energy gained by the resistor 147 00:05:27,011 --> 00:05:29,892 is coming from the loss of potential energy 148 00:05:29,892 --> 00:05:31,355 of these charges. 149 00:05:31,355 --> 00:05:33,689 So these charges are losing potential energy. 150 00:05:33,689 --> 00:05:35,547 They're losing electric potential energy, 151 00:05:35,547 --> 00:05:36,967 and that electric potential energy's 152 00:05:36,967 --> 00:05:38,211 turning into thermal energy, 153 00:05:38,211 --> 00:05:40,311 so the thermal energy this resistor gains 154 00:05:40,311 --> 00:05:44,092 is just equal to the amount of electric potential energy 155 00:05:44,092 --> 00:05:45,350 that these charges lose. 156 00:05:45,350 --> 00:05:46,698 So I can just rewrite this. 157 00:05:46,698 --> 00:05:48,805 I can just say that the power is gonna be equal 158 00:05:48,805 --> 00:05:51,886 to the change in electrical potential energy 159 00:05:51,886 --> 00:05:53,913 of these charges per time. 160 00:05:53,913 --> 00:05:55,806 And so I'll just continue down here. 161 00:05:55,806 --> 00:05:58,260 Power's gonna be equal to, how do we find the change 162 00:05:58,260 --> 00:05:59,665 in electric potential energy? 163 00:05:59,665 --> 00:06:01,691 Well remember, potential is defined 164 00:06:01,691 --> 00:06:04,235 to be the potential energy per charge. 165 00:06:04,235 --> 00:06:06,539 So that means the electric potential energy 166 00:06:06,539 --> 00:06:09,794 is just the charge times the electric potential. 167 00:06:09,794 --> 00:06:12,147 So if I want to find delta U, I can just say that 168 00:06:12,147 --> 00:06:16,278 that's gonna be U when they emerge, U two minus U one, 169 00:06:16,278 --> 00:06:18,689 and this is a way we can find the U values. 170 00:06:18,689 --> 00:06:21,841 So the U at two, since it's Q times V, 171 00:06:21,841 --> 00:06:25,358 is just gonna be the charge at two times V two. 172 00:06:25,358 --> 00:06:27,786 And then the U at one, so we'll do minus the U at one, 173 00:06:27,786 --> 00:06:29,078 is the charge at one. 174 00:06:29,078 --> 00:06:30,177 But that's the same charge. 175 00:06:30,177 --> 00:06:33,137 Whatever charge enters this resistor has to exit it, 176 00:06:33,137 --> 00:06:36,596 so it'd be the charge at one times the V at one. 177 00:06:36,596 --> 00:06:39,512 This is the change in electric potential energy. 178 00:06:39,512 --> 00:06:40,667 So I could rewrite this. 179 00:06:40,667 --> 00:06:43,040 I can pull out a common factor of Q 180 00:06:43,040 --> 00:06:44,468 in this expression right here. 181 00:06:44,468 --> 00:06:46,856 Then we get that the power's going to be equal to 182 00:06:46,856 --> 00:06:51,218 this common factor of Q times V two minus V one. 183 00:06:51,218 --> 00:06:54,088 So that's delta U, and that's what we're plugging in 184 00:06:54,088 --> 00:06:55,237 right here for delta U. 185 00:06:55,237 --> 00:06:58,604 Delta U is just the difference in these Q times V values. 186 00:06:58,604 --> 00:07:00,143 And then we still have to divide by time 187 00:07:00,143 --> 00:07:01,327 since we're talking about a power. 188 00:07:01,327 --> 00:07:03,183 But what is V two minus V one? 189 00:07:03,183 --> 00:07:05,873 That's simply the voltage across this resistor. 190 00:07:05,873 --> 00:07:08,532 Delta V is V two minus V one. 191 00:07:08,532 --> 00:07:09,638 So I could rewrite this. 192 00:07:09,638 --> 00:07:13,054 I could just say that this is Q times delta V, the voltage, 193 00:07:13,054 --> 00:07:15,829 across that resistor, divided by the time it took 194 00:07:15,829 --> 00:07:17,849 for the charge to pass through that resistor. 195 00:07:17,849 --> 00:07:20,206 And now something magical happens, check this out. 196 00:07:20,206 --> 00:07:23,699 So we got power equals, I've got charge the past 197 00:07:23,699 --> 00:07:27,085 through the resistor, divided by the time that it took 198 00:07:27,085 --> 00:07:29,450 for that charge to pass through the resistor, 199 00:07:29,450 --> 00:07:32,217 but charge per time is just the definition of current. 200 00:07:32,217 --> 00:07:33,514 So we get this beautiful formula. 201 00:07:33,514 --> 00:07:35,667 If I just factor out this Q over T, 202 00:07:35,667 --> 00:07:38,709 I get Q over T times delta V, 203 00:07:38,709 --> 00:07:40,085 the voltage across the resistor. 204 00:07:40,085 --> 00:07:41,931 But Q over T is just the current, 205 00:07:41,931 --> 00:07:44,351 so I get that the power is going to equal 206 00:07:44,351 --> 00:07:46,319 the current through that resistor 207 00:07:46,319 --> 00:07:48,981 times the voltage across that resistor. 208 00:07:48,981 --> 00:07:51,635 And this is the formula for electrical power. 209 00:07:51,635 --> 00:07:54,400 This tells you how many joules of thermal energy 210 00:07:54,400 --> 00:07:57,349 are being created in that resistor per time. 211 00:07:57,349 --> 00:07:59,491 So the units are joules per second, 212 00:07:59,491 --> 00:08:00,756 or in other words, watts. 213 00:08:00,756 --> 00:08:01,852 Because what this is telling you 214 00:08:01,852 --> 00:08:04,655 is the amount of thermal energy generated per second. 215 00:08:04,655 --> 00:08:07,389 If the power value came out to be 20 watts, 216 00:08:07,389 --> 00:08:09,455 that would mean there'd be 20 joules 217 00:08:09,455 --> 00:08:11,987 of thermal energy generated every second, 218 00:08:11,987 --> 00:08:13,454 which is a really useful thing to know. 219 00:08:13,454 --> 00:08:15,899 This formula's extremely useful when you want to figure out 220 00:08:15,899 --> 00:08:18,753 how much energy's gonna be used by a light bulb, 221 00:08:18,753 --> 00:08:21,728 or a toaster, or a TV, or whatever electronic device 222 00:08:21,728 --> 00:08:22,561 you want to use. 223 00:08:22,561 --> 00:08:25,019 This tells you how much energy that it's going to turn 224 00:08:25,019 --> 00:08:28,047 into either thermal energy, or light, or sound, 225 00:08:28,047 --> 00:08:30,463 or whatever other kind of energy that it's converting 226 00:08:30,463 --> 00:08:32,312 that electric potential energy into. 227 00:08:32,312 --> 00:08:34,052 'Cause notice, we never really assumed 228 00:08:34,052 --> 00:08:35,196 that this was thermal energy. 229 00:08:35,196 --> 00:08:37,691 We just called it E, and then we said that whatever energy 230 00:08:37,691 --> 00:08:41,455 got transformed came from the change in potential energy 231 00:08:41,455 --> 00:08:43,980 of those charges, and that's always gonna be true. 232 00:08:43,980 --> 00:08:46,238 It's always gonna be electric potential energy 233 00:08:46,238 --> 00:08:48,495 converting into something, whether it's heat, 234 00:08:48,495 --> 00:08:50,134 or light, or sound. 235 00:08:50,134 --> 00:08:52,581 So this doesn't just work for resistors. 236 00:08:52,581 --> 00:08:55,045 It works for almost all electrical components 237 00:08:55,045 --> 00:08:56,934 that turn electric potential energy 238 00:08:56,934 --> 00:08:58,539 into some other kind of energy. 239 00:08:58,539 --> 00:09:00,276 And you could rewrite this in different forms. 240 00:09:00,276 --> 00:09:01,800 Sometimes you'll see it a different way. 241 00:09:01,800 --> 00:09:04,165 There's this form, and then I'm just gonna copy this. 242 00:09:04,165 --> 00:09:05,875 I'm gonna show you there's a couple other forms 243 00:09:05,875 --> 00:09:06,933 you could see this in. 244 00:09:06,933 --> 00:09:09,475 Let me clean this up, we'll put this formula down here. 245 00:09:09,475 --> 00:09:12,159 This gives us the power, but we know from Ohm's law 246 00:09:12,159 --> 00:09:15,369 that delta V is equal to I R. 247 00:09:15,369 --> 00:09:17,209 So if both of these formulas are true, 248 00:09:17,209 --> 00:09:19,646 I can plug delta V as I R. 249 00:09:19,646 --> 00:09:23,699 So I could take this I R, I could plug it in for delta V, 250 00:09:23,699 --> 00:09:25,301 and I get an alternate expression 251 00:09:25,301 --> 00:09:28,416 for the power used by electrical component. 252 00:09:28,416 --> 00:09:31,975 I get that the power's gonna be I times I times R, 253 00:09:31,975 --> 00:09:34,647 which is just I squared times R. 254 00:09:34,647 --> 00:09:37,625 So this one might be more useful if you've got a situation 255 00:09:37,625 --> 00:09:39,141 where you don't know the voltage, 256 00:09:39,141 --> 00:09:41,664 but you happen to know the current and the resistance. 257 00:09:41,664 --> 00:09:42,497 But there's one more. 258 00:09:42,497 --> 00:09:44,793 I could've solved this Ohm's law formula for I, 259 00:09:44,793 --> 00:09:48,230 and I'd get that I equals delta V over R. 260 00:09:48,230 --> 00:09:52,414 And then if I plug delta V over R in for I up here, 261 00:09:52,414 --> 00:09:54,493 I'd get an alternate expression for the power. 262 00:09:54,493 --> 00:09:58,702 I'd get that the power equals delta V over R times delta V. 263 00:09:58,702 --> 00:10:00,866 It's just gonna be delta V squared. 264 00:10:00,866 --> 00:10:03,158 The voltage across that resistor squared 265 00:10:03,158 --> 00:10:05,824 divided by the resistance of that resistor. 266 00:10:05,824 --> 00:10:07,508 And this formula might be more useful 267 00:10:07,508 --> 00:10:10,045 if you know the voltages and the resistance, 268 00:10:10,045 --> 00:10:11,467 but you don't know the current. 269 00:10:11,467 --> 00:10:12,957 So depending on what you know, 270 00:10:12,957 --> 00:10:15,229 these can get you the power used by a resistor. 271 00:10:15,229 --> 00:10:16,551 They're all equivalent. 272 00:10:16,551 --> 00:10:19,112 They will all give you the correct and the same power. 273 00:10:19,112 --> 00:10:21,502 It's just a matter of what's more convenient 274 00:10:21,502 --> 00:10:23,244 for the actual problem that you're dealing with. 275 00:10:23,244 --> 00:10:25,968 So recapping, when current passes through a resistor, 276 00:10:25,968 --> 00:10:29,457 it converts electrical potential energy into thermal energy, 277 00:10:29,457 --> 00:10:30,888 and you can calculate the amount 278 00:10:30,888 --> 00:10:33,699 of electrical potential energy converted per second 279 00:10:33,699 --> 00:10:35,825 using current times voltage, 280 00:10:35,825 --> 00:10:37,827 current squared times the resistance, 281 00:10:37,827 --> 00:00:00,000 or voltage squared divided by the resistance.