1 00:00:00,227 --> 00:00:02,122 - [Narrator] So here's something that used to confuse me. 2 00:00:02,122 --> 00:00:03,504 If you had two charges, 3 00:00:03,504 --> 00:00:06,120 and we'll keep these straight by giving them a name. 4 00:00:06,120 --> 00:00:09,766 We'll call this one Q1 and I'll call this one Q2. 5 00:00:09,766 --> 00:00:12,182 If you've got these two charges sitting next to each other, 6 00:00:12,182 --> 00:00:15,206 and you let go of them, they're gonna fly apart 7 00:00:15,206 --> 00:00:16,507 because they repel each other. 8 00:00:16,507 --> 00:00:19,199 Like charges repel, so the Q2's gonna get pushed 9 00:00:19,199 --> 00:00:20,314 to the right, 10 00:00:20,314 --> 00:00:22,155 and the Q1's gonna get pushed to the left. 11 00:00:22,155 --> 00:00:24,097 They're gonna start gaining kinetic energy. 12 00:00:24,097 --> 00:00:25,765 They're gonna start speeding up. 13 00:00:25,765 --> 00:00:28,110 But if these charges are gaining kinetic energy, 14 00:00:28,110 --> 00:00:30,352 where is that energy coming from? 15 00:00:30,352 --> 00:00:32,610 I mean, if you believe in conservation of energy, 16 00:00:32,610 --> 00:00:34,855 this energy had to come from somewhere. 17 00:00:34,855 --> 00:00:37,039 So where is this energy coming from? 18 00:00:37,039 --> 00:00:39,807 What is the source of this kinetic energy? 19 00:00:39,807 --> 00:00:43,184 Well, the source is the electrical potential energy. 20 00:00:43,184 --> 00:00:45,886 We would say that electrical potential energy 21 00:00:45,886 --> 00:00:48,217 is turning into kinetic energy. 22 00:00:48,217 --> 00:00:49,676 So originally in this system, 23 00:00:49,676 --> 00:00:51,300 there was electrical potential energy, 24 00:00:51,300 --> 00:00:53,879 and then there was less electrical potential energy, 25 00:00:53,879 --> 00:00:55,450 but more kinetic energy. 26 00:00:55,450 --> 00:00:58,602 So as the electrical potential energy decreases, 27 00:00:58,602 --> 00:01:00,596 the kinetic energy increases. 28 00:01:00,596 --> 00:01:02,535 But the total energy in this system, 29 00:01:02,535 --> 00:01:04,843 this two-charge system, would remain the same. 30 00:01:04,843 --> 00:01:06,708 So this is where that kinetic energy's coming from. 31 00:01:06,708 --> 00:01:08,775 It's coming from the electrical potential energy. 32 00:01:08,775 --> 00:01:10,776 And the letter that physicists typically choose 33 00:01:10,776 --> 00:01:13,727 to represent potential energies is a u. 34 00:01:13,727 --> 00:01:16,095 So why u for potential energy? 35 00:01:16,095 --> 00:01:16,928 I don't know. 36 00:01:16,928 --> 00:01:18,833 Like PE would've made sense, too, 37 00:01:18,833 --> 00:01:20,506 because that's the first two letters 38 00:01:20,506 --> 00:01:22,228 of the words potential energy. 39 00:01:22,228 --> 00:01:24,016 But more often you see it like this. 40 00:01:24,016 --> 00:01:25,310 We'll put a little subscript e 41 00:01:25,310 --> 00:01:26,966 so that we know we're talking about 42 00:01:26,966 --> 00:01:28,678 electrical potential energy 43 00:01:28,678 --> 00:01:31,308 and not gravitational potential energy, say. 44 00:01:31,308 --> 00:01:32,957 So that's all fine and good. 45 00:01:32,957 --> 00:01:36,051 We've got potential energy turning into kinetic energy. 46 00:01:36,051 --> 00:01:37,942 Well, we know the formula for the kinetic energy 47 00:01:37,942 --> 00:01:39,191 of these charges. 48 00:01:39,191 --> 00:01:41,197 We can find the kinetic energy of these charges 49 00:01:41,197 --> 00:01:44,218 by taking one half the mass of one of the charges 50 00:01:44,218 --> 00:01:46,883 times the speed of one of those charges squared. 51 00:01:46,883 --> 00:01:49,813 What's the formula to find the electrical potential energy 52 00:01:49,813 --> 00:01:51,289 between these charges? 53 00:01:51,289 --> 00:01:53,052 So if you've got two or more charges 54 00:01:53,052 --> 00:01:54,376 sitting next to each other, 55 00:01:54,376 --> 00:01:56,513 Is there a nice formula to figure out 56 00:01:56,513 --> 00:01:58,590 how much electrical potential energy there is 57 00:01:58,590 --> 00:01:59,706 in that system? 58 00:01:59,706 --> 00:02:01,535 Well, the good news is, there is. 59 00:02:01,535 --> 00:02:02,672 There's a really nice formula 60 00:02:02,672 --> 00:02:04,170 that will let you figure this out. 61 00:02:04,170 --> 00:02:07,473 The bad news is, to derive it requires calculus. 62 00:02:07,473 --> 00:02:10,065 So I'm not gonna do the calculus derivation in this video. 63 00:02:10,065 --> 00:02:11,740 There's already a video on this. 64 00:02:11,740 --> 00:02:13,630 We'll put a link to that so you can find that. 65 00:02:13,630 --> 00:02:15,662 But in this video, I'm just gonna quote the result, 66 00:02:15,662 --> 00:02:17,028 show you how to use it, 67 00:02:17,028 --> 00:02:20,092 give you a tour so to speak of this formula. 68 00:02:20,092 --> 00:02:21,956 And the formula looks like this. 69 00:02:21,956 --> 00:02:23,547 So to find the electrical potential energy 70 00:02:23,547 --> 00:02:26,467 between two charges, we take K, the electric constant, 71 00:02:26,467 --> 00:02:28,690 multiplied by one of the charges, 72 00:02:28,690 --> 00:02:31,046 and then multiplied by the other charge, 73 00:02:31,046 --> 00:02:32,949 and then we divide by the distance 74 00:02:32,949 --> 00:02:34,455 between those two charges. 75 00:02:34,455 --> 00:02:35,806 We'll call that r. 76 00:02:35,806 --> 00:02:37,723 So this is the center to center distance. 77 00:02:37,723 --> 00:02:39,487 It would be from the center of one charge 78 00:02:39,487 --> 00:02:41,409 to the center of the other. 79 00:02:41,409 --> 00:02:44,149 That distance would be r, and we don't square it. 80 00:02:44,149 --> 00:02:45,745 So in a lot of these formulas, 81 00:02:45,745 --> 00:02:48,915 for instance Coulomb's law, the r is always squared. 82 00:02:48,915 --> 00:02:51,329 For electrical fields, the r is squared, 83 00:02:51,329 --> 00:02:53,901 but for potential energy, this r is not squared. 84 00:02:53,901 --> 00:02:56,671 Basically, to find this formula in this derivation, 85 00:02:56,671 --> 00:02:57,721 you do an integral. 86 00:02:57,721 --> 00:03:00,502 That integral turns the r squared into just an r 87 00:03:00,502 --> 00:03:01,419 on the bottom. 88 00:03:01,419 --> 00:03:02,555 So don't try to square this. 89 00:03:02,555 --> 00:03:03,921 It's just r this time. 90 00:03:03,921 --> 00:03:05,887 And that's it. That's the formula to find 91 00:03:05,887 --> 00:03:08,911 the electrical potential energy between two charges. 92 00:03:08,911 --> 00:03:10,855 And here's something that used to confuse me. 93 00:03:10,855 --> 00:03:13,589 I used to wonder, is this the electrical potential energy 94 00:03:13,589 --> 00:03:15,538 of that charge, Q1? 95 00:03:15,538 --> 00:03:18,917 Or is it the electrical potential energy of this charge, Q2? 96 00:03:18,917 --> 00:03:20,074 Well, the best way to think about this 97 00:03:20,074 --> 00:03:22,280 is that this is the electrical potential energy 98 00:03:22,280 --> 00:03:24,410 of the system of charges. 99 00:03:24,410 --> 00:03:25,722 So you need two of these charges 100 00:03:25,722 --> 00:03:27,724 to have potential energy at all. 101 00:03:27,724 --> 00:03:29,970 If you only had one, there would be no potential energy, 102 00:03:29,970 --> 00:03:32,839 so think of this potential energy as the potential energy 103 00:03:32,839 --> 00:03:35,220 that exists in this charge system. 104 00:03:35,220 --> 00:03:37,952 So since this is an electrical potential energy 105 00:03:37,952 --> 00:03:41,257 and all energy has units of joules if you're using SI units, 106 00:03:41,257 --> 00:03:43,141 this will also have units of joules. 107 00:03:43,141 --> 00:03:44,290 Something else that's important to know 108 00:03:44,290 --> 00:03:47,291 is that this electrical potential energy is a scalar. 109 00:03:47,291 --> 00:03:49,247 That is to say, it is not a vector. 110 00:03:49,247 --> 00:03:51,047 There's no direction of this energy. 111 00:03:51,047 --> 00:03:53,253 It's just a number with a unit that tells you 112 00:03:53,253 --> 00:03:55,551 how much potential energy is in that system. 113 00:03:55,551 --> 00:03:57,345 In other words, this is good news. 114 00:03:57,345 --> 00:03:58,593 When things are vectors, 115 00:03:58,593 --> 00:04:00,364 you have to break them into pieces. 116 00:04:00,364 --> 00:04:02,695 And potentially you've got component problems here, 117 00:04:02,695 --> 00:04:04,721 you got to figure out how much of that vector points right 118 00:04:04,721 --> 00:04:05,917 and how much points up. 119 00:04:05,917 --> 00:04:08,102 But that's not the case with electrical potential energy. 120 00:04:08,102 --> 00:04:09,771 There's no direction of this energy, 121 00:04:09,771 --> 00:04:12,682 so there will never be any components of this energy. 122 00:04:12,682 --> 00:04:15,143 It is simply just the electrical potential energy. 123 00:04:15,143 --> 00:04:16,298 So how do you use this formula? 124 00:04:16,298 --> 00:04:17,406 What do problems look like? 125 00:04:17,406 --> 00:04:19,912 Let's try a sample problem to give you some feel 126 00:04:19,913 --> 00:04:23,021 for how you might use this equation in a given problem. 127 00:04:23,021 --> 00:04:24,278 Okay, so for our sample problem, 128 00:04:24,278 --> 00:04:26,857 let's say we know the values of the charges. 129 00:04:26,857 --> 00:04:28,419 And let's say they start from rest, 130 00:04:28,419 --> 00:04:31,465 separated by a distance of three centimeters. 131 00:04:31,465 --> 00:04:33,967 And after you release them from rest, 132 00:04:33,967 --> 00:04:37,609 you let them fly to a distance 12 centimeters apart. 133 00:04:37,609 --> 00:04:39,025 And we need to know one more thing. 134 00:04:39,025 --> 00:04:41,277 We need to know the mass of each charge. 135 00:04:41,277 --> 00:04:44,784 So let's just say that each charge is one kilogram 136 00:04:44,784 --> 00:04:46,635 just to make the numbers come out nice. 137 00:04:46,635 --> 00:04:47,790 So the question we want to know is, 138 00:04:47,790 --> 00:04:50,977 how fast are these charges going to be moving 139 00:04:50,977 --> 00:04:54,013 once they've made it 12 centimeters away from each other? 140 00:04:54,013 --> 00:04:56,841 So the blue one here, Q1, is gonna be speeding to the left. 141 00:04:56,841 --> 00:04:58,774 Q2's gonna be speeding to the right. 142 00:04:58,774 --> 00:05:00,821 How fast are they gonna be moving? 143 00:05:00,821 --> 00:05:01,654 And to figure this out, 144 00:05:01,654 --> 00:05:03,165 we're gonna use conservation of energy. 145 00:05:03,165 --> 00:05:06,233 For our energy system, we'll include both charges, 146 00:05:06,233 --> 00:05:08,340 and we'll say that if we've included everything 147 00:05:08,340 --> 00:05:09,173 in our system, 148 00:05:09,173 --> 00:05:11,300 then the total initial energy of our system 149 00:05:11,300 --> 00:05:13,918 is gonna equal the total final energy of our system. 150 00:05:13,918 --> 00:05:16,119 What kind of energy did our system have initially? 151 00:05:16,119 --> 00:05:18,278 Well, the system started from rest initially, 152 00:05:18,278 --> 00:05:20,343 so there was no kinetic energy to start with. 153 00:05:20,343 --> 00:05:22,910 There would've only been electric potential energy 154 00:05:22,910 --> 00:05:23,856 to start with. 155 00:05:23,856 --> 00:05:25,429 So just call that u initial. 156 00:05:25,429 --> 00:05:27,608 And then that's gonna have to equal the final energy 157 00:05:27,608 --> 00:05:29,283 once they're 12 centimeters apart. 158 00:05:29,283 --> 00:05:31,313 So the farther apart, they're gonna have less 159 00:05:31,313 --> 00:05:33,512 electrical potential energy but they're still gonna have 160 00:05:33,512 --> 00:05:34,995 some potential energy. 161 00:05:34,995 --> 00:05:36,411 So we'll call that u final. 162 00:05:36,411 --> 00:05:38,007 And now they're gonna be moving. 163 00:05:38,007 --> 00:05:39,534 So since these charges are moving, 164 00:05:39,534 --> 00:05:40,776 they're gonna have kinetic energy. 165 00:05:40,776 --> 00:05:43,288 So plus the kinetic energy of our system. 166 00:05:43,288 --> 00:05:46,111 So we'll use our formula for electrical potential energy 167 00:05:46,111 --> 00:05:48,712 and we'll get that the initial electrical potential energy 168 00:05:48,712 --> 00:05:51,148 is gonna be nine times 10 to the ninth 169 00:05:51,148 --> 00:05:53,008 since that's the electric constant K 170 00:05:53,008 --> 00:05:55,121 multiplied by the charge of Q1. 171 00:05:55,121 --> 00:05:57,146 That's gonna be four microcoulombs. 172 00:05:57,146 --> 00:05:59,729 A micro is 10 to the negative sixth. 173 00:05:59,729 --> 00:06:02,626 So you gotta turn that into regular coulombs. 174 00:06:02,626 --> 00:06:06,150 And then multiplied by Q2, which is two microcoulombs. 175 00:06:06,150 --> 00:06:08,842 So that'd be two times 10 to the negative sixth 176 00:06:08,842 --> 00:06:10,393 divided by the distance. 177 00:06:10,393 --> 00:06:12,640 Well, this was the initial electrical potential energy 178 00:06:12,640 --> 00:06:15,107 so this would be the initial distance between them. 179 00:06:15,107 --> 00:06:17,731 That center to center distance was three centimeters, 180 00:06:17,731 --> 00:06:19,240 but I can't plug in three. 181 00:06:19,240 --> 00:06:20,381 This is in centimeters. 182 00:06:20,381 --> 00:06:22,729 If I want my units to be in joules, 183 00:06:22,729 --> 00:06:24,743 so that I get speeds in meters per second, 184 00:06:24,743 --> 00:06:26,624 I've got to convert this to meters, 185 00:06:26,624 --> 00:06:30,824 and three centimeters in meters is 0.03 meters. 186 00:06:30,824 --> 00:06:31,778 You divide by a hundred, 187 00:06:31,778 --> 00:06:34,139 because there's 100 centimeters in one meter. 188 00:06:34,139 --> 00:06:35,442 And I don't square this. 189 00:06:35,442 --> 00:06:37,787 The r in the bottom of here is not squared, 190 00:06:37,787 --> 00:06:39,186 so you don't square that r. 191 00:06:39,186 --> 00:06:40,903 So that's gonna be equal to 192 00:06:40,903 --> 00:06:42,338 it's gonna be equal to another term 193 00:06:42,338 --> 00:06:44,149 that looks just like this. 194 00:06:44,149 --> 00:06:45,680 So I'm gonna copy and paste that. 195 00:06:45,680 --> 00:06:47,891 The only difference is that now this is the final 196 00:06:47,891 --> 00:06:49,362 electrical potential energy. 197 00:06:49,362 --> 00:06:50,720 Well, the K value is the same. 198 00:06:50,720 --> 00:06:52,635 The value of each charge is the same. 199 00:06:52,635 --> 00:06:53,699 The only thing that's different is 200 00:06:53,699 --> 00:06:55,397 that after they've flown apart, 201 00:06:55,397 --> 00:06:57,320 they're no longer three centimeters apart, 202 00:06:57,320 --> 00:06:58,813 they're 12 centimeters apart. 203 00:06:58,813 --> 00:07:01,808 So we'll plug in 0.12 meters, 204 00:07:01,808 --> 00:07:04,610 since 12 centimeters is .12 meters. 205 00:07:04,610 --> 00:07:06,307 And then we have to add the kinetic energy. 206 00:07:06,307 --> 00:07:08,472 So I'm just gonna call this k for now. 207 00:07:08,472 --> 00:07:09,935 The total kinetic energy of the system 208 00:07:09,935 --> 00:07:12,274 after they've reached 12 centimeters. 209 00:07:12,274 --> 00:07:13,755 Well, if you calculate these terms, 210 00:07:13,755 --> 00:07:16,674 if you multiply all this out on the left-hand side, 211 00:07:16,674 --> 00:07:21,145 you get 2.4 joules of initial electrical potential energy. 212 00:07:21,145 --> 00:07:22,194 And that's gonna equal, 213 00:07:22,194 --> 00:07:24,267 if you calculate all of this in this term, 214 00:07:24,267 --> 00:07:26,624 multiply the charges, divide by .12 215 00:07:26,624 --> 00:07:28,801 and multiply by nine times 10 to the ninth, 216 00:07:28,801 --> 00:07:32,828 you get 0.6 joules of electrical potential energy 217 00:07:32,828 --> 00:07:34,519 after they're 12 centimeters apart 218 00:07:34,519 --> 00:07:36,481 plus the amount of kinetic energy in the system, 219 00:07:36,481 --> 00:07:38,978 so we can replace this kinetic energy of our system 220 00:07:38,978 --> 00:07:40,874 with the formula for kinetic energy, 221 00:07:40,874 --> 00:07:44,147 which is gonna be one half m-v squared. 222 00:07:44,147 --> 00:07:45,612 But here's the problem. 223 00:07:45,612 --> 00:07:47,929 Both of these charges are moving. 224 00:07:47,929 --> 00:07:49,258 So if we want to do this correctly, 225 00:07:49,258 --> 00:07:50,547 we're gonna have to take into account 226 00:07:50,547 --> 00:07:53,658 that both of these charges are gonna have kinetic energy, 227 00:07:53,658 --> 00:07:55,113 not just one of them. 228 00:07:55,113 --> 00:07:58,877 If I only put one half times one kilogram times v squared, 229 00:07:58,877 --> 00:08:00,076 I'd get the wrong answer 230 00:08:00,076 --> 00:08:02,325 because I would've neglected the fact that the other charge 231 00:08:02,325 --> 00:08:04,020 also had kinetic energy. 232 00:08:04,020 --> 00:08:05,157 So we could do one of two things. 233 00:08:05,157 --> 00:08:06,940 Since these masses are the same, 234 00:08:06,940 --> 00:08:08,477 they're gonna have the same speed, 235 00:08:08,477 --> 00:08:09,994 and that means we can write this mass here 236 00:08:09,994 --> 00:08:13,436 as two kilograms times the common speed squared 237 00:08:13,436 --> 00:08:17,068 or you could just write two terms, one for each charge. 238 00:08:17,068 --> 00:08:18,376 This is a little safer. 239 00:08:18,376 --> 00:08:19,285 I'm just gonna do that. 240 00:08:19,285 --> 00:08:21,064 Conceptually, it's a little easier to think about. 241 00:08:21,064 --> 00:08:22,149 Okay, so I solve this. 242 00:08:22,149 --> 00:08:25,149 2.4 minus .6 is gonna be 1.8 joules, 243 00:08:26,532 --> 00:08:30,146 and that's gonna equal one half times one kilogram 244 00:08:30,146 --> 00:08:33,225 times the speed of that second particle squared 245 00:08:33,225 --> 00:08:37,004 plus one half times one kilogram times the speed 246 00:08:37,004 --> 00:08:38,548 of the first particle squared. 247 00:08:38,548 --> 00:08:39,958 And here's where we have to make that argument. 248 00:08:39,958 --> 00:08:41,889 Since these have the same mass, 249 00:08:41,890 --> 00:08:44,316 they're gonna be moving with the same speed. 250 00:08:44,316 --> 00:08:46,844 One half v squared plus one half v squared 251 00:08:46,844 --> 00:08:49,096 which is really just v squared, 252 00:08:49,096 --> 00:08:51,445 because a half of v squared plus a half of v squared 253 00:08:51,445 --> 00:08:53,403 is a whole of v squared. 254 00:08:53,403 --> 00:08:55,197 Now if you're clever, you might be like, "Wait a minute. 255 00:08:55,197 --> 00:08:57,826 "This charge, even though it had the same mass, 256 00:08:57,826 --> 00:09:00,454 "it had more charge than this charge did. 257 00:09:00,454 --> 00:09:02,657 "Isn't this charge gonna be moving faster 258 00:09:02,657 --> 00:09:04,331 "since it had more charge?" 259 00:09:04,331 --> 00:09:05,309 No, it's not. 260 00:09:05,309 --> 00:09:07,521 The force that these charges are gonna exert on each other 261 00:09:07,521 --> 00:09:10,691 are always the same, even if they have different charges. 262 00:09:10,691 --> 00:09:12,775 That's counter-intuitive, but it's true. 263 00:09:12,775 --> 00:09:15,346 Newton's third law tells us that has to be true. 264 00:09:15,346 --> 00:09:17,059 So if they exert the same force on each other 265 00:09:17,059 --> 00:09:18,974 over the same amount of distance, 266 00:09:18,974 --> 00:09:21,360 then they will do the same amount of work on each other. 267 00:09:21,360 --> 00:09:22,844 And if they have the same mass, 268 00:09:22,844 --> 00:09:24,651 that means they're gonna end with the same speed 269 00:09:24,651 --> 00:09:25,661 as each other. 270 00:09:25,661 --> 00:09:28,756 So they'll have the same speed, a common speed we'll call v. 271 00:09:28,756 --> 00:09:30,166 So now to solve for v, I just take 272 00:09:30,166 --> 00:09:32,428 a square root of each side and I get that the speed 273 00:09:32,428 --> 00:09:36,413 of each charge is gonna be the square root of 1.8. 274 00:09:36,413 --> 00:09:38,177 Technically I'd have to divide that joules 275 00:09:38,177 --> 00:09:41,439 by kilograms first, because even though this was a 1, 276 00:09:41,439 --> 00:09:42,699 to make the units come out right 277 00:09:42,699 --> 00:09:44,540 I'd have to have joule per kilogram. 278 00:09:44,540 --> 00:09:48,493 And if I take the square root, I get 1.3 meters per second. 279 00:09:48,493 --> 00:09:50,821 That's how fast these charges are gonna be moving 280 00:09:50,821 --> 00:09:52,246 after they've moved to the point 281 00:09:52,246 --> 00:09:54,716 where they're 12 centimeters away from each other. 282 00:09:54,716 --> 00:09:57,682 Conceptually, potential energy was turning into 283 00:09:57,682 --> 00:09:58,748 kinetic energy. 284 00:09:58,748 --> 00:10:00,405 So the final potential energy was less 285 00:10:00,405 --> 00:10:01,972 than the initial potential energy, 286 00:10:01,972 --> 00:10:04,723 and all that energy went into the kinetic energies 287 00:10:04,723 --> 00:10:06,459 of these charges. 288 00:10:06,459 --> 00:10:07,525 So we solved this problem. 289 00:10:07,525 --> 00:10:08,700 Let's switch it up. 290 00:10:08,700 --> 00:10:10,667 Let's say instead of starting these charges from rest 291 00:10:10,667 --> 00:10:11,973 three centimeters apart, 292 00:10:11,973 --> 00:10:15,341 let's say we start them from rest 12 centimeters apart 293 00:10:15,341 --> 00:10:18,023 but we make this Q2 negative. 294 00:10:18,023 --> 00:10:20,425 So now instead of being positive 2 microcoulombs, 295 00:10:20,425 --> 00:10:23,021 we're gonna make this negative 2 microcoulombs. 296 00:10:23,021 --> 00:10:24,396 And now that this charge is negative, 297 00:10:24,396 --> 00:10:26,463 it's attracted to the positive charge, 298 00:10:26,463 --> 00:10:28,127 and likewise this positive charge is 299 00:10:28,127 --> 00:10:30,085 attracted to the negative charge. 300 00:10:30,085 --> 00:10:31,746 So let's say we released these from rest 301 00:10:31,746 --> 00:10:33,534 12 centimeters apart, 302 00:10:33,534 --> 00:10:36,167 and we allowed them to fly forward to each other 303 00:10:36,167 --> 00:10:38,542 until they're three centimeters apart. 304 00:10:38,542 --> 00:10:39,757 And we ask the same question, 305 00:10:39,757 --> 00:10:43,058 how fast are they gonna be going when they get to this point 306 00:10:43,058 --> 00:10:45,080 where they're three centimeters apart? 307 00:10:45,080 --> 00:10:47,413 Okay, so what would change in the math up here? 308 00:10:47,413 --> 00:10:49,132 Since they're still released from rest, 309 00:10:49,132 --> 00:10:51,227 we still start with no kinetic energy, 310 00:10:51,227 --> 00:10:52,260 so that doesn't change. 311 00:10:52,260 --> 00:10:54,774 But this time, they didn't start three centimeters apart. 312 00:10:54,774 --> 00:10:58,089 So instead of starting with three and ending with 12, 313 00:10:58,089 --> 00:11:00,451 they're gonna start 12 centimeters apart 314 00:11:00,451 --> 00:11:02,975 and end three centimeters apart. 315 00:11:02,975 --> 00:11:04,735 All right, so what else changes up here? 316 00:11:04,735 --> 00:11:07,638 The only other thing that changed was the sign of Q2. 317 00:11:07,638 --> 00:11:10,050 And you might think, I shouldn't plug in the signs 318 00:11:10,050 --> 00:11:11,276 of the charges in here, 319 00:11:11,276 --> 00:11:12,682 because that gets me mixed up. 320 00:11:12,682 --> 00:11:15,515 But that was for electric field and electric force. 321 00:11:15,515 --> 00:11:18,208 If these aren't vectors, you can plug in positives 322 00:11:18,208 --> 00:11:19,527 and negative signs. 323 00:11:19,527 --> 00:11:21,051 And you should. The easiest thing to do 324 00:11:21,051 --> 00:11:23,012 is just plug in those positives and negatives. 325 00:11:23,012 --> 00:11:24,603 And this equation will just tell you 326 00:11:24,603 --> 00:11:26,641 whether you end up with a positive potential energy 327 00:11:26,641 --> 00:11:28,202 or a negative potential energy. 328 00:11:28,202 --> 00:11:30,437 We don't like including this in the electric field 329 00:11:30,437 --> 00:11:32,260 and electric force formulas 330 00:11:32,260 --> 00:11:33,572 because those are vectors, 331 00:11:33,572 --> 00:11:35,197 and if they're vectors, we're gonna have to decide 332 00:11:35,197 --> 00:11:38,204 what direction they point and this negative can screw us up. 333 00:11:38,204 --> 00:11:39,772 But it's not gonna screw us up in this case. 334 00:11:39,772 --> 00:11:41,008 This negative is just gonna tell us 335 00:11:41,008 --> 00:11:43,166 whether we have positive potential energy 336 00:11:43,166 --> 00:11:44,485 or negative potential energy. 337 00:11:44,485 --> 00:11:46,244 There's no worry about breaking up a vector, 338 00:11:46,244 --> 00:11:47,932 because these are scalars. 339 00:11:47,932 --> 00:11:50,407 So long story short, we plug in the positive signs 340 00:11:50,407 --> 00:11:51,683 if it's a positive charge. 341 00:11:51,683 --> 00:11:54,017 We plug in the negative sign if it's a negative charge. 342 00:11:54,017 --> 00:11:56,105 This formula's smart enough to figure it out, 343 00:11:56,105 --> 00:11:57,771 since it's a scalar, we don't have to worry about 344 00:11:57,771 --> 00:11:59,427 breaking up any components. 345 00:11:59,427 --> 00:12:01,395 In other words, instead of two up here, 346 00:12:01,395 --> 00:12:04,239 we're gonna have negative two microcoulombs. 347 00:12:04,239 --> 00:12:06,142 And instead of positive two in this formula, 348 00:12:06,142 --> 00:12:08,308 we're gonna have negative two microcoulombs. 349 00:12:08,308 --> 00:12:10,444 So if we multiply out the left-hand side, 350 00:12:10,444 --> 00:12:11,984 it might not be surprising. 351 00:12:11,984 --> 00:12:15,617 All we're gonna get is negative 0.6 joules 352 00:12:15,617 --> 00:12:17,404 of initial potential energy. 353 00:12:17,404 --> 00:12:18,488 And this might worry you. 354 00:12:18,488 --> 00:12:19,635 You might be like, "Wait a minute, 355 00:12:19,635 --> 00:12:22,067 "we're starting with negative potential energy?" 356 00:12:22,067 --> 00:12:23,820 You might say, "That makes no sense. 357 00:12:23,820 --> 00:12:26,308 "How are we gonna get kinetic energy out of a system 358 00:12:26,308 --> 00:12:29,190 "that starts with less than zero potential energy?" 359 00:12:29,190 --> 00:12:30,023 So it seems kind of weird. 360 00:12:30,023 --> 00:12:33,340 How can I start with less than zero or zero potential energy 361 00:12:33,340 --> 00:12:35,498 and still get kinetic energy out? 362 00:12:35,498 --> 00:12:37,154 Well, it's just because this term, 363 00:12:37,154 --> 00:12:39,131 your final potential energy term, 364 00:12:39,131 --> 00:12:41,009 is gonna be even more negative. 365 00:12:41,009 --> 00:12:45,019 If I calculate this term, I end up with negative 2.4 joules. 366 00:12:45,019 --> 00:12:47,475 And then we add to that the kinetic energy of the system. 367 00:12:47,475 --> 00:12:48,438 So in other words, our system 368 00:12:48,438 --> 00:12:50,702 is still gaining kinetic energy 369 00:12:50,702 --> 00:12:53,437 because it's still losing potential energy. 370 00:12:53,437 --> 00:12:55,271 Just because you've got negative potential energy 371 00:12:55,271 --> 00:12:57,739 doesn't mean you can't have less potential energy 372 00:12:57,739 --> 00:12:58,865 than you started with. 373 00:12:58,865 --> 00:13:00,506 It's kind of like finances. 374 00:13:00,506 --> 00:13:03,013 Trust me, if you start with less than zero money, 375 00:13:03,013 --> 00:13:04,083 if you start in debt, 376 00:13:04,083 --> 00:13:06,260 that doesn't mean you can't spend money. 377 00:13:06,260 --> 00:13:09,555 You can still get a credit card and become more in debt. 378 00:13:09,555 --> 00:13:12,582 You can still get stuff, even if you have no money 379 00:13:12,582 --> 00:13:14,405 or less than zero money. 380 00:13:14,405 --> 00:13:17,057 It just means you're gonna go more and more in debt. 381 00:13:17,057 --> 00:13:18,954 And that's what this electric potential is doing. 382 00:13:18,954 --> 00:13:20,883 It's becoming more and more in debt 383 00:13:20,883 --> 00:13:24,378 so that it can finance an increase in kinetic energy. 384 00:13:24,378 --> 00:13:26,688 Not the best financial decision, but this is physics, 385 00:13:26,688 --> 00:13:27,646 so they don't care. 386 00:13:27,646 --> 00:13:29,312 All right, so we solve this for the kinetic energy 387 00:13:29,312 --> 00:13:30,145 of the system. 388 00:13:30,145 --> 00:13:32,550 We add 2.4 joules to both sides 389 00:13:32,550 --> 00:13:36,397 and we get positive 1.8 joules on the left hand side 390 00:13:36,397 --> 00:13:37,729 equals 391 00:13:37,729 --> 00:13:39,697 We'll have two terms because they're both gonna be moving. 392 00:13:39,697 --> 00:13:42,234 We'll have the one half times one kilogram 393 00:13:42,234 --> 00:13:44,315 times the speed of one of the charges squared 394 00:13:44,315 --> 00:13:47,434 plus one half times one kilogram times the speed 395 00:13:47,434 --> 00:13:49,119 of the other charge squared, 396 00:13:49,119 --> 00:13:51,219 which again just gives us v squared. 397 00:13:51,219 --> 00:13:53,626 And if we solve this for v, we're gonna get the same value 398 00:13:53,626 --> 00:13:57,663 we got last time, 1.3 meters per second. 399 00:13:57,663 --> 00:14:00,443 So recapping the formula for the electrical potential energy 400 00:14:00,443 --> 00:14:02,145 between two charges is gonna be 401 00:14:02,145 --> 00:14:03,395 k Q1 Q2 over r. 402 00:14:04,856 --> 00:14:06,377 And since the energy is a scalar, 403 00:14:06,377 --> 00:14:08,002 you can plug in those negative signs 404 00:14:08,002 --> 00:14:11,187 to tell you if the potential energy is positive or negative. 405 00:14:11,187 --> 00:14:13,424 Since this is energy, you could use it in conservation 406 00:14:13,424 --> 00:14:14,257 of energy. 407 00:14:14,257 --> 00:14:15,874 And it's possible for systems to have 408 00:14:15,874 --> 00:14:17,499 negative electric potential energy, 409 00:14:17,499 --> 00:14:19,589 and those systems can still convert energy 410 00:14:19,589 --> 00:14:20,651 into kinetic energy. 411 00:14:20,651 --> 00:14:21,660 They would just have to make sure 412 00:14:21,660 --> 00:14:23,285 that their electric potential energy becomes 413 00:14:23,285 --> 00:00:00,000 even more negative.