1 00:00:00,242 --> 00:00:01,106 - [Instructor] Okay, so we know that 2 00:00:01,106 --> 00:00:03,622 electric charges create electric fields. 3 00:00:03,622 --> 00:00:05,455 And we know the definition of the electric field 4 00:00:05,455 --> 00:00:07,622 is the amount of force per charge. 5 00:00:07,622 --> 00:00:08,660 What charge? 6 00:00:08,660 --> 00:00:11,338 Some charge that finds its way into this region, 7 00:00:11,338 --> 00:00:13,141 let's say this charge right here. 8 00:00:13,141 --> 00:00:14,857 If we took the force on this charge, 9 00:00:14,857 --> 00:00:15,706 and let's give this a name. 10 00:00:15,706 --> 00:00:17,373 Let's call this Q2, so we can keep these 11 00:00:17,373 --> 00:00:20,038 all straight, and I'll call it Q2 up here. 12 00:00:20,038 --> 00:00:21,841 If we took the force on this charge Q2 13 00:00:21,841 --> 00:00:24,406 divided by Q2, that would be the electric field 14 00:00:24,406 --> 00:00:26,239 at that point in space, but something that would 15 00:00:26,239 --> 00:00:28,055 be useful to have is a formula 16 00:00:28,055 --> 00:00:30,389 that would let us figure out what's the electric field 17 00:00:30,389 --> 00:00:32,973 being created at that point in space, 18 00:00:32,973 --> 00:00:35,240 without even referring to Q2. 19 00:00:35,240 --> 00:00:36,622 It'd be useful to have a formula 20 00:00:36,622 --> 00:00:38,490 that would let us figure out what's the electric field 21 00:00:38,490 --> 00:00:41,688 that this Q1 is creating over here at this point 22 00:00:41,688 --> 00:00:45,006 in space, without even referring to Q2 at all. 23 00:00:45,006 --> 00:00:46,539 Is there a formula for that? 24 00:00:46,539 --> 00:00:48,655 There is, and it's not that hard to find, 25 00:00:48,655 --> 00:00:50,439 but the first thing I'll caution you about 26 00:00:50,439 --> 00:00:52,423 is that the formula we're about to find here 27 00:00:52,423 --> 00:00:55,222 is gonna be for the magnitude of the electric field. 28 00:00:55,222 --> 00:00:58,971 So I'm gonna erase these vector crowns on these variables. 29 00:00:58,971 --> 00:01:00,239 This formula we get will just be 30 00:01:00,239 --> 00:01:02,554 for the magnitude of the electric field, 31 00:01:02,554 --> 00:01:03,921 and I'll tell you why in a second. 32 00:01:03,921 --> 00:01:06,571 The way we'll find a formula for the magnitude 33 00:01:06,571 --> 00:01:09,071 of the electric field is simply by inserting 34 00:01:09,071 --> 00:01:12,708 what we already know is the formula for the electric force. 35 00:01:12,708 --> 00:01:15,462 Coulomb's Law gives us the force between two charges, 36 00:01:15,462 --> 00:01:17,351 and we're just gonna put that right in here. 37 00:01:17,351 --> 00:01:18,799 Coulomb's Law says that the electric force 38 00:01:18,799 --> 00:01:22,140 between two charges is gonna be k, the electric constant, 39 00:01:22,140 --> 00:01:24,025 which is always nine times 10 to the ninth, 40 00:01:24,025 --> 00:01:26,423 multiplied by Q1, the first charge 41 00:01:26,423 --> 00:01:29,106 that's interacting, and that'd be this Q1 over here, 42 00:01:29,106 --> 00:01:32,189 multiplied by Q2, the other charge interacting, 43 00:01:32,189 --> 00:01:34,573 divided by the center to center distance 44 00:01:34,573 --> 00:01:37,089 between them squared, and then because we're finding 45 00:01:37,089 --> 00:01:40,105 electric field in here, we're dividing by Q2. 46 00:01:40,105 --> 00:01:41,288 Notice what happens here. 47 00:01:41,288 --> 00:01:44,005 Q2 is canceling, and we get that the magnitude 48 00:01:44,005 --> 00:01:46,022 of the electric field is gonna be equal 49 00:01:46,022 --> 00:01:49,055 to k, this electric constant, and I'll write that 50 00:01:49,055 --> 00:01:50,555 down over here so we know what it is. 51 00:01:50,555 --> 00:01:53,305 K is nine times 10 to the ninth, 52 00:01:53,305 --> 00:01:54,438 and it's got kinda weird units, 53 00:01:54,438 --> 00:01:55,638 but it makes sure that all the units 54 00:01:55,638 --> 00:01:57,371 come out okay when you multiply. 55 00:01:57,371 --> 00:01:58,405 And then, what do we still have up here? 56 00:01:58,405 --> 00:02:01,105 We've still got a Q1 divided by the center 57 00:02:01,105 --> 00:02:04,255 to center distance between those two charges squared, 58 00:02:04,255 --> 00:02:06,790 and you might be like, well, the other charge went away. 59 00:02:06,790 --> 00:02:07,837 We canceled it out. 60 00:02:07,837 --> 00:02:09,993 Centered between which two charges? 61 00:02:09,993 --> 00:02:12,022 Well, this could be to any point in space, really. 62 00:02:12,022 --> 00:02:14,738 So you imagine your test charge at any point you want. 63 00:02:14,738 --> 00:02:16,854 I could put it here, I can move it over to here. 64 00:02:16,854 --> 00:02:18,437 The r would just be the distance 65 00:02:18,437 --> 00:02:21,853 from the first charge, Q1, to wherever I wanna figure out 66 00:02:21,853 --> 00:02:23,504 what the electric field would be. 67 00:02:23,504 --> 00:02:25,254 But since this Q2 always divides out, 68 00:02:25,254 --> 00:02:26,655 we don't even need to talk about that. 69 00:02:26,655 --> 00:02:28,486 We can just figure out the electric field 70 00:02:28,486 --> 00:02:32,293 that's created by Q1 at any point in space, 71 00:02:32,293 --> 00:02:34,871 so this r is just the distance from the center 72 00:02:34,871 --> 00:02:37,157 of the charge creating the field to the point 73 00:02:37,157 --> 00:02:39,770 in space where you wanna determine the electric field. 74 00:02:39,770 --> 00:02:40,603 And now we've got it. 75 00:02:40,603 --> 00:02:42,336 This is a formula for the electric field 76 00:02:42,336 --> 00:02:44,897 created by a charge Q1. 77 00:02:44,897 --> 00:02:46,361 Technically, though, this is only true 78 00:02:46,361 --> 00:02:48,028 if this is a point charge. 79 00:02:48,028 --> 00:02:49,615 In other words, if it's really, really small 80 00:02:49,615 --> 00:02:51,877 compared to the other dimensions in the problem. 81 00:02:51,877 --> 00:02:53,896 Or, if this is spherically symmetric, 82 00:02:53,896 --> 00:02:54,827 then it doesn't matter. 83 00:02:54,827 --> 00:02:56,194 If you're outside of this charge 84 00:02:56,194 --> 00:02:57,783 and you've got a spherically symmetrical 85 00:02:57,783 --> 00:03:00,132 charge distribution, where all the charges 86 00:03:00,132 --> 00:03:02,009 are lumped on one side of this sphere, 87 00:03:02,009 --> 00:03:04,207 or anything like that, if it's evenly distributed 88 00:03:04,207 --> 00:03:05,907 throughout, then this formula also works 89 00:03:05,907 --> 00:03:08,174 just as well when you're outside the sphere. 90 00:03:08,174 --> 00:03:09,319 And what's this formula saying? 91 00:03:09,319 --> 00:03:11,141 It's saying that the absolute value, 92 00:03:11,141 --> 00:03:13,140 or the magnitude of the electric field 93 00:03:13,140 --> 00:03:16,257 created at a point in space is equal to k, 94 00:03:16,257 --> 00:03:19,508 the electric constant, times the charge creating the field. 95 00:03:19,508 --> 00:03:20,341 This is important. 96 00:03:20,341 --> 00:03:23,690 This charge, Q1, is creating this electric field. 97 00:03:23,690 --> 00:03:25,744 And then you plug in the distance away from that charge 98 00:03:25,744 --> 00:03:28,107 that you wanna determine the electric field, r, 99 00:03:28,107 --> 00:03:29,422 you square it, and that'll tell you 100 00:03:29,422 --> 00:03:31,906 what the magnitude of the electric field 101 00:03:31,906 --> 00:03:35,190 is created by Q1 at any point in space around it. 102 00:03:35,190 --> 00:03:36,591 Now why are we being so careful, 103 00:03:36,591 --> 00:03:38,541 saying that this is just the magnitude? 104 00:03:38,541 --> 00:03:39,374 Here's why. 105 00:03:39,374 --> 00:03:41,357 Imagine we plugged in this charge as positive 106 00:03:41,357 --> 00:03:43,490 because the charge creating it is positive. 107 00:03:43,490 --> 00:03:45,824 You'd get a positive value for the electric field, 108 00:03:45,824 --> 00:03:47,757 and you might think, oh, that means positive. 109 00:03:47,757 --> 00:03:48,873 That means to the right. 110 00:03:48,873 --> 00:03:50,157 And in this case, it works out. 111 00:03:50,157 --> 00:03:51,773 It does go to the right at this point. 112 00:03:51,773 --> 00:03:53,357 But let's say you put those same calculations 113 00:03:53,357 --> 00:03:55,407 for a point over here, and you wanna determine 114 00:03:55,407 --> 00:03:58,093 what's the value of the electric field at this point? 115 00:03:58,093 --> 00:04:00,373 Well, if you plugged in k, it's a positive number, 116 00:04:00,373 --> 00:04:02,240 your Q is a positive number, r is gonna 117 00:04:02,240 --> 00:04:03,922 be a positive number, even if you tried 118 00:04:03,922 --> 00:04:05,717 to plus in r as a negative, it'd square 119 00:04:05,717 --> 00:04:07,156 away that negative anyway. 120 00:04:07,156 --> 00:04:08,539 This would all come out positive. 121 00:04:08,539 --> 00:04:11,261 So you might think, oh, over here points to the right 122 00:04:11,261 --> 00:04:14,189 as well because that's the positive direction, right? 123 00:04:14,189 --> 00:04:15,022 Well, no. 124 00:04:15,022 --> 00:04:16,789 This formula's not gonna tell you 125 00:04:16,789 --> 00:04:19,362 whether this electric field goes up, down, left, or right. 126 00:04:19,362 --> 00:04:21,427 If you really wanted to include a positive 127 00:04:21,427 --> 00:04:22,977 or negative sign for this charge, 128 00:04:22,977 --> 00:04:24,788 all that positive would be telling you is 129 00:04:24,788 --> 00:04:28,296 that the field's pointing radially away from the charge. 130 00:04:28,296 --> 00:04:32,027 But radially away could mean left, it could mean up, 131 00:04:32,027 --> 00:04:34,460 it could mean right, or it could mean down, 132 00:04:34,460 --> 00:04:36,260 and all of those count as a positive 133 00:04:36,260 --> 00:04:38,343 as far as this formula would be concerned. 134 00:04:38,343 --> 00:04:39,609 So that's why we say be careful. 135 00:04:39,609 --> 00:04:41,478 You're just finding the magnitude 136 00:04:41,478 --> 00:04:43,095 when you find this formula. 137 00:04:43,095 --> 00:04:44,309 The way you find the direction is just 138 00:04:44,309 --> 00:04:46,643 by knowing that the field created by a positive 139 00:04:46,643 --> 00:04:49,609 is always radially away from that positive. 140 00:04:49,609 --> 00:04:50,809 But even though this formula just gives you 141 00:04:50,809 --> 00:04:52,743 the magnitude, that's still really useful. 142 00:04:52,743 --> 00:04:53,576 So we're gonna use this. 143 00:04:53,576 --> 00:04:55,659 This gives you the magnitude of the electric field 144 00:04:55,659 --> 00:04:57,297 from a point charge at any point 145 00:04:57,297 --> 00:04:59,027 away from that point charge. 146 00:04:59,027 --> 00:05:00,204 Let's solve some examples here. 147 00:05:00,204 --> 00:05:01,592 Let's use this thing. 148 00:05:01,592 --> 00:05:04,576 Let's say you had a positive two nanoCoulomb charge, 149 00:05:04,576 --> 00:05:06,792 and you wanted to determine the size 150 00:05:06,792 --> 00:05:09,393 and direction of the electric field 151 00:05:09,393 --> 00:05:12,358 at a point three meters below that charge. 152 00:05:12,358 --> 00:05:13,558 We wanna know, what's the size 153 00:05:13,558 --> 00:05:16,059 and direction of the electric field right there. 154 00:05:16,059 --> 00:05:18,576 To get the size, we could use the new formula we've got, 155 00:05:18,576 --> 00:05:20,609 which says that the electric field created 156 00:05:20,609 --> 00:05:22,711 by a charge, Q, is gonna be equal 157 00:05:22,711 --> 00:05:24,742 to k times that Q over r squared. 158 00:05:24,742 --> 00:05:26,143 We'll use that down here. 159 00:05:26,143 --> 00:05:29,075 We get k, which is always nine times 10 to the ninth, 160 00:05:29,075 --> 00:05:31,562 and then we multiply by the charge creating the field, 161 00:05:31,562 --> 00:05:33,009 which in this case, it's this 162 00:05:33,009 --> 00:05:35,525 positive two nanoCoulomb charge. 163 00:05:35,525 --> 00:05:38,442 Nano stands for 10 to the negative ninth, 164 00:05:38,442 --> 00:05:39,910 and then we divide by the distance 165 00:05:39,910 --> 00:05:42,742 from our charge to the point where we wanna find the field. 166 00:05:42,742 --> 00:05:44,059 That's three meters. 167 00:05:44,059 --> 00:05:45,741 You can't forget to square this. 168 00:05:45,741 --> 00:05:47,609 People forget to square this all the time. 169 00:05:47,609 --> 00:05:49,192 It doesn't come out right, so you gotta remember 170 00:05:49,192 --> 00:05:51,341 to square the r, and if we solve this, 171 00:05:51,341 --> 00:05:54,809 10 to the ninth times 10 to the negative ninth goes away. 172 00:05:54,809 --> 00:05:56,075 That's pretty nice. 173 00:05:56,075 --> 00:05:59,408 And then nine divided by three squared is just nine divided 174 00:05:59,408 --> 00:06:01,725 by nine, so all of that's gonna go away. 175 00:06:01,725 --> 00:06:03,524 And all we really get for the electric field is that 176 00:06:03,524 --> 00:06:07,907 it's gonna be two Newtons per Coulomb at this point here. 177 00:06:07,907 --> 00:06:09,007 So that is the magnitude. 178 00:06:09,007 --> 00:06:11,908 This gives us the magnitude of the electric field 179 00:06:11,908 --> 00:06:14,774 at this point in space, that's how you get the size of it. 180 00:06:14,774 --> 00:06:16,257 How do we get the direction? 181 00:06:16,257 --> 00:06:18,641 We just ask, what was creating this field? 182 00:06:18,641 --> 00:06:20,007 It was a positive charge. 183 00:06:20,007 --> 00:06:21,690 Positive charges always create fields 184 00:06:21,690 --> 00:06:23,956 that point radially away from them, 185 00:06:23,956 --> 00:06:25,826 and at this point, radially away 186 00:06:25,826 --> 00:06:28,139 from this positive is gonna point straight down. 187 00:06:28,139 --> 00:06:29,023 So we've got an electric field 188 00:06:29,023 --> 00:06:31,890 from this two nanoCoulomb charge that points straight down, 189 00:06:31,890 --> 00:06:34,791 and has a value of two Newtons per Coulomb. 190 00:06:34,791 --> 00:06:36,077 What did this number mean? 191 00:06:36,077 --> 00:06:37,572 It means if we put another charge 192 00:06:37,572 --> 00:06:40,457 at that point in space, some little charge q, 193 00:06:40,457 --> 00:06:41,756 then there would be two Newtons 194 00:06:41,756 --> 00:06:44,073 for every Coulomb of charge that you put there, 195 00:06:44,073 --> 00:06:45,389 since we know that electric field 196 00:06:45,389 --> 00:06:48,142 is the amount of force per charge. 197 00:06:48,142 --> 00:06:50,722 Notice that even though this electric field came out 198 00:06:50,722 --> 00:06:53,271 to be positive, it was pointing down 199 00:06:53,271 --> 00:06:55,090 because all we're getting out of this calculation 200 00:06:55,090 --> 00:06:57,605 is the magnitude of the electric field. 201 00:06:57,605 --> 00:06:59,088 Let's do one more. 202 00:06:59,088 --> 00:06:59,921 Let's try this one out. 203 00:06:59,921 --> 00:07:03,121 Let's say you had a negative four microCoulomb charge, 204 00:07:03,121 --> 00:07:04,872 and you wanted to determine the size 205 00:07:04,872 --> 00:07:07,205 and direction of the electric field 206 00:07:07,205 --> 00:07:11,288 at a point six meters away, to the left of that charge. 207 00:07:11,288 --> 00:07:12,638 We use the same formula. 208 00:07:12,638 --> 00:07:14,987 We'll say that the electric field created 209 00:07:14,987 --> 00:07:17,571 by this negative charge is gonna equal k, 210 00:07:17,571 --> 00:07:19,788 which is always nine times 10 to the ninth. 211 00:07:19,788 --> 00:07:23,171 And then multiplied by the charge creating that field, 212 00:07:23,171 --> 00:07:26,087 which in this case is negative four microCoulombs, 213 00:07:26,087 --> 00:07:28,403 but I am not gonna plug in this negative sign 214 00:07:28,403 --> 00:07:30,653 because I know all this formula's giving me 215 00:07:30,653 --> 00:07:33,122 is the magnitude of the electric field. 216 00:07:33,122 --> 00:07:35,388 I'm not gonna get tricked into thinking that this negative 217 00:07:35,388 --> 00:07:37,670 sign would tell me the direction of the electric field. 218 00:07:37,670 --> 00:07:40,072 I mean, it does tell you the direction. 219 00:07:40,072 --> 00:07:42,403 It tells you that it points radially inward, 220 00:07:42,403 --> 00:07:44,353 but it's safest, gotta take my word on this, 221 00:07:44,353 --> 00:07:46,103 safest to just leave that negative sign out 222 00:07:46,103 --> 00:07:48,103 and know that this is just the magnitude 223 00:07:48,103 --> 00:07:49,205 of the electric field. 224 00:07:49,205 --> 00:07:50,772 So we have four microCoulombs. 225 00:07:50,772 --> 00:07:53,788 Micro stands for 10 to the negative sixth. 226 00:07:53,788 --> 00:07:54,955 And then we divide by the distance 227 00:07:54,955 --> 00:07:56,224 from the center of that charge 228 00:07:56,224 --> 00:07:58,538 to the point we wanna determine the electric field at, 229 00:07:58,538 --> 00:08:00,288 which is right here, and we square it. 230 00:08:00,288 --> 00:08:03,205 That's six meters, and we cannot forget to square. 231 00:08:03,205 --> 00:08:04,889 If we solve this for the electric field, 232 00:08:04,889 --> 00:08:07,554 we're gonna get, well, six squared is 36, 233 00:08:07,554 --> 00:08:09,889 and nine over 36 is 1/4. 234 00:08:09,889 --> 00:08:12,087 1/4 of four is just one, so all we're left 235 00:08:12,087 --> 00:08:15,548 with is 10 to the ninth times 10 to the negative sixth, 236 00:08:15,548 --> 00:08:18,154 but that's just 10 to the third, which is 1000. 237 00:08:18,154 --> 00:08:21,027 So this electric field's gonna be 1000 Newtons 238 00:08:21,027 --> 00:08:23,286 per Coulomb at that point in space. 239 00:08:23,286 --> 00:08:25,554 That's the magnitude of the electric field. 240 00:08:25,554 --> 00:08:27,122 Or, in other words, that's the size 241 00:08:27,122 --> 00:08:28,986 of the electric field at that point. 242 00:08:28,986 --> 00:08:30,388 How do we get the direction? 243 00:08:30,388 --> 00:08:32,687 We're gonna decide this by thinking carefully about it. 244 00:08:32,687 --> 00:08:34,975 In other words, we don't include this negative, 245 00:08:34,975 --> 00:08:37,120 not because direction isn't important. 246 00:08:37,120 --> 00:08:38,318 We don't include this negative 247 00:08:38,318 --> 00:08:40,085 because direction is so important, 248 00:08:40,086 --> 00:08:41,919 we're gonna make sure we get this right. 249 00:08:41,919 --> 00:08:43,669 What I say is that I've got a negative charge. 250 00:08:43,669 --> 00:08:45,594 I know that negative charges create fields 251 00:08:45,594 --> 00:08:47,977 that point radially into them. 252 00:08:47,977 --> 00:08:49,993 And that means over here at this point to the left, 253 00:08:49,993 --> 00:08:52,869 the electric field is pointing radially toward 254 00:08:52,869 --> 00:08:55,718 that negative charge, it would point to the right. 255 00:08:55,718 --> 00:08:57,518 So we'd have 1000 Newtons per Coulomb 256 00:08:57,518 --> 00:09:00,336 of electric field, and it would point to the right. 257 00:09:00,336 --> 00:09:02,402 Note, if I would've just naively plugged 258 00:09:02,402 --> 00:09:04,836 this negative sign in over to here, 259 00:09:04,836 --> 00:09:06,602 I would've come out with a negative value 260 00:09:06,602 --> 00:09:08,669 for my electric field, and I might've thought, 261 00:09:08,669 --> 00:09:10,685 well, negative, that means leftward, right? 262 00:09:10,685 --> 00:09:12,554 So that means it points to the left. 263 00:09:12,554 --> 00:09:13,852 And I would've got the wrong direction. 264 00:09:13,852 --> 00:09:15,235 So that's why we don't do that. 265 00:09:15,235 --> 00:09:17,351 All this negative sign is representing, if you were 266 00:09:17,351 --> 00:09:20,618 to plug it in, is that it's pointing radially inward, 267 00:09:20,618 --> 00:09:22,318 but radially in could mean right 268 00:09:22,318 --> 00:09:23,667 if you're over here to the left. 269 00:09:23,667 --> 00:09:25,785 It could mean left if you're over here to the right. 270 00:09:25,785 --> 00:09:28,000 It could mean up if you're underneath the charge, 271 00:09:28,000 --> 00:09:30,051 and it could mean down if you're above the charge. 272 00:09:30,051 --> 00:09:32,001 In other words, it doesn't mean anything 273 00:09:32,001 --> 00:09:33,651 for a given particular problem. 274 00:09:33,651 --> 00:09:34,851 It often just screws you up. 275 00:09:34,851 --> 00:09:35,850 Leave that outta there. 276 00:09:35,850 --> 00:09:37,817 Don't put the negative signs in. 277 00:09:37,817 --> 00:09:39,863 Just use this formula to get the magnitude, 278 00:09:39,863 --> 00:09:41,250 and once you have that magnitude, 279 00:09:41,250 --> 00:09:43,067 just know which direction negative charges 280 00:09:43,067 --> 00:09:44,750 create their fields, and that'll tell you 281 00:09:44,750 --> 00:09:46,900 which direction the field points. 282 00:09:46,900 --> 00:09:48,549 So recapping, this is the formula 283 00:09:48,549 --> 00:09:51,583 for the electric field created by a charge, Q. 284 00:09:51,583 --> 00:09:52,933 And it tells you that the magnitude 285 00:09:52,933 --> 00:09:54,549 of the electric field is gonna be equal 286 00:09:54,549 --> 00:09:58,384 to k, the electric constant, times the charge creating 287 00:09:58,384 --> 00:10:00,583 that field, divided by the distance 288 00:10:00,583 --> 00:10:02,082 from the center of that charge 289 00:10:02,082 --> 00:00:00,000 to the point where you wanna find the field, squared.