1 00:00:00,000 --> 00:00:00,740 2 00:00:00,740 --> 00:00:04,150 We learned several videos ago that if I had an infinite 3 00:00:04,150 --> 00:00:08,284 uniformly charged plane-- let me draw one right here, and I 4 00:00:08,284 --> 00:00:09,960 won't draw it infinite and I'll tell you why in a 5 00:00:09,960 --> 00:00:13,450 second-- that if we had an infinite uniformly charged 6 00:00:13,450 --> 00:00:19,600 plane, and let's say this one's positive, that the 7 00:00:19,600 --> 00:00:23,935 electric field generated by it is constant. 8 00:00:23,935 --> 00:00:29,030 9 00:00:29,030 --> 00:00:30,290 Those are the field lines. 10 00:00:30,290 --> 00:00:32,890 They should all be the same size. 11 00:00:32,890 --> 00:00:38,210 And the strength of the field, or the magnitude of the field, 12 00:00:38,210 --> 00:00:45,820 is equal to 2 times Coulomb's constant times pi times the 13 00:00:45,820 --> 00:00:47,750 charge density of the plate. 14 00:00:47,750 --> 00:00:50,130 So if this is infinite-- so what was charge density? 15 00:00:50,130 --> 00:00:54,450 We defined it when we proved that this truly is a uniform 16 00:00:54,450 --> 00:00:57,140 electric field, but what is charge density? 17 00:00:57,140 --> 00:01:03,280 Charge density is just the total amount of charge divided 18 00:01:03,280 --> 00:01:05,290 by the area, or charge per area. 19 00:01:05,290 --> 00:01:10,560 Well, if we have an infinite plane, the area's going to be 20 00:01:10,560 --> 00:01:13,110 infinite, and so if this is a constant number, this is also 21 00:01:13,110 --> 00:01:16,740 going to be infinite, so it's kind of hard to work with. 22 00:01:16,740 --> 00:01:22,350 But what we also know is that when we have a non-infinite 23 00:01:22,350 --> 00:01:27,210 plane that has some finite area, that near the center of 24 00:01:27,210 --> 00:01:32,250 it and fairly close to it, it approximates an infinite 25 00:01:32,250 --> 00:01:34,780 uniformly charged plane. 26 00:01:34,780 --> 00:01:37,180 So with that said, let's see if we can figure out some of 27 00:01:37,180 --> 00:01:40,830 the properties of the voltage and how the voltage relates to 28 00:01:40,830 --> 00:01:42,350 the charge. 29 00:01:42,350 --> 00:01:48,130 If we were to have two parallel-- let me draw it 30 00:01:48,130 --> 00:01:50,040 before I say it, because I think saying it'll just 31 00:01:50,040 --> 00:01:51,850 confuse it. 32 00:01:51,850 --> 00:02:02,350 So let's say I have two plates, that plate-- and then 33 00:02:02,350 --> 00:02:05,290 I'll do this one in a different color-- and I have 34 00:02:05,290 --> 00:02:11,650 that plate, and let's say they're the same size and they 35 00:02:11,650 --> 00:02:13,470 both have area A. 36 00:02:13,470 --> 00:02:19,860 Let's say that I place plus Q worth of charges here. 37 00:02:19,860 --> 00:02:22,460 So this is plus Q so this is positively charged, right? 38 00:02:22,460 --> 00:02:24,010 I could draw a bunch of charges here. 39 00:02:24,010 --> 00:02:30,590 Let's say this is minus Q, so this is negatively charged. 40 00:02:30,590 --> 00:02:32,860 So what's the electric field going to look like 41 00:02:32,860 --> 00:02:34,730 between these two? 42 00:02:34,730 --> 00:02:37,070 Well, it's essentially going to be the combination of the 43 00:02:37,070 --> 00:02:40,400 electric field generated by this plate on top of the 44 00:02:40,400 --> 00:02:42,400 electric field generated by this plate. 45 00:02:42,400 --> 00:02:45,290 And they're both going to be constant close to the center, 46 00:02:45,290 --> 00:02:46,970 assuming that they're reasonably-- and let's say 47 00:02:46,970 --> 00:02:49,220 that they're d apart. 48 00:02:49,220 --> 00:02:52,140 Assuming that d isn't too big, near the center we're going to 49 00:02:52,140 --> 00:02:53,660 have a constant electric field. 50 00:02:53,660 --> 00:02:58,070 For example, this green one is going to be generating-- its 51 00:02:58,070 --> 00:03:00,470 field lines are going to look something like this. 52 00:03:00,470 --> 00:03:02,200 Near the center, it's constant. 53 00:03:02,200 --> 00:03:03,450 These are meant to look constant. 54 00:03:03,450 --> 00:03:06,500 55 00:03:06,500 --> 00:03:10,250 Near the center, it'll look like that, and it'll start to 56 00:03:10,250 --> 00:03:12,250 bulge out when you get to the edges. 57 00:03:12,250 --> 00:03:14,770 58 00:03:14,770 --> 00:03:18,440 Once again, near the center, it's constant. 59 00:03:18,440 --> 00:03:19,635 That one should've been at an angle. 60 00:03:19,635 --> 00:03:22,330 They start to bulge out, and it'll look 61 00:03:22,330 --> 00:03:23,830 something like that. 62 00:03:23,830 --> 00:03:28,120 And similarly, this purple plate will generate a constant 63 00:03:28,120 --> 00:03:30,630 electric field, and since it's negatively charged, the field 64 00:03:30,630 --> 00:03:33,470 lines will be going towards it, not away from it, so its 65 00:03:33,470 --> 00:03:35,020 field lines are going to look something like this. 66 00:03:35,020 --> 00:03:40,890 Near the center, they'll be constant, and its field lines 67 00:03:40,890 --> 00:03:42,140 are going to look something like that. 68 00:03:42,140 --> 00:03:44,640 69 00:03:44,640 --> 00:03:48,380 As you can see, they're going to be of the same magnitude 70 00:03:48,380 --> 00:03:52,730 and in the same direction, and they also 71 00:03:52,730 --> 00:03:54,820 will bulge out there. 72 00:03:54,820 --> 00:03:59,090 So the big picture is that you just kind of have twice the 73 00:03:59,090 --> 00:04:02,530 electric field as you would have if you just had one of 74 00:04:02,530 --> 00:04:03,920 these plates. 75 00:04:03,920 --> 00:04:06,690 So let's say we're operating near the center of these, 76 00:04:06,690 --> 00:04:09,940 where we have roughly a constant electric field and 77 00:04:09,940 --> 00:04:12,620 see if we can figure out the relationship between the 78 00:04:12,620 --> 00:04:17,760 voltage across these two plates and the area, and maybe 79 00:04:17,760 --> 00:04:20,120 the distance between the two plates. 80 00:04:20,120 --> 00:04:23,260 So we know that the electric field generated by any one of 81 00:04:23,260 --> 00:04:27,100 these charge plates-- I'll do it in the blue of this color. 82 00:04:27,100 --> 00:04:29,190 So for the bottom plate right here, what is the electric 83 00:04:29,190 --> 00:04:29,810 field generated? 84 00:04:29,810 --> 00:04:34,930 It's 2K pi times sigma. 85 00:04:34,930 --> 00:04:40,270 Sigma is just the total charge divided by the area. 86 00:04:40,270 --> 00:04:45,640 So Q/A, right? 87 00:04:45,640 --> 00:04:48,570 And we know that the total electric field generated by 88 00:04:48,570 --> 00:04:50,890 this one is going to be essentially the same thing. 89 00:04:50,890 --> 00:04:53,370 I mean, we could say it's a minus, a negative, because 90 00:04:53,370 --> 00:04:56,110 it's going towards it, but it's 91 00:04:56,110 --> 00:04:57,140 essentially the same thing. 92 00:04:57,140 --> 00:04:58,800 Because we see that they overlap just drawing the field 93 00:04:58,800 --> 00:05:02,750 line, so the electric field from that one, and we know 94 00:05:02,750 --> 00:05:04,570 that they go in the same direction. 95 00:05:04,570 --> 00:05:08,155 If this was somehow-- well, this is negative, so the field 96 00:05:08,155 --> 00:05:08,850 lines go towards it. 97 00:05:08,850 --> 00:05:14,300 So plus 2K pi and this is Q/A. 98 00:05:14,300 --> 00:05:16,640 We could have said minus and then had a minus Q/A, but we 99 00:05:16,640 --> 00:05:18,730 know that they go into same direction, so we know that 100 00:05:18,730 --> 00:05:19,820 they're going to be additive. 101 00:05:19,820 --> 00:05:22,950 And so we know that the total electric field is 102 00:05:22,950 --> 00:05:29,625 going to be 4K pi Q/A. 103 00:05:29,625 --> 00:05:32,670 104 00:05:32,670 --> 00:05:34,320 So now we know the exact strength of 105 00:05:34,320 --> 00:05:35,490 the electric field. 106 00:05:35,490 --> 00:05:41,220 Let's see if we can figure out the voltage difference between 107 00:05:41,220 --> 00:05:46,850 this point and this point. 108 00:05:46,850 --> 00:05:49,420 What was voltage difference, just as a review? 109 00:05:49,420 --> 00:05:53,680 Well, voltage difference is the electrical potential 110 00:05:53,680 --> 00:05:59,200 energy per charge if the charge was here versus here. 111 00:05:59,200 --> 00:06:02,500 So how much more potential energy per coulomb is there 112 00:06:02,500 --> 00:06:04,390 for a charge to be here relative to here? 113 00:06:04,390 --> 00:06:07,220 So another way to view it is a charge here, a positive charge 114 00:06:07,220 --> 00:06:10,170 here, because by default we're always assuming a positive 115 00:06:10,170 --> 00:06:14,100 charge when we talk about positive numbers and the 116 00:06:14,100 --> 00:06:15,820 direction of the field lines or what the positive 117 00:06:15,820 --> 00:06:16,840 charge would do. 118 00:06:16,840 --> 00:06:21,180 So by default, a positive charge here really wants to go 119 00:06:21,180 --> 00:06:24,500 up to this negative plate, although we later learned that 120 00:06:24,500 --> 00:06:28,500 most of the movement in electronics and electricity, 121 00:06:28,500 --> 00:06:30,080 it's actually the negative charge that's moving. 122 00:06:30,080 --> 00:06:31,385 It's the electrons moving. 123 00:06:31,385 --> 00:06:33,370 But let's say we did have a positive ion 124 00:06:33,370 --> 00:06:34,500 or a positive charge. 125 00:06:34,500 --> 00:06:39,170 The voltage is a measure of if any charge is here, how badly 126 00:06:39,170 --> 00:06:42,550 does it want to move to this point if it has a way to move? 127 00:06:42,550 --> 00:06:46,500 If we have air here or if we have a vacuum here, it might 128 00:06:46,500 --> 00:06:49,740 be difficult or impossible for it to move up here. 129 00:06:49,740 --> 00:06:52,310 But maybe if we were to connect a wire where the 130 00:06:52,310 --> 00:06:55,360 charges could freely conduct, then it will move. 131 00:06:55,360 --> 00:06:57,420 And the voltage is just kind of how badly 132 00:06:57,420 --> 00:06:58,210 does it want to move? 133 00:06:58,210 --> 00:07:00,990 You could almost view it as electrical pressure. 134 00:07:00,990 --> 00:07:03,290 And maybe I'll do a whole video on trying to get an 135 00:07:03,290 --> 00:07:05,330 intuitive understanding of voltage, because that really 136 00:07:05,330 --> 00:07:08,040 is probably the most important thing to get an intuitive 137 00:07:08,040 --> 00:07:10,240 understanding of, if you ever want to study electrical 138 00:07:10,240 --> 00:07:11,900 engineering or whatever. 139 00:07:11,900 --> 00:07:13,190 But anyway, back to the problem. 140 00:07:13,190 --> 00:07:17,060 We know that the combined electric field is this, right? 141 00:07:17,060 --> 00:07:19,590 It goes upwards in that direction. 142 00:07:19,590 --> 00:07:25,840 So what is the electric potential at this point 143 00:07:25,840 --> 00:07:28,430 relative to this point or the potential difference 144 00:07:28,430 --> 00:07:30,380 from here to here? 145 00:07:30,380 --> 00:07:34,220 Well, that's the amount of energy per charge it would 146 00:07:34,220 --> 00:07:42,650 take to move a positive charge from here to there, right? 147 00:07:42,650 --> 00:07:44,810 Remember, electric potential energy is the amount of work 148 00:07:44,810 --> 00:07:48,210 necessary to move a charge from there to there, and then 149 00:07:48,210 --> 00:07:52,810 the voltage is how much to do it per charge. 150 00:07:52,810 --> 00:07:53,910 Let me write that down. 151 00:07:53,910 --> 00:07:58,730 So the work necessary to move a charge from there to there-- 152 00:07:58,730 --> 00:08:07,010 let's say a 1-coulomb charge, it will be 1 coulomb times the 153 00:08:07,010 --> 00:08:09,060 electric field, because we're always going to have to be 154 00:08:09,060 --> 00:08:10,660 going against the electric field. 155 00:08:10,660 --> 00:08:13,260 So we have to apply an equal and opposite force. 156 00:08:13,260 --> 00:08:17,360 So the force that is going to be the electric field-- so 157 00:08:17,360 --> 00:08:19,750 far, this just generates this force. coulomb times electric 158 00:08:19,750 --> 00:08:23,430 field, charge times electric field, tells us the force on 159 00:08:23,430 --> 00:08:25,200 the charge, right? 160 00:08:25,200 --> 00:08:26,410 That's force, and then we have to 161 00:08:26,410 --> 00:08:28,380 multiply that times distance. 162 00:08:28,380 --> 00:08:31,020 Force times distance. 163 00:08:31,020 --> 00:08:36,590 So we see the work necessary is going to be the electric 164 00:08:36,590 --> 00:08:42,929 field times d joules-- the J is joules-- and so what is the 165 00:08:42,929 --> 00:08:48,210 voltage difference or the electric potential difference 166 00:08:48,210 --> 00:08:50,560 between this point and this point? 167 00:08:50,560 --> 00:08:54,720 Let's call that point a. 168 00:08:54,720 --> 00:08:56,450 Let's call that point b. 169 00:08:56,450 --> 00:09:03,040 So Va minus Vb, which is the voltage difference, that's 170 00:09:03,040 --> 00:09:06,000 essentially the electric potential energy difference 171 00:09:06,000 --> 00:09:08,580 divided by the charge. 172 00:09:08,580 --> 00:09:09,330 Or, per charge. 173 00:09:09,330 --> 00:09:11,900 Well, here, the charge was just 1, so we can just divide 174 00:09:11,900 --> 00:09:18,380 by 1, and we see that it is equal to the electric field 175 00:09:18,380 --> 00:09:20,410 times the distance. 176 00:09:20,410 --> 00:09:23,200 And the units are going to be joules because we divided both 177 00:09:23,200 --> 00:09:27,570 sides by charge joules per coulomb, or volts, right? 178 00:09:27,570 --> 00:09:29,600 That's just the units. 179 00:09:29,600 --> 00:09:30,380 So what does that equal? 180 00:09:30,380 --> 00:09:36,250 So the voltage difference-- so we can say change in voltage. 181 00:09:36,250 --> 00:09:39,130 The voltage difference is equal to the electric field, 182 00:09:39,130 --> 00:09:51,720 which we know is constant 4K pi Q over A times distance. 183 00:09:51,720 --> 00:09:53,020 Or we could rewrite this. 184 00:09:53,020 --> 00:10:00,300 Let's see if we could write Q as a function of V. 185 00:10:00,300 --> 00:10:03,220 So if we just do a little bit of algebraic manipulation, we 186 00:10:03,220 --> 00:10:08,110 can get Q is equal to what? 187 00:10:08,110 --> 00:10:13,800 We would essentially divide both sides by 4 pi Kd and 188 00:10:13,800 --> 00:10:19,350 multiply both sides by A, so we would get A 189 00:10:19,350 --> 00:10:29,680 over 4K pi d voltage. 190 00:10:29,680 --> 00:10:30,960 And why is this interesting? 191 00:10:30,960 --> 00:10:33,030 Why did I go through all of this work to get this 192 00:10:33,030 --> 00:10:33,870 relationship? 193 00:10:33,870 --> 00:10:37,780 Well, what it shows you, if you look at this, if we assume 194 00:10:37,780 --> 00:10:40,150 that the area of the plates aren't changing-- that's a 195 00:10:40,150 --> 00:10:43,630 constant; this is definitely a constant-- and if we assume 196 00:10:43,630 --> 00:10:46,980 that the distance between the plates don't change, what we 197 00:10:46,980 --> 00:10:49,690 see is that there's a proportional difference 198 00:10:49,690 --> 00:10:54,910 between the voltage and the amount of the combined charge 199 00:10:54,910 --> 00:10:56,080 in the plates. 200 00:10:56,080 --> 00:10:59,680 And that's interesting because, before doing this, 201 00:10:59,680 --> 00:11:03,240 maybe voltage is somehow proportional to the square of 202 00:11:03,240 --> 00:11:06,435 the charges or to the square root, but now we know that 203 00:11:06,435 --> 00:11:08,280 it's directly proportionally. 204 00:11:08,280 --> 00:11:14,430 And actually, this term right here has a name, and it is 205 00:11:14,430 --> 00:11:15,290 called capacitance. 206 00:11:15,290 --> 00:11:18,220 And so another way of rewriting this, if we divide 207 00:11:18,220 --> 00:11:33,660 both sides by voltage, we get Q/V is equal to 1 over 4K pi 208 00:11:33,660 --> 00:11:37,270 area over distance. 209 00:11:37,270 --> 00:11:41,340 And so what it essentially says is that the amount of 210 00:11:41,340 --> 00:11:43,730 energy that-- well, actually, I don't want to 211 00:11:43,730 --> 00:11:45,520 go into that yet. 212 00:11:45,520 --> 00:11:48,630 But for a given configuration, and the configuration is 213 00:11:48,630 --> 00:11:53,030 defined by the area of the plates and the distance-- for 214 00:11:53,030 --> 00:11:55,890 a given configuration, if I know the amount of charge that 215 00:11:55,890 --> 00:11:59,000 I put onto the plates, if I did a minus Q here and a plus 216 00:11:59,000 --> 00:12:00,900 Q here, I know the voltage across the 217 00:12:00,900 --> 00:12:02,875 plates or vice versa. 218 00:12:02,875 --> 00:12:05,090 If I know the voltage across the plates and I know its 219 00:12:05,090 --> 00:12:09,300 configuration, I know how much charge there is, and this is 220 00:12:09,300 --> 00:12:12,010 called capacitance, and the unit for capacitance 221 00:12:12,010 --> 00:12:13,990 is called the Farad. 222 00:12:13,990 --> 00:12:15,590 And if you become an electrical engineer or even 223 00:12:15,590 --> 00:12:17,350 take a couple of electrical engineering courses, you'll 224 00:12:17,350 --> 00:12:19,370 become very familiar with this. 225 00:12:19,370 --> 00:12:22,650 And one other thing to point out; this term right here, 226 00:12:22,650 --> 00:12:26,200 just so you know a little bit of terminology. 227 00:12:26,200 --> 00:12:27,360 This term right here. 228 00:12:27,360 --> 00:12:36,630 This 1 over 4K pi, this is often called epsilon nought, 229 00:12:36,630 --> 00:12:39,220 or just epsilon, and that's called the permittivity of 230 00:12:39,220 --> 00:12:41,560 free space or permittivity of the vacuum. 231 00:12:41,560 --> 00:12:45,460 And maybe in a future lecture or a future video, I'll talk 232 00:12:45,460 --> 00:12:47,010 more about why it's called that. 233 00:12:47,010 --> 00:12:49,910 But anyway, I'm already well over the time limit. 234 00:12:49,910 --> 00:12:55,260 So I just wanted to give you a sense of, one, that you can 235 00:12:55,260 --> 00:12:58,230 calculate the voltage across what we call, in 236 00:12:58,230 --> 00:12:59,600 this case, a capacitor. 237 00:12:59,600 --> 00:13:01,430 It has capacitance. 238 00:13:01,430 --> 00:13:03,385 That voltage, you can kind of view it as 239 00:13:03,385 --> 00:13:04,170 the electric pressure. 240 00:13:04,170 --> 00:13:06,800 How bad does the charge here want to move here? 241 00:13:06,800 --> 00:13:09,750 And if you put a wire here, you'll learn in a second-- not 242 00:13:09,750 --> 00:13:11,380 in a second, in several videos-- that 243 00:13:11,380 --> 00:13:12,575 that charge will flow. 244 00:13:12,575 --> 00:13:15,610 Or actually the negative charges will flow this way and 245 00:13:15,610 --> 00:13:16,310 generate current. 246 00:13:16,310 --> 00:13:17,780 And we'll do that when we start learning a little bit 247 00:13:17,780 --> 00:13:19,360 more about electricity. 248 00:13:19,360 --> 00:13:24,120 For any given configuration, it has a corresponding 249 00:13:24,120 --> 00:13:27,030 capacitance, and then given that capacitance, if I put 250 00:13:27,030 --> 00:13:28,910 some amount of charge, I can figure out the voltage, or if 251 00:13:28,910 --> 00:13:31,070 I know there's some voltage, I can figure out the charge. 252 00:13:31,070 --> 00:00:00,000 Anyway, I will see you in the next video.