1 00:00:00,210 --> 00:00:06,720 In this lecture, I want to present to you how you can calculate the energy in electricity based on 2 00:00:06,720 --> 00:00:12,090 our established equations for the electric field and for the electrostatic potential. 3 00:00:14,410 --> 00:00:21,310 So to derive the equation for the energy and electrostatic, we need to go back to one of the very first 4 00:00:21,310 --> 00:00:27,100 slides of this cause this was in one of the first sections on the Maxwell's equations where we have 5 00:00:27,100 --> 00:00:30,960 established here this continuity equation for the energy density. 6 00:00:31,780 --> 00:00:38,350 And here we have found out that the energy density is given by such a term where you have the absolute 7 00:00:38,350 --> 00:00:43,610 value of the electric field squared, plus the absolute value of the magnetic field squared. 8 00:00:44,950 --> 00:00:50,530 But since we are in electrostatic, we do not have these magnetic fields, so we can focus only here 9 00:00:50,530 --> 00:00:51,310 on this term. 10 00:00:52,600 --> 00:00:59,110 Now, since this is the energy density, we need to integrate over the whole space of the whole volume 11 00:00:59,410 --> 00:01:01,180 to get the total energy. 12 00:01:02,050 --> 00:01:03,720 So this is the expression. 13 00:01:03,850 --> 00:01:10,780 And if we know what the electric field independence of the position vector art looks like, then we 14 00:01:10,780 --> 00:01:15,040 can quite easily determine the energy of the whole system and we are done. 15 00:01:16,420 --> 00:01:21,970 But also, I want to give you here an alternative method or an alternative expression on how you can 16 00:01:21,970 --> 00:01:29,170 calculate the energy if you do not know the electric field yet and if you only know the charged density 17 00:01:29,170 --> 00:01:30,010 distribution. 18 00:01:31,100 --> 00:01:38,090 And to do this, we use this and that's here where we have introduced the electrostatic potential and 19 00:01:38,090 --> 00:01:42,740 we ride the electric field in terms of this gradient of this scale scalar potential. 20 00:01:43,640 --> 00:01:47,230 So he can see I have only replaced one of these eyes. 21 00:01:47,240 --> 00:01:48,190 So this is square. 22 00:01:48,200 --> 00:01:49,810 So it's E times E, obviously. 23 00:01:50,150 --> 00:01:54,560 And so I have really expressed one of these ease in terms of minus Knobler five. 24 00:01:56,330 --> 00:02:05,390 So the next step, what I do here is I first show you the results, so when you look at this term here, 25 00:02:05,390 --> 00:02:11,930 you have to let Nabala act on both of these terms here on the field and on the scale of potential. 26 00:02:11,930 --> 00:02:12,390 Fine. 27 00:02:13,010 --> 00:02:16,700 So you have to apply the product rule, which gives you two terms. 28 00:02:17,480 --> 00:02:23,480 So the first term would be e scalar product, not acting on Fi's. 29 00:02:23,510 --> 00:02:25,060 This is exactly this term here. 30 00:02:25,430 --> 00:02:34,310 But then you get another term, which is five times or five times the divergence of this electric field 31 00:02:34,610 --> 00:02:35,600 with a minus sign. 32 00:02:36,230 --> 00:02:41,210 So since this term is not here, we have to get rid of it again to compensate it. 33 00:02:41,210 --> 00:02:42,610 So we add this term here. 34 00:02:43,310 --> 00:02:45,740 So we know now that these two equations are equal. 35 00:02:47,060 --> 00:02:52,670 Now, in the next step, I use this relation here, which is one of the maximal equations that the divergence 36 00:02:52,670 --> 00:02:56,990 of the electric field can be expressed in terms of the charged density. 37 00:02:58,040 --> 00:03:00,680 And also so let's hear this right term. 38 00:03:01,010 --> 00:03:05,960 And also I change I express this left term here. 39 00:03:07,260 --> 00:03:13,500 And so if we go to one dimension, maybe you better understand what I did here, some what you see is 40 00:03:13,500 --> 00:03:20,790 we have an integral of a derivative and if you integrate a derivative, you get the initial function 41 00:03:20,820 --> 00:03:21,990 at a certain position. 42 00:03:23,280 --> 00:03:27,660 And so what this does here is we basically. 43 00:03:28,750 --> 00:03:33,700 Get rid of one of the integration's, so this is a three dimensional integration, so these are actually 44 00:03:33,700 --> 00:03:34,740 three integrals here. 45 00:03:35,350 --> 00:03:40,020 And when we get rid of this gradient, we are left with only two integrals. 46 00:03:40,120 --> 00:03:42,100 So this here is a surface integral. 47 00:03:43,390 --> 00:03:50,440 And this is maybe a bit too complicated, but let me tell you that this term here describes the surface 48 00:03:50,440 --> 00:03:51,830 charges of a system. 49 00:03:52,300 --> 00:03:58,840 So, for example, if we have a sphere and we have some charged separation, then this term contributes 50 00:03:58,840 --> 00:03:59,680 an energy. 51 00:04:00,670 --> 00:04:06,820 But if we make our system infinitely large or if we have a system that is finite but does not have any 52 00:04:06,820 --> 00:04:10,030 surface charges, then this term here is zero. 53 00:04:10,480 --> 00:04:13,000 And for simplicity, we want to consider this case. 54 00:04:13,270 --> 00:04:18,100 So we are only left with this right hand term, which is quite easy to derive when you look here. 55 00:04:18,370 --> 00:04:20,200 We have just used the Maxwell equation. 56 00:04:20,320 --> 00:04:21,780 So I hope that you understand this. 57 00:04:23,290 --> 00:04:28,750 So if we have no surface charges, then this is here the energy of the whole system. 58 00:04:29,170 --> 00:04:36,070 And you can see we do not need the electric field anymore, but we only need to charge distribution 59 00:04:36,310 --> 00:04:38,230 and the electrostatic potential. 60 00:04:39,960 --> 00:04:45,240 So now we can go even one step further, because we already know what the electrostatic potential looks 61 00:04:45,240 --> 00:04:49,370 like and that it only depends on the charged distribution. 62 00:04:50,190 --> 00:04:55,890 So we take this expression and put it in here and you see that this is the general expression for our 63 00:04:55,890 --> 00:05:01,990 energy, and it only depends on the charged distribution. 64 00:05:02,220 --> 00:05:09,120 So you have such a product and please notice that you have to you cannot simply square this term here, 65 00:05:09,120 --> 00:05:12,300 but you have to integrate of a different position vectors. 66 00:05:12,330 --> 00:05:18,960 So you have to do two integrations, one after another, the first integration of R and the other one 67 00:05:18,960 --> 00:05:19,840 over the other arm. 68 00:05:20,310 --> 00:05:27,120 And the reason is that we have this term here where we have a difference, quite a distance of two precision 69 00:05:27,120 --> 00:05:27,600 vectors. 70 00:05:29,810 --> 00:05:37,220 Now, if we have parking charges, we can integrate over these charge distributions because these charge 71 00:05:37,220 --> 00:05:42,910 distributions will then be zero almost everywhere and will only be finite at certain points. 72 00:05:43,460 --> 00:05:46,940 And what we get is then that we integrate of our role. 73 00:05:46,940 --> 00:05:56,780 And Debe, so we get just total charge of this point and we have to add up all of these powers and we 74 00:05:56,780 --> 00:06:03,800 cannot consider himself into action because then the position vector distance would be zero and we would 75 00:06:03,800 --> 00:06:04,790 divide by zero. 76 00:06:04,880 --> 00:06:06,070 So that's not possible. 77 00:06:07,220 --> 00:06:12,740 And if we only have two charges, then we get this famous expression that you probably know already 78 00:06:13,350 --> 00:06:13,920 is here. 79 00:06:13,940 --> 00:06:16,640 Now, the energy of two point charges.