1 00:00:00,360 --> 00:00:05,920 So let us now come to our very first important example, which is the electric dipole. 2 00:00:07,680 --> 00:00:13,380 So in the previous lecture, we have established the equation on how we can calculate the electrostatic 3 00:00:13,380 --> 00:00:15,900 potential for multiple point charges. 4 00:00:16,470 --> 00:00:22,320 And it turns out that we can just write it down as to some of the individual electrostatic potentials 5 00:00:22,320 --> 00:00:23,920 of the individual charges. 6 00:00:25,470 --> 00:00:31,530 So now that we want to consider an electric dipole, we first have to understand what this dipole means 7 00:00:31,530 --> 00:00:32,460 and what it looks like. 8 00:00:33,300 --> 00:00:40,700 So a dipole describes two electric charges with opposite signs that are separated by some distance away. 9 00:00:42,060 --> 00:00:48,480 And so here I have defined this vector AI and this vector always points from the negative charge to 10 00:00:48,480 --> 00:00:49,560 the positive charge. 11 00:00:50,750 --> 00:00:56,600 And so this allows us also to introduce here to dipole moment, which is given by the individual in 12 00:00:56,600 --> 00:01:00,170 charge, and then we multiply by this factor. 13 00:01:01,340 --> 00:01:09,140 So this means dipole moment P is also a vector and it is a vector because, of course, it's very important 14 00:01:09,290 --> 00:01:11,120 how this dipole is oriented. 15 00:01:13,370 --> 00:01:20,360 So next, we want to consider this electrostatic potential for this dipole, so we, of course, get 16 00:01:20,360 --> 00:01:26,900 to terms from the sun because we have two charges and both of these charges are equal in magnitude. 17 00:01:26,900 --> 00:01:28,910 So we can hear pull out this cue. 18 00:01:29,990 --> 00:01:35,690 And now I have to find that our zero positions here at this position of the positive charge. 19 00:01:36,110 --> 00:01:41,750 So we get the first turn, which is plus one divided by hour. 20 00:01:42,380 --> 00:01:44,570 And then we have here the negative charge. 21 00:01:44,570 --> 00:01:52,040 So we get a minus sign and then we get instead of, ah, we get our plasma because the position here 22 00:01:52,040 --> 00:01:53,100 is at minus eight. 23 00:01:54,470 --> 00:01:55,910 And here we have the other minus sign. 24 00:01:56,540 --> 00:02:00,980 So this is what our electrostatic potential looks like for this dipole. 25 00:02:02,120 --> 00:02:04,940 So we did not really have to calculate it here. 26 00:02:04,940 --> 00:02:11,990 We could just use our result from the single charge and just add up to two electrostatic potentials. 27 00:02:13,480 --> 00:02:22,510 Now, what we can do next is we can do a Taylor expansion, so it's not not a big deal if you don't 28 00:02:22,510 --> 00:02:30,240 perfectly remember what a Taylor expansion is and how it works, maybe you only know it from one dimension. 29 00:02:30,610 --> 00:02:36,520 So in multiple dimensions, in our case, in three dimensions, a Taylor expansion looks like this. 30 00:02:36,740 --> 00:02:42,760 So you have here it is derivatives, which in three dimensions is, of course, the Nahlah operator. 31 00:02:43,390 --> 00:02:48,550 Then you have here as one of our faculty and then we have to function that we want to evolve. 32 00:02:49,240 --> 00:02:53,490 So we have here that one divided by our Arpels eight. 33 00:02:53,560 --> 00:02:56,480 So this term here can be expressed like this. 34 00:02:58,000 --> 00:03:06,040 So what we can do now is that we can say that we want to look at the electric field and at the electrostatic 35 00:03:06,040 --> 00:03:09,640 potential far away from this dipole. 36 00:03:10,660 --> 00:03:18,070 So, for example, if you consider some very small molecule, which is has two two atoms, one of them 37 00:03:18,070 --> 00:03:20,650 is positively charged, the other one is negatively charged. 38 00:03:21,110 --> 00:03:23,140 So this gives us an electric dipole. 39 00:03:23,650 --> 00:03:30,280 And then we want to find out what the electrostatic potential and the electric field look like far away 40 00:03:30,280 --> 00:03:31,630 from this molecule. 41 00:03:32,590 --> 00:03:39,450 So this means that this position R is much, much larger than this distance away. 42 00:03:40,960 --> 00:03:47,890 And what this means is we don't have to take here this whole row, so we don't have to take into account 43 00:03:47,890 --> 00:03:53,940 infinite numbers of terms, but we just have to take into account the first two terms. 44 00:03:54,460 --> 00:03:57,080 So this is an equal zero and a new one. 45 00:03:58,090 --> 00:04:04,330 So this means our electrostatic potential in this case is approximately one of her, which is this one 46 00:04:04,330 --> 00:04:04,620 here. 47 00:04:05,080 --> 00:04:09,000 Then we get another one of our which is this term here, four and equals zero. 48 00:04:09,970 --> 00:04:11,230 So this gives us one. 49 00:04:11,380 --> 00:04:14,070 This gives us one and then just one over. 50 00:04:14,980 --> 00:04:18,420 And then we get another term, which gives us again here one. 51 00:04:18,430 --> 00:04:21,870 This one gives us eight times Nabala and this one gives us one of our. 52 00:04:22,000 --> 00:04:23,210 So that's this term here. 53 00:04:24,490 --> 00:04:31,960 So you see that these two terms cancel out and we are left with just this pretty factor and then this 54 00:04:31,960 --> 00:04:32,590 term here. 55 00:04:34,700 --> 00:04:42,380 Now, this one is still quite difficult, we have used a number of operator acting on one of our and 56 00:04:42,620 --> 00:04:45,650 you can as an exercise, calculate what this looks like. 57 00:04:46,160 --> 00:04:48,220 Here you can also already see the solution. 58 00:04:48,230 --> 00:04:53,060 It's the position vector of our divided by our to the power of three. 59 00:04:54,110 --> 00:04:58,460 So please post a video and do the exercise and calculate this yourself. 60 00:05:01,310 --> 00:05:03,390 So this is what you should have come up with. 61 00:05:03,470 --> 00:05:08,470 This is not the operator acting on one of our to make it really, really simple. 62 00:05:08,870 --> 00:05:14,930 I write it down in the coordinates so we have X squared plus Y square policy square and then add to 63 00:05:14,930 --> 00:05:22,760 the power of minus one over two, because this position vector is the square root of X Square skyscrapers, 64 00:05:22,790 --> 00:05:23,300 Z Square. 65 00:05:24,830 --> 00:05:31,940 Now you can do this for every individual coordinate because you know there's not an operator. 66 00:05:31,940 --> 00:05:36,090 It's a vector of partial derivatives with respect to X, Y and Z. 67 00:05:36,920 --> 00:05:41,090 So let's start out with calculating the derivative with respect to X.. 68 00:05:41,630 --> 00:05:44,300 So this means we got this outer derivative. 69 00:05:44,690 --> 00:05:48,550 We have to pull this one in front and decrease the exponent by one. 70 00:05:48,920 --> 00:05:55,220 So we get minus one half times this one and the exponent is now minus three 1/2. 71 00:05:55,700 --> 00:06:02,960 And then we have to also consider the inner derivative with which four X is two X and the other terms 72 00:06:02,960 --> 00:06:03,470 give zero. 73 00:06:03,620 --> 00:06:04,940 So we have here to X. 74 00:06:05,750 --> 00:06:10,520 And now of course, if you do it for the other coordinates, you will get the same thing. 75 00:06:10,520 --> 00:06:14,180 But for the inner derivative, you will get to Y and 2C. 76 00:06:15,410 --> 00:06:17,880 So this means we can pull this one out front. 77 00:06:18,410 --> 00:06:21,830 This one is just one of our to the power of three. 78 00:06:22,400 --> 00:06:25,310 And here this one is just a position vector. 79 00:06:25,520 --> 00:06:29,510 So we get minus position nektar divided by our to the power of three. 80 00:06:31,580 --> 00:06:39,350 So this means our electrostatic potential far away from the dipole is given by this prefecture times 81 00:06:39,650 --> 00:06:46,190 to charge, and then here we have eight times our scale, our product, and then divide it by half of 82 00:06:46,190 --> 00:06:47,070 the power of three. 83 00:06:48,260 --> 00:06:53,720 And of course, also we can use here just dipole moment P and write it down like this. 84 00:06:57,560 --> 00:07:00,380 So this is all a result for the electrostatic potential. 85 00:07:00,740 --> 00:07:03,350 Now, what does the electric field look like? 86 00:07:04,930 --> 00:07:10,810 So, of course, we know that we can calculate the electric field from the electrostatic potential and 87 00:07:10,810 --> 00:07:13,820 it's very easy, we just have to calculate the gradient. 88 00:07:14,650 --> 00:07:16,300 So this is what it looks like in general. 89 00:07:16,310 --> 00:07:21,200 And now we have to, of course, calculate this term here, which is not so simple. 90 00:07:21,820 --> 00:07:26,040 So once again, it's a nice exercise, quite similar to the previous exercise. 91 00:07:26,410 --> 00:07:29,470 So if you want to solve it, please post a video and do it yourself. 92 00:07:32,380 --> 00:07:39,610 And here is the solution, so we calculate this gradient of P times are divided by R to the power of 93 00:07:39,610 --> 00:07:48,340 three, and most often it's very simple or most simple to just write it down in terms of coordinates. 94 00:07:49,330 --> 00:07:53,650 So we write it down as times X plus D, Y and Z terms. 95 00:07:54,250 --> 00:08:02,380 And what we get is we first do it for the the first component, which is the partial derivative with 96 00:08:02,380 --> 00:08:03,730 respect to X.. 97 00:08:04,300 --> 00:08:12,820 So what we get is that we differentiate here with respect to X, so we get P X divided by R to the power 98 00:08:12,820 --> 00:08:13,290 of three. 99 00:08:14,320 --> 00:08:18,250 But then we also have to remember that this R here is also X dependent. 100 00:08:18,830 --> 00:08:21,390 So we have to use the product rule and get another term. 101 00:08:22,180 --> 00:08:28,570 So that's our first term P X provided by our to the power of three and then the other term would be 102 00:08:29,170 --> 00:08:30,790 P times X. 103 00:08:32,390 --> 00:08:37,010 Divided by the derivative of one of our to the power of three. 104 00:08:39,600 --> 00:08:47,250 And yeah, this we can also do for the other coordinates, which gives us this in total, and then of 105 00:08:47,250 --> 00:08:54,450 course, we have to still calculate the derivative of one divided by our to the power of three. 106 00:08:55,880 --> 00:09:02,840 And this is just as crazy gradient acting on this one here, and this is very, very similar to this 107 00:09:02,840 --> 00:09:04,410 exercise on the previous slide. 108 00:09:04,880 --> 00:09:09,430 Now we just get here as a as a prefect, a minus three divided by two. 109 00:09:09,740 --> 00:09:15,920 And here we get it as an exponent minus five, divided by two because we have to decrease this one by 110 00:09:15,920 --> 00:09:16,190 one. 111 00:09:17,210 --> 00:09:22,320 Then we also have to account for the inner derivative, which is again, to X, to Y, to Z. 112 00:09:22,910 --> 00:09:26,780 So this means this gradient here is equal to minus three. 113 00:09:28,250 --> 00:09:33,860 Divided by the opposition, Victor must be better positioned to divide it by ah to the power of five. 114 00:09:35,310 --> 00:09:44,010 So now we put all of these things together and we have our electric field, which is given by this prefecture, 115 00:09:44,010 --> 00:09:46,980 and then we get to terms here due to the product rule. 116 00:09:47,510 --> 00:09:51,900 The first term is the dipole moment divided by our to the power of three. 117 00:09:52,650 --> 00:09:59,370 And then we have you this other term where we have to project the dipole moment on the position vector 118 00:09:59,610 --> 00:10:06,900 and then we get this dependence of R divided by R to the power of five so we can write it down like 119 00:10:06,900 --> 00:10:07,260 this. 120 00:10:07,860 --> 00:10:13,170 And on the next slide I will show you what the electrostatic field and also the electric field look 121 00:10:13,170 --> 00:10:13,500 like. 122 00:10:16,600 --> 00:10:22,700 So we have now figured out what the electrostatic field and electric field look like far away from the 123 00:10:22,700 --> 00:10:23,270 dipole. 124 00:10:24,350 --> 00:10:31,880 And so what we have actually done here is we have used the approximation that is only valid for our 125 00:10:31,880 --> 00:10:34,280 as much larger to aim for. 126 00:10:34,280 --> 00:10:39,000 This means we have to look at our dipole and have to sort of say zoom out. 127 00:10:39,410 --> 00:10:44,720 So this is when we zoom in and when we zoom out these to come closer to each other. 128 00:10:45,410 --> 00:10:51,400 And if we zoom out very, very far, we cannot really resolve the distance anymore between them. 129 00:10:51,410 --> 00:10:54,050 But please remember, there will always be this distance A. 130 00:10:57,020 --> 00:11:06,950 So another important thing have to realize is that if we want to find out how this electric field behaves 131 00:11:06,950 --> 00:11:11,300 far from the dipole, we have to consider this fraction here. 132 00:11:11,570 --> 00:11:15,080 And you can see that in these two terms we have here, our square. 133 00:11:15,080 --> 00:11:18,030 And he also our teams are which is also a square. 134 00:11:18,890 --> 00:11:21,020 And then here we have our to the par five. 135 00:11:21,620 --> 00:11:28,910 So this means essentially the electric field decreases with the power of one divided by our to the power 136 00:11:28,910 --> 00:11:29,480 of three. 137 00:11:33,100 --> 00:11:39,700 So here you can see again, yeah, what the electric field looks like for such a dipole, and if you 138 00:11:39,700 --> 00:11:45,610 remember correctly, we have already figured out this electric field in one of the very first lecture's. 139 00:11:46,830 --> 00:11:53,010 So this is just the addition of the two individual electric fields and on the previous slides, we have 140 00:11:53,160 --> 00:11:59,850 simplified this or generalised this dipole to the so-called mathematical dipole, where we have considered 141 00:11:59,850 --> 00:12:06,370 the approximation that the position vector is much larger than the distance between these two charges. 142 00:12:07,620 --> 00:12:15,690 And another thing that I want to mention here is that we can find cuts along these electric field lines 143 00:12:16,110 --> 00:12:18,770 where the electrostatic potential is constant. 144 00:12:19,470 --> 00:12:25,530 And, for example, if he cuts right between these two charges, we will find that in on this red line 145 00:12:25,530 --> 00:12:32,700 here, the electrostatic potential is zero in all cases because the individual electrostatic potentials 146 00:12:32,700 --> 00:12:36,660 of these two charges cancel out because they have opposite sides. 147 00:12:37,800 --> 00:12:42,670 And so this line where the electrostatic potential to zero will be important later. 148 00:12:43,020 --> 00:12:44,760 So maybe you remember this later.