1 00:00:00,300 --> 00:00:05,490 OK, that was really difficult, but we have done it, we have calculated the electric field and the 2 00:00:05,490 --> 00:00:12,030 magnetic field for our Hatzius Dipole, so let's go ahead and discuss the results. 3 00:00:12,960 --> 00:00:19,160 So what we had considered is a time dependent current in an infinitesimally small wire. 4 00:00:19,560 --> 00:00:26,250 So basically we have here a tiny segment of an antenna and our results where this be field with the 5 00:00:26,250 --> 00:00:30,120 two terms and the electric field with these six terms here. 6 00:00:30,660 --> 00:00:37,410 And in both cases, we still have our -- time argument, which comes, of course, from the -- 7 00:00:37,410 --> 00:00:38,130 potentials. 8 00:00:39,630 --> 00:00:41,940 So first of all, we can do something very simple. 9 00:00:42,390 --> 00:00:50,520 We can check if this result is equal to our results from electrostatic, where we have considered the, 10 00:00:50,670 --> 00:00:51,810 you know, the static dipole. 11 00:00:52,740 --> 00:00:58,360 And this we can do by considering considering a time independent dipole moment. 12 00:00:59,070 --> 00:01:03,390 So this means that pitot a zero and also P is zero. 13 00:01:04,200 --> 00:01:09,610 So this means we get a magnetic field of zero, which is, of course, clear because we have yeah, 14 00:01:09,630 --> 00:01:15,120 we have static electric charges and we just got an electric field with these two terms. 15 00:01:15,570 --> 00:01:22,200 And when we compare it to our previous results from Electrostatic, we see it's exactly the same result. 16 00:01:23,100 --> 00:01:24,510 That's really satisfying. 17 00:01:24,510 --> 00:01:28,110 I think because we had this difficult calculation. 18 00:01:28,410 --> 00:01:33,060 We have first had this difficult lecture on proving that the -- potentials are right. 19 00:01:33,480 --> 00:01:40,350 Then we had the difficult lecture on calculating the electric field and it turns out that's in the limit 20 00:01:40,350 --> 00:01:41,940 of time, independent type of moments. 21 00:01:41,940 --> 00:01:46,380 We get the same result as previously, which means we did a good job. 22 00:01:46,380 --> 00:01:47,580 And that is correct. 23 00:01:49,000 --> 00:01:54,100 But now, of course, our aim was not to calculate time independent problems, because then we could 24 00:01:54,100 --> 00:01:56,840 have done it in a much more simple way. 25 00:01:57,490 --> 00:02:04,960 Instead, we want to calculate time dependent on, for example, harmonically oscillating dipoles. 26 00:02:06,520 --> 00:02:14,920 And so what we can assume here is some some cosine dependence on the time or as you would typically 27 00:02:14,920 --> 00:02:19,340 do it, you write it down in terms of an exponential function with an imaginary argument. 28 00:02:20,200 --> 00:02:28,150 So you have your electric dipole moment at the time, zero, and then it oscillates and thus you do 29 00:02:28,150 --> 00:02:36,040 by multiplying such an E to the eye on McAtee, where Armacost the frequency of the oscillation. 30 00:02:37,570 --> 00:02:44,260 Now you can go ahead and calculate p dot and people dot dot is just minus eight times, oh my God, 31 00:02:44,320 --> 00:02:51,520 times this whole function and then for people that you do it one more time, so you get plus I square 32 00:02:51,520 --> 00:02:57,150 Iomega which plus square Omega's Square, which is of course minus Omega Square Times Pete. 33 00:02:58,150 --> 00:03:03,120 And now we can go ahead and take these and put them into B and E. 34 00:03:04,030 --> 00:03:11,230 So when we start with B what we get is the following we gets here is P in both cases and then of course 35 00:03:11,230 --> 00:03:18,640 here the different prefectures and you can see we have all we got of a C, and this is typically something 36 00:03:18,640 --> 00:03:25,570 that you introduce as a as the K vector, which is which was for example, for an electromagnetic wave, 37 00:03:25,900 --> 00:03:27,250 the so-called wave factor. 38 00:03:29,020 --> 00:03:36,940 Now what we must also consider is that we have in both cases here our P, which was P zero times E to 39 00:03:36,940 --> 00:03:44,380 the Omega T, but it's also really important to remember that this T argument is, in fact, not simply 40 00:03:44,380 --> 00:03:49,200 our T, but it's also has to take into account the retardation. 41 00:03:49,720 --> 00:03:54,010 So we have to write instead of t t minus are divided by C. 42 00:03:55,550 --> 00:04:08,810 So we get here I omega T minus, I only got T plus I Omega are divided by sea and this is our K.R. minus 43 00:04:08,810 --> 00:04:13,590 Omega T and this is something that you already know from our electromagnetic waves. 44 00:04:13,910 --> 00:04:21,110 So in some sense we get to hear again an electromagnetic wave, but it has some difficult space dependence 45 00:04:21,110 --> 00:04:21,950 and some difficult. 46 00:04:24,680 --> 00:04:30,260 Now, one other thing that you can do is you can introduce the wavelength of this of this wave. 47 00:04:30,560 --> 00:04:36,350 And this is just, you know, two pi, one of akei or vice versa as it's written down here. 48 00:04:37,490 --> 00:04:43,040 And what I have done here is I have ordered these terms with respect to their power. 49 00:04:43,040 --> 00:04:47,580 In a SO later, we can discuss the near field and the far field. 50 00:04:48,980 --> 00:04:56,810 So here we get as our sold for B that we have some unit vector along R and then the cross products of 51 00:04:56,810 --> 00:05:01,040 these terms and the vectorial orientation is along P0. 52 00:05:03,070 --> 00:05:06,500 All right, so we have the results for B, let's do the same thing for E! 53 00:05:07,720 --> 00:05:14,020 So here what we can can do or what we must do is we must take P dot and P double dots and put it in. 54 00:05:14,950 --> 00:05:20,600 So in all cases we get no P and we get these three factors here. 55 00:05:22,210 --> 00:05:23,890 So this is what we get here. 56 00:05:23,890 --> 00:05:27,850 You get the term four for these two, you get a term for these two. 57 00:05:27,850 --> 00:05:38,650 And here this one is for these two and you can see the prefectures minus I omegle divided by C and minus 58 00:05:38,800 --> 00:05:44,260 Omega squared, divided by C squared, because here we have C Square, here we have C, here we have 59 00:05:44,260 --> 00:05:44,650 nothing. 60 00:05:46,090 --> 00:05:51,310 All right, so now we once again take this expression for. 61 00:05:52,060 --> 00:05:54,120 So we get P0 in all cases. 62 00:05:54,400 --> 00:06:02,680 So yeah, this one also has to be p0 in fact, and then we pull out this exponential function and also 63 00:06:02,680 --> 00:06:04,780 have to consider the -- argument. 64 00:06:06,370 --> 00:06:09,580 So this is what we what we get, we get. 65 00:06:11,910 --> 00:06:16,320 Yeah, OK, we get the same thing as here, so nothing has really changed, but the only thing that 66 00:06:16,320 --> 00:06:25,950 I have changed is this term here, because this one looks exactly like this double, double, double 67 00:06:25,950 --> 00:06:30,780 vector product where we used to be a C minus C, a B rule. 68 00:06:31,320 --> 00:06:34,140 So we can write this one down as this one. 69 00:06:35,370 --> 00:06:40,440 So to make it just a bit more look more simple and a bit more beautiful, but it's actually not really 70 00:06:40,440 --> 00:06:41,790 that important, to be honest. 71 00:06:43,230 --> 00:06:49,620 OK, now we are almost done, the only thing that I want to do is I want to order these terms with respect 72 00:06:49,620 --> 00:06:53,710 to their powers and are just in the same way as we did for the magnetic field. 73 00:06:54,390 --> 00:07:02,700 So this means we have to replace all of these actors are with the unit, Victor E. R. and have to pull 74 00:07:02,700 --> 00:07:05,190 the legs are out of this vector. 75 00:07:05,970 --> 00:07:14,010 So what we get is we get basically the same thing, but we have pulled out the the RCA everywhere and 76 00:07:14,010 --> 00:07:17,060 have canceled them with our to the power of five. 77 00:07:17,070 --> 00:07:21,420 So now we got here on the R to the power of three because here we have these two hours, here we have 78 00:07:21,420 --> 00:07:25,970 these two hours and then the similar thing we can do for the other terms here. 79 00:07:26,520 --> 00:07:33,270 So we finally have arrived at the results for a harmonically oscillating electric dipole. 80 00:07:33,840 --> 00:07:38,570 And we have also ordered our terms with respect to the powers of R. 81 00:07:39,750 --> 00:07:48,810 So what this means is in the very close vicinity to our antenna, we have terms that are dominating 82 00:07:49,170 --> 00:07:50,250 which are these ones here. 83 00:07:50,310 --> 00:07:51,880 So it's just a single term. 84 00:07:52,410 --> 00:07:59,250 This is because for for, you know, in the vicinity we have an R that is very small. 85 00:07:59,550 --> 00:08:06,360 So this means that one of our to the power of free will be very large compared to these other two terms 86 00:08:06,480 --> 00:08:08,130 and also to these two terms. 87 00:08:09,210 --> 00:08:18,120 And then far away we have a small R, so this means here to terms with the power, um, our to the minus 88 00:08:18,120 --> 00:08:21,780 one, they dominate with respect to all of the other terms. 89 00:08:22,020 --> 00:08:24,960 So this means here we will mainly get these two terms. 90 00:08:26,130 --> 00:08:27,130 So what does it mean. 91 00:08:27,780 --> 00:08:34,290 So in the near field, alien blue, what we get is that the electric field dominates because the electric 92 00:08:34,290 --> 00:08:35,990 field is dominated by this term. 93 00:08:36,450 --> 00:08:41,820 And this is much larger than this term of the magnetic field, because here we have our to the power 94 00:08:41,820 --> 00:08:43,290 of two instead of three. 95 00:08:44,760 --> 00:08:52,890 And it also means that the electric field in the near field is rather similar to our dipole from electrostatic 96 00:08:52,890 --> 00:08:53,980 so that you can see here. 97 00:08:54,900 --> 00:09:02,310 So along the Z axis, we get a field profile that is pointing along the radio direction. 98 00:09:02,670 --> 00:09:09,540 So in this case, Homsi and especially close to the x y plane, we get a field that is pointing along 99 00:09:09,540 --> 00:09:12,780 this Edyta direction, which is here along this one. 100 00:09:13,960 --> 00:09:19,900 So in the near field, it looks rather similar to what we already know, and this is why you could also 101 00:09:19,900 --> 00:09:23,700 call it a very slowly oscillating dipole in the near field. 102 00:09:25,390 --> 00:09:29,920 Now, in the far field, it looks very different because here we have these two terms to dominate. 103 00:09:30,520 --> 00:09:37,540 And you can also or you can immediately see that in the far field, the electric field is perpendicular 104 00:09:37,720 --> 00:09:39,160 to the electric field. 105 00:09:40,000 --> 00:09:46,600 And also you can see that the electric field, as large as near the x y plane is because of this triple 106 00:09:46,870 --> 00:09:47,890 vector product here. 107 00:09:49,030 --> 00:09:55,270 And in the X Y plane, once again, like here, the electric field is pointing along the etha to the 108 00:09:55,270 --> 00:09:55,850 direction. 109 00:09:57,070 --> 00:10:03,520 The magnetic field is also largest near the XY plane, but it's pointing along the fire direction, 110 00:10:03,520 --> 00:10:04,780 as is indicated here. 111 00:10:06,540 --> 00:10:14,700 So this means in the far field, the majority of the energy is transmitted near or close to the X Y 112 00:10:14,700 --> 00:10:22,440 plane, and it turns out that the data point in Vector s, which is essentially a prefect at times to 113 00:10:22,440 --> 00:10:29,580 vector product of E and B, is pointing along the radial direction because the product of Edyta and 114 00:10:29,580 --> 00:10:34,970 if you feel if I is E r, which is to radial vector. 115 00:10:35,460 --> 00:10:43,200 So this means all of the energy is transferred along the radial direction close to the x y plane.