1 00:00:00,210 --> 00:00:08,190 Let us start a section by driving the wave equation, so in the previous section, we have introduced 2 00:00:08,190 --> 00:00:13,150 the Maxwells equation and now we want to consider these equations in a vacuum. 3 00:00:14,070 --> 00:00:15,270 So what does that mean? 4 00:00:16,380 --> 00:00:20,020 Vacuum means we have no charges and no currents. 5 00:00:20,490 --> 00:00:22,830 So both of these quantities will be zero. 6 00:00:24,030 --> 00:00:30,960 So what you can see will happen is there is some kind of symmetry in the electric and magnetic fields 7 00:00:30,960 --> 00:00:39,690 because now both the divergence of E and the divergence of B will be zero and the rotation of E will 8 00:00:39,690 --> 00:00:47,280 be proportional to the time derivative of B and vice versa, because this term is zero and this term 9 00:00:47,280 --> 00:00:47,800 is zero. 10 00:00:49,050 --> 00:00:51,480 So we have these four Maxwell equations. 11 00:00:51,930 --> 00:00:58,320 And also something I will introduce in the following is the speed of light, which is actually this 12 00:00:58,500 --> 00:01:02,910 factor here, the square root and then the power of minus one. 13 00:01:04,500 --> 00:01:11,880 So let's take our whiteboard here and I want to show you how we can derive the wave equation for E and 14 00:01:11,880 --> 00:01:15,000 also for B based on these maximal equations. 15 00:01:17,160 --> 00:01:20,340 So let us take this maximal equation here. 16 00:01:20,350 --> 00:01:26,100 The rotation of E is equal to the time derivative of B and A minus sign. 17 00:01:26,640 --> 00:01:29,450 So this is what is written here and here. 18 00:01:30,300 --> 00:01:37,890 So of course what we can do is we can multiply a KNOBLER operator from the left using a vector product 19 00:01:38,580 --> 00:01:40,670 and the equation is still true. 20 00:01:41,460 --> 00:01:47,970 So we have the rotation of the rotation of E is equal to the rotation of minus the time derivative of 21 00:01:47,970 --> 00:01:54,010 B, so we can now do is we can switch around these two derivatives. 22 00:01:54,370 --> 00:02:02,370 Is something you can basically always do if you here you have the derivative with respect to the coordinates 23 00:02:02,370 --> 00:02:03,930 and to you with respect to the time. 24 00:02:04,290 --> 00:02:08,430 So these are independent properties coordinates. 25 00:02:08,730 --> 00:02:15,060 So you can switch around these derivatives so we can write down that this is equal to minus the time 26 00:02:15,060 --> 00:02:18,110 derivative of the rotation of B. 27 00:02:19,620 --> 00:02:25,680 Now you can see here we have the rotation of B so we can take the second Maxwell equation. 28 00:02:25,680 --> 00:02:31,970 The rotation of B is equal to this prefectly here, which is equal to one of the C Square. 29 00:02:33,030 --> 00:02:35,550 And then we have the time derivative of E. 30 00:02:37,460 --> 00:02:43,610 And since we have here already another time derivative, this is equal to the second derivative of E 31 00:02:43,610 --> 00:02:45,140 with respect to the time. 32 00:02:47,350 --> 00:02:53,380 OK, so here you can see I have still these three factors, and here I have introduced a velocity of 33 00:02:53,380 --> 00:02:53,710 light. 34 00:02:54,580 --> 00:02:56,230 So we have here an equation. 35 00:02:56,230 --> 00:03:02,140 The rotation of the rotation of E is equal to minus one of a C square and then the second derivative 36 00:03:02,140 --> 00:03:03,580 of E with respect to time. 37 00:03:04,930 --> 00:03:11,380 Now, this is almost the Maxwell equation, sorry, almost a wave equation, but we need one more thing. 38 00:03:11,830 --> 00:03:16,210 We need to transform this first term here and here. 39 00:03:16,210 --> 00:03:23,920 We use something, some relation using these Nabala operators, which is true for every vector field. 40 00:03:24,160 --> 00:03:29,180 So this is not a particular relation for E, but it is true for every vector. 41 00:03:30,160 --> 00:03:37,960 So the rotation of the rotation can be expressed as this gradient of the divergence minus a dollar plus 42 00:03:37,960 --> 00:03:38,920 operator of each. 43 00:03:39,910 --> 00:03:47,410 So if you don't believe me, write down this equation in all of the coordinates, X, Y and Z, and 44 00:03:47,410 --> 00:03:52,590 then you will see that you can transform it to this expression here now. 45 00:03:54,210 --> 00:04:00,340 So so far, this is true for every week, as I said, and now we use a special property of the electric 46 00:04:00,340 --> 00:04:00,710 field. 47 00:04:01,360 --> 00:04:08,560 So here you can see we have the divergence of E and we know from the Maxwell's equations that the virgins 48 00:04:08,560 --> 00:04:10,010 of E is zero. 49 00:04:10,570 --> 00:04:13,660 So we are left only with this term here. 50 00:04:14,740 --> 00:04:22,630 And this is something where it's really important that we consider vacuum, because if we would not 51 00:04:22,630 --> 00:04:29,320 be in a vacuum, then this term you would not be equal to zero, because then we would have here a term 52 00:04:29,320 --> 00:04:31,600 proportional to the charged density. 53 00:04:31,720 --> 00:04:35,200 And then this whole derivation of the WAF equation would not be true. 54 00:04:36,460 --> 00:04:43,510 But since we consider here a vacuum, we have now that this term is equal to that one here and also 55 00:04:43,510 --> 00:04:46,790 it is equal to minus the plus operator of E! 56 00:04:47,440 --> 00:04:51,390 So we can write down this term is equal to this term. 57 00:04:51,880 --> 00:04:54,220 So this is our wave equation. 58 00:04:56,060 --> 00:05:05,560 Now, to find a solution, it's most easy to use only a one dimensional problem, so then this class 59 00:05:05,570 --> 00:05:10,840 operator here becomes just the second derivative with respect to the coordinate, let's say X. 60 00:05:11,210 --> 00:05:17,640 So we have here one of a C squared, the second derivative of E with respect to time, minus the second 61 00:05:17,720 --> 00:05:25,620 privative of E with respect to the coordinate, X is equal to zero and now we can guess the solution. 62 00:05:26,360 --> 00:05:33,590 So the solution will be some cosine or sine function or more general and exponential function with an 63 00:05:33,590 --> 00:05:35,430 imaginary argument here. 64 00:05:36,350 --> 00:05:44,660 So the solution is some vector is zero times this exponential function e to the power of AI and then 65 00:05:44,660 --> 00:05:49,770 here we have omega T minus the scale of product of a vector K and a vector. 66 00:05:51,860 --> 00:05:56,480 So in the following we will learn what these quantities Omega and K are. 67 00:05:56,870 --> 00:05:59,070 But so far they are just coefficients. 68 00:06:00,650 --> 00:06:08,870 So if we calculate the second derivatives, you will see that since this is here in E function and an 69 00:06:08,870 --> 00:06:15,110 exponential function, this exponential function will remain and we will just get the derivative. 70 00:06:15,650 --> 00:06:23,510 So for this term here, we will get a square which is minus one, and then we will have Omega Square. 71 00:06:24,170 --> 00:06:30,640 And for this term here we get a square and then we get K Square. 72 00:06:31,730 --> 00:06:37,640 And as I mentioned in the following, we will relate Omega and K to see and then we will understand 73 00:06:37,640 --> 00:06:38,980 what Omega and K are. 74 00:06:39,620 --> 00:06:43,770 But as you can see, this is an exponential function within an imaginary argument. 75 00:06:44,330 --> 00:06:52,610 So the real part, as you probably know, is then a cosine function and it is fluctuating in time and 76 00:06:52,610 --> 00:06:59,390 it is fluctuating in the coordinated R, so it is a wave and this is an electromagnetic wave. 77 00:06:59,870 --> 00:07:06,950 And as it turns out, this is light because we are now in a vacuum, which means we have no charges, 78 00:07:07,220 --> 00:07:08,180 we have no currents. 79 00:07:08,450 --> 00:07:12,010 So this is the assumption that we put into these maximal equations here. 80 00:07:12,530 --> 00:07:18,020 So the only thing that we can have is light, because light is not a charge and is not a current. 81 00:07:18,500 --> 00:07:25,860 It's just photons and it's just some excitation in forms in terms of an electromagnetic wave. 82 00:07:27,440 --> 00:07:33,140 So this is the wave equation for light in vacuum and this is the solution for the electric field. 83 00:07:34,310 --> 00:07:40,070 Next, we want to find the same equation and a similar solution for the magnetic field. 84 00:07:40,910 --> 00:07:47,300 And this is actually a nice exercise for you to see if you really understood what we did on the previous 85 00:07:47,300 --> 00:07:47,740 slide. 86 00:07:48,410 --> 00:07:54,740 So I think it would be best if you would pause the video now and take a sheet of paper and a pen and 87 00:07:55,010 --> 00:07:59,690 write down a very similar derivation of the wave equation for beep. 88 00:08:01,310 --> 00:08:04,440 But of course, I will now show you also the solution. 89 00:08:05,210 --> 00:08:11,120 So what you have to do is you once again have to consider here this Maxwell equation. 90 00:08:11,120 --> 00:08:16,540 So we have the rotation of P is essentially you the time derivative of E with some perfecta. 91 00:08:16,970 --> 00:08:18,290 So we have this one here. 92 00:08:18,950 --> 00:08:23,780 And once again we can switch around these two derivatives, which I have done here. 93 00:08:24,260 --> 00:08:31,640 And now we see that this is the rotation of E, which is essentially the time derivative of be so we 94 00:08:31,640 --> 00:08:33,470 can get another time derivative. 95 00:08:33,470 --> 00:08:37,050 And here the vector field of work, the field of B. 96 00:08:38,360 --> 00:08:40,340 So this is the first term that we want to have. 97 00:08:40,340 --> 00:08:41,570 And now the second term. 98 00:08:41,840 --> 00:08:46,970 First of all, we use this vector identity, which gives us these two terms here. 99 00:08:47,360 --> 00:08:53,870 And then we see that we have here the divergence of B, which is zero, which is true generally not 100 00:08:53,870 --> 00:08:54,730 only in vacuum. 101 00:08:55,940 --> 00:08:58,400 So we have that these two terms here are equal. 102 00:08:58,550 --> 00:09:05,780 And this gives us the wave equation for B, which looks exactly the same as for E, and therefore our 103 00:09:05,780 --> 00:09:07,580 solution will be the same. 104 00:09:08,810 --> 00:09:13,960 But to please note that B zero can still be a different vector to e0. 105 00:09:14,960 --> 00:09:21,200 But like the electric field, also the magnetic field is an electromagnetic wave. 106 00:09:21,620 --> 00:09:25,100 It fluctuates in time and in the coordinates. 107 00:09:26,630 --> 00:09:31,360 And then the following section we will determine the dispersion relation. 108 00:09:31,370 --> 00:09:35,480 And I want to show you what Omega means and what K means.