1 00:00:00,000 --> 00:00:25,000 2 00:00:25,000 --> 00:00:27,802 Today we are going to talk about the interactions of 3 00:00:27,802 --> 00:00:30,000 electromagnetic waves with conductors. 4 00:00:30,000 --> 00:00:34,278 No electric fields can exist inside an ideal conductor, 5 00:00:34,278 --> 00:00:38,398 so when electromagnetic waves are incident to an idea 6 00:00:38,398 --> 00:00:42,360 conductor somehow that electromagnetic wave must be 7 00:00:42,360 --> 00:00:45,292 reflected. And so we are going to look 8 00:00:45,292 --> 00:00:49,174 this morning at the boundary conditions and at the 9 00:00:49,174 --> 00:00:52,423 consequences of those boundary conditions. 10 00:00:52,423 --> 00:00:56,306 We have four Maxwell's equations so we expect four 11 00:00:56,306 --> 00:01:01,644 boundary conditions. The key is that inside an ideal 12 00:01:01,644 --> 00:01:05,141 conductor the electric field must be zero. 13 00:01:05,141 --> 00:01:08,808 The magnetic field does not have to be zero. 14 00:01:08,808 --> 00:01:13,414 A static magnetic field is possible inside a conductor. 15 00:01:13,414 --> 00:01:18,276 However, you cannot have a change in magnetic field inside 16 00:01:18,276 --> 00:01:22,370 a conductor because the curl of E is minus dB/dt. 17 00:01:22,370 --> 00:01:28,000 And so, if E is zero inside, dB/dt is also zero inside. 18 00:01:28,000 --> 00:01:32,086 Now, before we work this out, the consequences, 19 00:01:32,086 --> 00:01:36,972 I have to be fair to you and tell you that in reality no 20 00:01:36,972 --> 00:01:41,236 conductors are ideal. And this is what Bekefi and 21 00:01:41,236 --> 00:01:44,700 Barrett discuss on page 442 of the book. 22 00:01:44,700 --> 00:01:49,675 The electric field falls off exponentially in a very thin 23 00:01:49,675 --> 00:01:52,695 layer which we call the skin depth. 24 00:01:52,695 --> 00:01:58,025 And that skin depth's delta is defined in such a way that the 25 00:01:58,025 --> 00:02:03,000 electric field falls off by a factor of E. 26 00:02:03,000 --> 00:02:08,833 And that skin depth, which I will not derive here, 27 00:02:08,833 --> 00:02:15,380 is the square root of two divided by the frequency omega 28 00:02:15,380 --> 00:02:20,380 times mu zero times the conductivity sigma. 29 00:02:20,380 --> 00:02:26,452 And the conductivity for an ideal conductor would be 30 00:02:26,452 --> 00:02:33,000 infinitely high so that the skin depth is zero. 31 00:02:33,000 --> 00:02:37,456 But even if you take something like copper, which is an 32 00:02:37,456 --> 00:02:41,334 extremely good conductor, then sigma, of course, 33 00:02:41,334 --> 00:02:45,791 is not infinitely high, but sigma is something like 5.8 34 00:02:45,791 --> 00:02:49,669 times 10 to the 7. And the units of conductivity 35 00:02:49,669 --> 00:02:53,878 are ohmic meters to the power minus one is SI units. 36 00:02:53,878 --> 00:02:58,334 If you use this sigma and you take a thousand megahertz 37 00:02:58,334 --> 00:03:02,543 radiation, which has a wavelength of 30 centimeters, 38 00:03:02,543 --> 00:03:07,000 then the skin depth is only two microns. 39 00:03:07,000 --> 00:03:10,872 It is very, very small. If you take optical light, 40 00:03:10,872 --> 00:03:15,457 which has a frequency of 5 times 10 to the 14 hertz and you 41 00:03:15,457 --> 00:03:19,962 put that in here then you will find the skin depth is only 42 00:03:19,962 --> 00:03:23,597 three nanometers, but it is 30 atomic distances 43 00:03:23,597 --> 00:03:28,261 that the E field decays into the layer and reduces them by a 44 00:03:28,261 --> 00:03:33,042 factor of E. The wavelength of visible light 45 00:03:33,042 --> 00:03:38,440 is 500 nanometers roughly, and so the skin depth is only 46 00:03:38,440 --> 00:03:44,328 something like three nanometers. I want to start now deriving 47 00:03:44,328 --> 00:03:49,726 the boundary conditions, and I will start with the first 48 00:03:49,726 --> 00:03:55,320 Maxwell's equation which is the divergence of E equals rho 49 00:03:55,320 --> 00:04:00,129 divided by epsilon zero, or this rho is the volume 50 00:04:00,129 --> 00:04:04,546 charge density, so this rho is in Coulombs per 51 00:04:04,546 --> 00:04:10,474 cubic meter. You should remember this from 52 00:04:10,474 --> 00:04:15,506 your 8.02 days. I will write it in integral 53 00:04:15,506 --> 00:04:19,820 form. If I have the integral of E dot 54 00:04:19,820 --> 00:04:26,410 dA, I have a closed surface. And you will see it at work 55 00:04:26,410 --> 00:04:30,723 very shortly. A close surface is very 56 00:04:30,723 --> 00:04:37,913 essential. This is the electric flux that 57 00:04:37,913 --> 00:04:45,448 emerges from that surface. It is a dot product. 58 00:04:45,448 --> 00:04:53,474 There is a dot here. This is one over epsilon zero 59 00:04:53,474 --> 00:05:02,974 times the volume integral of all the charge that is inside. 60 00:05:02,974 --> 00:05:12,965 You have called that earlier q inside divided by epsilon zero, 61 00:05:12,965 --> 00:05:20,008 but this is a more mature way of writing it. 62 00:05:20,008 --> 00:05:26,887 So you integrate it over that whole volume. 63 00:05:26,887 --> 00:05:39,121 05:00 I am decomposing this now in a 64 00:05:39,121 --> 00:05:56,452 component which is tangential, that is in the plane of the 65 00:05:56,452 --> 00:06:07,702 surface, and I call that E tangential. 66 00:06:07,702 --> 00:06:21,689 And a component which is normal to the surface, 67 00:06:21,689 --> 00:06:35,979 and so I give that an n which stands for normal. 68 00:06:35,979 --> 00:06:48,445 And so now I am going to make my pill box. 69 00:06:48,445 --> 00:07:00,000 Here is my pill box. 06:00 70 00:07:00,000 --> 00:07:02,604 That is the flux leaving that surface. 71 00:07:02,604 --> 00:07:06,827 Then I have an electric flux which I will call just phi of E. 72 00:07:06,827 --> 00:07:10,487 This is also an electric flux, but I call it phi of E 73 00:07:10,487 --> 00:07:14,288 shorthand notation for everything that goes through the 74 00:07:14,288 --> 00:07:17,667 cylindrical surface. It is what I call the curved 75 00:07:17,667 --> 00:07:20,131 surface. Let's call it the cylinder. 76 00:07:20,131 --> 00:07:23,932 Well, when I am going to make dl zero, that cylindrical 77 00:07:23,932 --> 00:07:29,000 surface goes to zero so no electric flux can escape there. 78 00:07:29,000 --> 00:07:32,169 And so this one then equals zero. 79 00:07:32,169 --> 00:07:37,913 Now, there is no electric field here, so this is the entire 80 00:07:37,913 --> 00:07:42,370 electric flux as it escapes from this surface, 81 00:07:42,370 --> 00:07:46,728 from this box. And so that is now one divided 82 00:07:46,728 --> 00:07:52,769 by epsilon zero times the volume charge density that I have in 83 00:07:52,769 --> 00:07:56,038 here. And so that is rho times the 84 00:07:56,038 --> 00:08:02,318 volume which is dl times dA. This is now Coulombs per cubic 85 00:08:02,318 --> 00:08:04,096 meter. You will say now, 86 00:08:04,096 --> 00:08:07,497 when dl goes to zero, that that goes to zero. 87 00:08:07,497 --> 00:08:10,666 But that is not true because there can be, 88 00:08:10,666 --> 00:08:15,227 and there will be charge right at the surface of a conductor 89 00:08:15,227 --> 00:08:19,787 which is not Coulomb's per cubic meter but it is the surface 90 00:08:19,787 --> 00:08:22,570 density in Coulombs per square meter. 91 00:08:22,570 --> 00:08:26,357 Totally different dimension. And that is rho of S, 92 00:08:26,357 --> 00:08:32,000 how much charge there is per square meter on that surface. 93 00:08:32,000 --> 00:08:37,744 And so in the case when dl goes to zero this does not go to zero 94 00:08:37,744 --> 00:08:43,215 but this becomes rho s divided by epsilon zero times that dA. 95 00:08:43,215 --> 00:08:47,501 And this rho s now is Coulombs per square meter. 96 00:08:47,501 --> 00:08:50,601 That is the surface charge density. 97 00:08:50,601 --> 00:08:55,799 And so you see that we have derived now our first boundary 98 00:08:55,799 --> 00:09:02,000 condition which I will write down on the blackboard there. 99 00:09:02,000 --> 00:09:05,903 And that is that the normal component of any changing 100 00:09:05,903 --> 00:09:10,257 electric field that is due to the electromagnetic radiation 101 00:09:10,257 --> 00:09:14,536 equals the surface charge density divided by epsilon zero. 102 00:09:14,536 --> 00:09:18,739 Now when you see this result you will say that is nothing 103 00:09:18,739 --> 00:09:21,292 new. We have seen that also in 8.02 104 00:09:21,292 --> 00:09:23,694 for static charges. That is true. 105 00:09:23,694 --> 00:09:26,246 It is also true for static charges. 106 00:09:26,246 --> 00:09:30,000 But here there is something special. 107 00:09:30,000 --> 00:09:34,758 And that is that this E vector is changing all the time with 108 00:09:34,758 --> 00:09:38,709 the frequency of the incident wave and, therefore, 109 00:09:38,709 --> 00:09:41,532 rho S must also change all the time. 110 00:09:41,532 --> 00:09:45,564 That is all the time rearrangements of that surface 111 00:09:45,564 --> 00:09:50,645 charge density to make sure that this boundary condition is met. 112 00:09:50,645 --> 00:09:54,032 But, yes, it also holds for static charges. 113 00:09:54,032 --> 00:09:56,854 The next one is the divergence of B. 114 00:09:56,854 --> 00:10:02,153 You are going to do that. You are going to make a pill 115 00:10:02,153 --> 00:10:06,230 box and you are going to demonstrate to me that if you 116 00:10:06,230 --> 00:10:10,230 use the divergence of B, which is the second equation 117 00:10:10,230 --> 00:10:14,692 Maxwell, that now you find that the normal component of the 118 00:10:14,692 --> 00:10:18,230 changing magnetic field at the surface is zero. 119 00:10:18,230 --> 00:10:22,230 Keep in mind it is time variable, the magnetic field, 120 00:10:22,230 --> 00:10:25,461 as it comes in. And it is the time variable 121 00:10:25,461 --> 00:10:30,000 component, the normal one that has to be zero. 122 00:10:30,000 --> 00:10:36,202 The next one in line is the curl of E minus dB/dt, 123 00:10:36,202 --> 00:10:43,291 the famous Faraday's law. And then we have the curl of B. 124 00:10:43,291 --> 00:10:50,126 I will do the curl of E, and I will leave the curl of B 125 00:10:50,126 --> 00:10:55,063 up to you. The curl of E is minus dB/dt. 126 00:10:55,063 --> 00:11:02,532 This equation runs our economy. This is why we have light. 127 00:11:02,532 --> 00:11:07,305 This is why we have energy. This is Faraday's law. 128 00:11:07,305 --> 00:11:11,493 And I am going to write it in integral form. 129 00:11:11,493 --> 00:11:15,389 Of course, you are probably used to that. 130 00:11:15,389 --> 00:11:20,649 This is a closed loop now. Any closed loop that you may 131 00:11:20,649 --> 00:11:24,448 choose of E dot dl, it is a dot product, 132 00:11:24,448 --> 00:11:30,000 it is a line integral equals minus d phi B/dt. 133 00:11:30,000 --> 00:11:34,318 And this is a magnetic flux that goes through the surface 134 00:11:34,318 --> 00:11:37,172 that you attached to that closed loop. 135 00:11:37,172 --> 00:11:41,259 It is an open surface. Any surface that you may choose 136 00:11:41,259 --> 00:11:43,573 is fine. It is an open surface. 137 00:11:43,573 --> 00:11:47,660 I cannot stress that enough. You choose a closed loop. 138 00:11:47,660 --> 00:11:50,205 You can do it right here in space. 139 00:11:50,205 --> 00:11:54,832 You can attest to that an open surface, which could bulge out 140 00:11:54,832 --> 00:12:00,000 into the audience any moment in time, this will hold. 141 00:12:00,000 --> 00:12:05,239 And so whenever you deal with curls you do line integrals. 142 00:12:05,239 --> 00:12:10,479 And so let's do a line integral and then see what boundary 143 00:12:10,479 --> 00:12:15,259 condition that leads to. We do the same thing that we 144 00:12:15,259 --> 00:12:18,935 did there. We have here the conductor and 145 00:12:18,935 --> 00:12:23,164 here we have vacuum. And so the E field here is 146 00:12:23,164 --> 00:12:27,484 everywhere zero. And here is your electric field 147 00:12:27,484 --> 00:12:33,000 vector coming in from your electromagnetic wave. 148 00:12:33,000 --> 00:12:37,552 Here is your Et and here is your En exactly in the way that 149 00:12:37,552 --> 00:12:41,162 I defined them. This is the normal component to 150 00:12:41,162 --> 00:12:44,537 the surface. Now I am going to make a closed 151 00:12:44,537 --> 00:12:46,735 loop. This is my closed loop. 152 00:12:46,735 --> 00:12:50,738 It shouldn't surprise you that I choose it this way. 153 00:12:50,738 --> 00:12:54,898 And I am going to march around this closed loop in any 154 00:12:54,898 --> 00:13:00,000 direction that I want to, but I chose this direction. 155 00:13:00,000 --> 00:13:03,959 It makes no difference which direction you choose. 156 00:13:03,959 --> 00:13:06,949 And let this be dl and let this be dB. 157 00:13:06,949 --> 00:13:10,424 And I am going to do exactly the same thing. 158 00:13:10,424 --> 00:13:14,626 I am going to go in the limiting case when dl goes to 159 00:13:14,626 --> 00:13:19,151 zero so that I get the boundary condition at the surface. 160 00:13:19,151 --> 00:13:23,757 If you are ready for this, I start here and I move in this 161 00:13:23,757 --> 00:13:26,505 direction. The E of t has no impact 162 00:13:26,505 --> 00:13:32,000 because it is perpendicular to my direction of motion. 163 00:13:32,000 --> 00:13:35,927 The dot product is zero. All I worry about is this one. 164 00:13:35,927 --> 00:13:38,836 And so I get E of n times one-half of dl. 165 00:13:38,836 --> 00:13:42,618 Let's suppose that this is one-half of dl and this is 166 00:13:42,618 --> 00:13:45,454 one-half. That is when I go from here to 167 00:13:45,454 --> 00:13:48,072 here. And now I go from here to here. 168 00:13:48,072 --> 00:13:51,709 And now, of course, E of n has no effect because it 169 00:13:51,709 --> 00:13:55,636 is perpendicular to dl. dl is a vector in the direction 170 00:13:55,636 --> 00:14:00,000 that I am moving. Now I only deal with E of t. 171 00:14:00,000 --> 00:14:04,538 Now I get plus E of t times dB. Now I am here and I go down. 172 00:14:04,538 --> 00:14:07,692 Now E of t is up, but I go down so the dot 173 00:14:07,692 --> 00:14:12,076 product creates a minus sign. Now I get minus E of n times 174 00:14:12,076 --> 00:14:14,076 one-half dl. I said E of t, 175 00:14:14,076 --> 00:14:17,692 but I meant E of n. E of n is in this direction, 176 00:14:17,692 --> 00:14:20,461 and you are moving in this direction. 177 00:14:20,461 --> 00:14:23,461 And so that is why you get a minus sign. 178 00:14:23,461 --> 00:14:28,000 And E of t has no effect here because it is perpendicular to 179 00:14:28,000 --> 00:14:33,989 my direction of motion. And the electric field here is 180 00:14:33,989 --> 00:14:37,078 zero. This now becomes minus the 181 00:14:37,078 --> 00:14:41,861 change of the magnetic flux through that surface, 182 00:14:41,861 --> 00:14:47,441 this surface if you wish, or a surface that bulges out in 183 00:14:47,441 --> 00:14:50,032 the audience, I don't care. 184 00:14:50,032 --> 00:14:55,412 It is minus d phi B/dt. And that goes to zero because I 185 00:14:55,412 --> 00:15:01,389 am going to make dl zero. There is no surface left, 186 00:15:01,389 --> 00:15:06,453 and so that one goes to zero. And notice that this one 187 00:15:06,453 --> 00:15:11,901 cancels out against this one. And so now we find the third 188 00:15:11,901 --> 00:15:17,061 condition, and that is that at the boundary condition E 189 00:15:17,061 --> 00:15:20,979 tangential is zero. That is the third one. 190 00:15:20,979 --> 00:15:25,375 I want to remind you, though, there is the skin 191 00:15:25,375 --> 00:15:29,257 depth. In reality, there is an 192 00:15:29,257 --> 00:15:34,754 exponential decay of this E vector that is in the surface 193 00:15:34,754 --> 00:15:40,546 that decays exponentially over that skin depth delta that we 194 00:15:40,546 --> 00:15:44,668 discussed earlier. That is that skin depth. 195 00:15:44,668 --> 00:15:49,184 Now there is the curl of B, which is your turn. 196 00:15:49,184 --> 00:15:54,484 You are going to do a closed lope integral of B dot dl. 197 00:15:54,484 --> 00:15:59,000 And you work your way through that. 198 00:15:59,000 --> 00:16:04,269 And then you will find that the magnitude of B of t, 199 00:16:04,269 --> 00:16:10,263 that is the component of the magnetic field in the plane of 200 00:16:10,263 --> 00:16:14,293 the conductor, the tangential component, 201 00:16:14,293 --> 00:16:20,080 that that magnitude is mu zero times what we call surface 202 00:16:20,080 --> 00:16:24,523 current density. J of s is a surface current 203 00:16:24,523 --> 00:16:30,000 density, and it has units. MP is per meter. 204 00:16:30,000 --> 00:16:34,025 And so there are actually oscillatory currents on the 205 00:16:34,025 --> 00:16:38,438 surface that occur when the electromagnetic wave comes in. 206 00:16:38,438 --> 00:16:42,387 These currents change with the frequency, of course, 207 00:16:42,387 --> 00:16:46,722 of the incident radiation. Not only is there all the time 208 00:16:46,722 --> 00:16:51,522 a readjustment of rho of s which is the surface charge density, 209 00:16:51,522 --> 00:16:55,703 but all the time as that electromagnetic wave interacts 210 00:16:55,703 --> 00:16:59,419 with the conductor is there going to be a current 211 00:16:59,419 --> 00:17:03,987 oscillating in order to make sure that we meet this boundary 212 00:17:03,987 --> 00:17:08,764 condition. Real things are really 213 00:17:08,764 --> 00:17:13,790 happening. We can use these four boundary 214 00:17:13,790 --> 00:17:19,947 conditions now to start looking into some results, 215 00:17:19,947 --> 00:17:26,984 and I will start with a very simple example because I can 216 00:17:26,984 --> 00:17:36,343 demonstrate that. I take linearly polarized 217 00:17:36,343 --> 00:17:44,716 radiation. This is my z direction. 218 00:17:44,716 --> 00:17:53,850 It is going to make a traveling wave. 219 00:17:53,850 --> 00:18:06,283 This is my x direction and this is my y direction. 220 00:18:06,283 --> 00:18:15,164 I have linearly polarized radiation. 221 00:18:15,164 --> 00:18:29,373 It only has an x component. I am going to make it move in 222 00:18:29,373 --> 00:18:38,000 the z direction. 18:00 223 00:18:38,000 --> 00:18:40,636 It is propagating in this direction. 224 00:18:40,636 --> 00:18:44,479 This is the k vector. And that is the incident wave, 225 00:18:44,479 --> 00:18:48,171 and it is in the x direction. They are plane waves 226 00:18:48,171 --> 00:18:51,561 perpendicular to z infinite in all directions. 227 00:18:51,561 --> 00:18:55,404 And the E vector is then everywhere in those planes. 228 00:18:55,404 --> 00:19:00,000 This value changes with z and it changes with time. 229 00:19:00,000 --> 00:19:04,913 Now what I am going to do is put here a perfect conductor. 230 00:19:04,913 --> 00:19:07,758 And so the E vector arrives there. 231 00:19:07,758 --> 00:19:12,931 There is no normal component to the E vector so I don't worry 232 00:19:12,931 --> 00:19:16,551 about E of n. There is no normal component. 233 00:19:16,551 --> 00:19:20,775 I only have E of t, and that E of t must be called 234 00:19:20,775 --> 00:19:25,258 zero at the conductor. And so this reminds you of the 235 00:19:25,258 --> 00:19:30,603 string that we had when we were wiggling the string at one side 236 00:19:30,603 --> 00:19:35,000 and we fixed that side of the string. 237 00:19:35,000 --> 00:19:39,554 That in order to make sure that there was no motion of the end 238 00:19:39,554 --> 00:19:42,914 of the string, the reflectivity was minus one, 239 00:19:42,914 --> 00:19:46,945 a mountain came back as a valley to make sure that this 240 00:19:46,945 --> 00:19:50,454 point never moved. And so exactly the same thing 241 00:19:50,454 --> 00:19:53,814 is happening now. We must make sure that right 242 00:19:53,814 --> 00:19:58,070 here the electric vector is always, at all moment in time, 243 00:19:58,070 --> 00:20:00,160 zero. If I just call this for 244 00:20:00,160 --> 00:20:04,640 convenience z equals zero at this location then the reflected 245 00:20:04,640 --> 00:20:10,307 wave must be minus E zero pi. That is the mountain becomes a 246 00:20:10,307 --> 00:20:12,692 valley. And then it moves in the 247 00:20:12,692 --> 00:20:17,615 opposite direction so you get kz plus omega t in the x direction. 248 00:20:17,615 --> 00:20:20,923 Now, you will see that the sum of these two, 249 00:20:20,923 --> 00:20:24,769 at any moment in time, will make the E vector right 250 00:20:24,769 --> 00:20:28,769 here in this entire plane, which is infinitely large, 251 00:20:28,769 --> 00:20:32,571 zero. That is the boundary condition 252 00:20:32,571 --> 00:20:36,514 that we have to meet. The total electric field, 253 00:20:36,514 --> 00:20:41,314 which is the sum of the two, one moving in this direction 254 00:20:41,314 --> 00:20:45,000 and this one moving backwards is, of course, 255 00:20:45,000 --> 00:20:48,857 the factorial sum. I have the cosine of alpha. 256 00:20:48,857 --> 00:20:52,542 I call that alpha, minus the cosine of beta. 257 00:20:52,542 --> 00:20:57,514 Your high school days tell you that it is twice the sine of 258 00:20:57,514 --> 00:21:03,000 half the sum times the sine of half the difference. 259 00:21:03,000 --> 00:21:09,000 Although, I always have to look that up because I forget that 260 00:21:09,000 --> 00:21:14,200 just as well as you do. And when I add them up then I 261 00:21:14,200 --> 00:21:19,900 get that the total electric field, which is the sum of the 262 00:21:19,900 --> 00:21:25,299 two, is going to be 2E0i, and then I get the sine of kz 263 00:21:25,299 --> 00:21:32,000 and I get the sine of omega t. And that is a standing wave. 264 00:21:32,000 --> 00:21:36,524 All the spatial information is in here and all the time 265 00:21:36,524 --> 00:21:39,540 information is in there. And, indeed, 266 00:21:39,540 --> 00:21:44,400 if you substitute in there z equals zero, notice that the E 267 00:21:44,400 --> 00:21:47,583 field is always zero. This, by the way, 268 00:21:47,583 --> 00:21:51,940 is in the x direction. And so we have a standing wave 269 00:21:51,940 --> 00:21:54,537 now. And what that means is that 270 00:21:54,537 --> 00:21:59,816 there are surfaces perpendicular to the direction of z where the 271 00:21:59,816 --> 00:22:04,465 E vector is always zero. An entire surface. 272 00:22:04,465 --> 00:22:08,537 We have now a nodal surface like this, and they are 273 00:22:08,537 --> 00:22:11,550 separated by distance one-half lambda. 274 00:22:11,550 --> 00:22:15,459 This is a nodal surface, but this is also a nodal 275 00:22:15,459 --> 00:22:18,472 surface. I am trying to draw the curve 276 00:22:18,472 --> 00:22:21,159 of the E vector, sinusoidal curve, 277 00:22:21,159 --> 00:22:26,127 so the E vector may be up here. And then the E vector would be 278 00:22:26,127 --> 00:22:29,978 down here. But it is all in the plane 279 00:22:29,978 --> 00:22:32,945 perpendicular to the blackboard to same. 280 00:22:32,945 --> 00:22:36,826 So this is a nodal surface, this is a nodal surface, 281 00:22:36,826 --> 00:22:40,934 this is a nodal surface, this is the wavelength and the 282 00:22:40,934 --> 00:22:44,815 E vectors go like this. Nothing is moving anymore in 283 00:22:44,815 --> 00:22:47,326 this direction. It goes like this, 284 00:22:47,326 --> 00:22:49,989 a standing wave with nodal surfaces. 285 00:22:49,989 --> 00:22:54,630 This standing electromagnetic wave has an electric field which 286 00:22:54,630 --> 00:22:57,902 is standing. There is an associated magnetic 287 00:22:57,902 --> 00:23:01,097 field with it, which is a standing magnetic 288 00:23:01,097 --> 00:23:04,895 field. And I would like you to work 289 00:23:04,895 --> 00:23:07,604 that out. I may even have put that in one 290 00:23:07,604 --> 00:23:10,380 of my problem sets. And what you find out, 291 00:23:10,380 --> 00:23:13,833 much to your surprise, certainly much to my surprise 292 00:23:13,833 --> 00:23:17,354 when I saw this first, that the magnetic field is now 293 00:23:17,354 --> 00:23:21,687 90 degrees out of phase with the electric field both in space and 294 00:23:21,687 --> 00:23:24,192 in time. That means where you have the 295 00:23:24,192 --> 00:23:27,645 nodal surfaces in E, you have the antinodal surfaces 296 00:23:27,645 --> 00:23:30,977 in B. And where the E fields reach 297 00:23:30,977 --> 00:23:35,002 the maximum you will have that the magnetic field is zero. 298 00:23:35,002 --> 00:23:38,956 Unlike a traveling wave where the magnetic fields and the 299 00:23:38,956 --> 00:23:42,769 electric fields are in phase both in space and in time. 300 00:23:42,769 --> 00:23:45,312 They are now 90 degrees out of phase. 301 00:23:45,312 --> 00:23:49,478 When you think about it that is perhaps not so weird because 302 00:23:49,478 --> 00:23:53,785 there cannot be any net energy transport in one direction with 303 00:23:53,785 --> 00:23:56,892 a standing wave. And so the mean value of the 304 00:23:56,892 --> 00:24:02,086 pointing vector must be zero. And, if you make E and B 90 305 00:24:02,086 --> 00:24:05,642 degrees out of phase then there is always mass. 306 00:24:05,642 --> 00:24:09,971 The mean vector of the pointing vector then becomes zero. 307 00:24:09,971 --> 00:24:13,603 But, in any case, if you use Maxwell's equations 308 00:24:13,603 --> 00:24:18,241 you will see that the magnetic fields and the E fields are 90 309 00:24:18,241 --> 00:24:22,570 degrees out of phase and 90 degrees in space and in time. 310 00:24:22,570 --> 00:24:26,202 I have here our famous 88 megahertz transmitter. 311 00:24:26,202 --> 00:24:30,376 This is an antenna where we accelerate charges back and 312 00:24:30,376 --> 00:24:34,491 forth. Currents are running at a 313 00:24:34,491 --> 00:24:39,772 frequency of 88 megahertz, 88 million oscillations per 314 00:24:39,772 --> 00:24:42,064 second. We are creating, 315 00:24:42,064 --> 00:24:47,843 because we went through this, that accelerated charges give 316 00:24:47,843 --> 00:24:51,131 rise to electromagnetic radiation. 317 00:24:51,131 --> 00:24:54,818 We are creating electromagnetic waves. 318 00:24:54,818 --> 00:25:00,000 Lambda equals c divided by the frequency. 319 00:25:00,000 --> 00:25:04,510 You will find that the wavelength is about 3.4 meters. 320 00:25:04,510 --> 00:25:09,276 And I have here a receiving antenna, which are two cooper 321 00:25:09,276 --> 00:25:12,851 wires that are not connected to each other. 322 00:25:12,851 --> 00:25:16,340 Well, they are. They are already connected 323 00:25:16,340 --> 00:25:19,404 through the filament of a light bulb. 324 00:25:19,404 --> 00:25:24,595 And so when the electric field reaches this receiving antenna, 325 00:25:24,595 --> 00:25:29,361 current will start to flow in this direction and then the 326 00:25:29,361 --> 00:25:33,476 light will go on. I have demonstrated this 327 00:25:33,476 --> 00:25:36,923 before, but I want to do it again, that if I hold up this 328 00:25:36,923 --> 00:25:40,738 receiving antenna like this then you will see the light bulb go 329 00:25:40,738 --> 00:25:42,646 on. If I do it like this then it 330 00:25:42,646 --> 00:25:45,784 will not go on because not currents can flow in this 331 00:25:45,784 --> 00:25:48,123 direction. But that is something that I 332 00:25:48,123 --> 00:25:51,692 have already shown you before. That is not the reason why I 333 00:25:51,692 --> 00:25:54,707 want to do this again. The reason why I want to do 334 00:25:54,707 --> 00:25:57,046 this again is that the blackboard here, 335 00:25:57,046 --> 00:26:01,480 behind the blackboard is metal. And so the electromagnetic 336 00:26:01,480 --> 00:26:04,667 waves that go out in your direction and to the blackboard 337 00:26:04,667 --> 00:26:07,969 are going to reflect off here and they are going to reflect 338 00:26:07,969 --> 00:26:10,531 off the walls. It is a very chaotic way that I 339 00:26:10,531 --> 00:26:13,377 could not predict nor calculate. But there will be, 340 00:26:13,377 --> 00:26:16,110 in this lecture hall, locations of nodal surfaces 341 00:26:16,110 --> 00:26:18,500 where the E field is very low or near zero. 342 00:26:18,500 --> 00:26:21,688 And certainly that has to be the case near the blackboard 343 00:26:21,688 --> 00:26:24,592 because this whole plane behind here is a conductor. 344 00:26:24,592 --> 00:26:27,893 And so the sum of the incoming wave and the reflecting wave 345 00:26:27,893 --> 00:26:31,950 must be zero here. So I don't want to see that 346 00:26:31,950 --> 00:26:35,128 light at all. And then I will walk out in the 347 00:26:35,128 --> 00:26:39,029 lecture hall and see whether there are locations in the 348 00:26:39,029 --> 00:26:42,857 lecture hall where we actually receive electromagnetic 349 00:26:42,857 --> 00:26:47,047 radiation, we will see the light, and where we will not see 350 00:26:47,047 --> 00:26:50,009 the light. Make sure this is not too close 351 00:26:50,009 --> 00:26:53,331 to the transmitter. Otherwise, we will blow the 352 00:26:53,331 --> 00:26:55,932 bulb. Let me first show you then that 353 00:26:55,932 --> 00:27:00,049 when I stand here that I am receiving this electromagnetic 354 00:27:00,049 --> 00:27:04,228 radiation. Remember that the E field is 355 00:27:04,228 --> 00:27:06,854 proportional to the sign of theta. 356 00:27:06,854 --> 00:27:10,912 And so I am here in an excellent position so I get a 357 00:27:10,912 --> 00:27:14,732 maximum electric field. If I rotate it like this, 358 00:27:14,732 --> 00:27:18,153 it does nothing, and that has to do with the 359 00:27:18,153 --> 00:27:21,177 polarization. It has nothing to do with 360 00:27:21,177 --> 00:27:23,962 standing waves, simply polarization. 361 00:27:23,962 --> 00:27:27,145 Now look. I claim that there must be here 362 00:27:27,145 --> 00:27:32,000 a nodal surface. There are no electric fields. 363 00:27:32,000 --> 00:27:34,501 There it is. Isn't that amazing? 364 00:27:34,501 --> 00:27:38,132 This is a standing wave that I am building up. 365 00:27:38,132 --> 00:27:42,651 I am now in that surface that I pointed out to you on the 366 00:27:42,651 --> 00:27:45,636 blackboard. Now I can also try to walk 367 00:27:45,636 --> 00:27:50,074 through the audience and see whether there are locations 368 00:27:50,074 --> 00:27:53,948 where I see light. Right here I see light so I am 369 00:27:53,948 --> 00:27:57,095 receiving. Now, how it bounces off these 370 00:27:57,095 --> 00:28:01,951 walls, I have no idea. That is probably one of the 371 00:28:01,951 --> 00:28:05,349 main contributors. But the walls also have metal 372 00:28:05,349 --> 00:28:08,168 in them. I walk further in and the light 373 00:28:08,168 --> 00:28:09,686 goes out. You may say, 374 00:28:09,686 --> 00:28:13,590 yeah, it goes out because you are further away from the 375 00:28:13,590 --> 00:28:16,192 transmitter. That is a good argument, 376 00:28:16,192 --> 00:28:20,674 but if there are standing waves in this room perhaps it will go 377 00:28:20,674 --> 00:28:23,060 back on again. And when I am here, 378 00:28:23,060 --> 00:28:25,879 I would like to see it go back on again. 379 00:28:25,879 --> 00:28:30,000 It doesn't want to do that. There it is. 380 00:28:30,000 --> 00:28:34,193 Because I tested it out, of course, before you came. 381 00:28:34,193 --> 00:28:37,317 You see the light again? Do you see it? 382 00:28:37,317 --> 00:28:41,511 Just tell me that you see it. It is not very strong, 383 00:28:41,511 --> 00:28:45,622 we are very far away, but it is clear this light is 384 00:28:45,622 --> 00:28:48,746 substantially brighter than it is here. 385 00:28:48,746 --> 00:28:51,788 Nothing here, but here it is stronger. 386 00:28:51,788 --> 00:28:54,584 There it is. And so there is a huge 387 00:28:54,584 --> 00:28:59,764 incredibly difficult pattern in this room of locations where the 388 00:28:59,764 --> 00:29:04,697 E fields are very weak and where the E fields are very strong 389 00:29:04,697 --> 00:29:09,039 like here. And here is even weaker. 390 00:29:09,039 --> 00:29:13,460 I think the most spectacular one is what I did first, 391 00:29:13,460 --> 00:29:18,644 and that is to show you that it becomes zero at the blackboard 392 00:29:18,644 --> 00:29:24,000 which is really a very well controlled plane of conductor. 393 00:29:24,000 --> 00:29:40,000 394 00:29:40,000 --> 00:29:47,209 I know would like to discuss with you another example which 395 00:29:47,209 --> 00:29:51,559 we give the name transmission lines. 396 00:29:51,559 --> 00:29:56,903 We are going to try to have traveling waves, 397 00:29:56,903 --> 00:30:02,000 just like we have with this system. 398 00:30:02,000 --> 00:30:07,486 The reason why I chose this example is because I can also 399 00:30:07,486 --> 00:30:11,797 demonstrate this. Here we have a copper wire. 400 00:30:11,797 --> 00:30:15,422 You see it right in front of you here. 401 00:30:15,422 --> 00:30:19,831 It is 425 centimeters long. And there are two. 402 00:30:19,831 --> 00:30:25,513 And here I have a power supply, so I am driving this with a 403 00:30:25,513 --> 00:30:31,000 variable voltage, v0 times the cosine omega t. 404 00:30:31,000 --> 00:30:35,691 And I am going to use my same transmitter for that. 405 00:30:35,691 --> 00:30:40,008 It is going to be the 88 megahertz transmitter. 406 00:30:40,008 --> 00:30:44,042 I call this y a one and I call this y a two. 407 00:30:44,042 --> 00:30:48,922 We are getting a wave of voltage cosinusoidal that is 408 00:30:48,922 --> 00:30:52,018 going to propagate in these wires. 409 00:30:52,018 --> 00:30:56,616 And what is going to happen is not what you think. 410 00:30:56,616 --> 00:31:00,463 First of all, there cannot be any electric 411 00:31:00,463 --> 00:31:06,000 field in the wires because it is a conductor. 412 00:31:06,000 --> 00:31:09,965 The electric field in this direction does not exist, 413 00:31:09,965 --> 00:31:14,163 so there is no potential difference from here to there. 414 00:31:14,163 --> 00:31:18,750 But, if there is any potential difference, it is going to be 415 00:31:18,750 --> 00:31:22,871 between here and there. If I take this wire and I make 416 00:31:22,871 --> 00:31:27,302 a cross-section of that wire, this is wire one and this is 417 00:31:27,302 --> 00:31:32,201 wire two, then at any moment in time, as the voltage wave passes 418 00:31:32,201 --> 00:31:36,885 by this may be positive. That is that row of s. 419 00:31:36,885 --> 00:31:41,000 And this will be negative. That is that row of s. 420 00:31:41,000 --> 00:31:44,942 And the electric field lines then go like this. 421 00:31:44,942 --> 00:31:48,799 That means if this is, at this moment in time, 422 00:31:48,799 --> 00:31:53,942 positive and this is negative then there is an electric field 423 00:31:53,942 --> 00:31:57,457 in this direction between these two wires. 424 00:31:57,457 --> 00:32:02,000 Not along the wires but between the wires. 425 00:32:02,000 --> 00:32:05,079 But then, of course, there are also going to be 426 00:32:05,079 --> 00:32:09,097 locations where this is minus and this is plus and then where 427 00:32:09,097 --> 00:32:13,181 this is plus and this is minus. And so then the electric field 428 00:32:13,181 --> 00:32:16,730 is in this direction and the electric field is in this 429 00:32:16,730 --> 00:32:19,006 direction. The potential difference 430 00:32:19,006 --> 00:32:21,215 between here and here is positive. 431 00:32:21,215 --> 00:32:24,429 The potential difference between here and here is 432 00:32:24,429 --> 00:32:27,107 negative because it is integral E dot dl. 433 00:32:27,107 --> 00:32:30,857 The potential difference v1 minus v2 is going from 1 to 2 434 00:32:30,857 --> 00:32:35,274 from E dot dl. And so if the electric field is 435 00:32:35,274 --> 00:32:38,576 in this direction, this has a higher potential 436 00:32:38,576 --> 00:32:41,290 than this. If it is in this direction, 437 00:32:41,290 --> 00:32:43,932 this has a lower potential than this. 438 00:32:43,932 --> 00:32:46,866 This is how this voltage wave propagates. 439 00:32:46,866 --> 00:32:50,755 The E field is between the wires, not along the wires, 440 00:32:50,755 --> 00:32:55,084 so there is only an electric field potential between the two 441 00:32:55,084 --> 00:32:57,652 wires. There are oscillating surface 442 00:32:57,652 --> 00:33:04,208 charge densities. In order to meet those boundary 443 00:33:04,208 --> 00:33:09,502 conditions as the voltage flow comes by. 444 00:33:09,502 --> 00:33:17,511 It is positive at this moment, but a little later in time it 445 00:33:17,511 --> 00:33:23,212 will be negative and this will be positive. 446 00:33:23,212 --> 00:33:28,778 And of that moves with the speed of light. 447 00:33:28,778 --> 00:33:35,565 And then there are also oscillating surface current 448 00:33:35,565 --> 00:33:41,131 densities, this j of s. Let us assume now, 449 00:33:41,131 --> 00:33:48,597 I call this z equals zero, that I am going to short this 450 00:33:48,597 --> 00:33:52,805 out. I am simply putting there a 451 00:33:52,805 --> 00:34:00,000 wire to totally short it out. 33:37 452 00:34:00,000 --> 00:34:05,440 And so here is the situation that the end of the lines which, 453 00:34:05,440 --> 00:34:08,704 by the way, is there, is shorted out. 454 00:34:08,704 --> 00:34:13,418 If it is shorted out, it acts like a conducting wall, 455 00:34:13,418 --> 00:34:16,501 just like what we discussed before. 456 00:34:16,501 --> 00:34:22,032 There cannot be any potential difference between here and here 457 00:34:22,032 --> 00:34:26,565 because the E field in this direction must be zero. 458 00:34:26,565 --> 00:34:31,552 And so you get a similar situation that the reflectivity 459 00:34:31,552 --> 00:34:36,591 is minus one. That a mountain comes back as a 460 00:34:36,591 --> 00:34:39,349 valley. So that the E field of the 461 00:34:39,349 --> 00:34:44,198 incident one and the E field of the reflected one kill each 462 00:34:44,198 --> 00:34:49,213 other exactly at this location. That means you get a standing 463 00:34:49,213 --> 00:34:51,972 wave. So that means this here is a 464 00:34:51,972 --> 00:34:55,148 nodal line. Everywhere in this line the 465 00:34:55,148 --> 00:35:01,000 electric vector will always be zero at all moments in time. 466 00:35:01,000 --> 00:35:05,601 But, since I get a standing wave, incident wave and 467 00:35:05,601 --> 00:35:09,558 reflective wave, right here there is another 468 00:35:09,558 --> 00:35:12,871 nodal line. And this distance then is 469 00:35:12,871 --> 00:35:17,196 one-half lambda. And right here there is another 470 00:35:17,196 --> 00:35:20,785 nodal line. And the electric field here, 471 00:35:20,785 --> 00:35:24,006 along this line, is everywhere zero. 472 00:35:24,006 --> 00:35:28,883 There is no potential difference between y and one and 473 00:35:28,883 --> 00:35:32,318 two here. There is no potential 474 00:35:32,318 --> 00:35:36,209 difference here and no potential difference here at all moments 475 00:35:36,209 --> 00:35:38,595 in time, because it is a standing wave. 476 00:35:38,595 --> 00:35:41,168 And in between you see the electric field. 477 00:35:41,168 --> 00:35:43,867 Let's assume it is in a down direction here, 478 00:35:43,867 --> 00:35:46,064 so it is smaller here, smaller here, 479 00:35:46,064 --> 00:35:48,889 smaller here and then zero here and zero here. 480 00:35:48,889 --> 00:35:51,777 And here the electric field would be like this. 481 00:35:51,777 --> 00:35:54,664 And that whole pattern then is a standing wave, 482 00:35:54,664 --> 00:35:58,493 and so this is down and this is up and then it oscillates like 483 00:35:58,493 --> 00:36:04,193 that. And these are always nodal 484 00:36:04,193 --> 00:36:09,677 lines. Imagine now that this end is 485 00:36:09,677 --> 00:36:14,677 open. So I just remove the short 486 00:36:14,677 --> 00:36:23,870 circuit and I leave it open. Well, I hope you remember the 487 00:36:23,870 --> 00:36:32,121 experiment we did with strings. This is really the equivalent 488 00:36:32,121 --> 00:36:36,439 of the string whereby the end is attached, so you get a node 489 00:36:36,439 --> 00:36:40,609 there, reflectivity is minus one, mountain comes back as a 490 00:36:40,609 --> 00:36:42,073 valley. But this now, 491 00:36:42,073 --> 00:36:46,243 when the end is open so there is completely nothing at the 492 00:36:46,243 --> 00:36:49,682 end, that is the situation that it is right now, 493 00:36:49,682 --> 00:36:52,243 by the way, this is completely open. 494 00:36:52,243 --> 00:36:55,609 This wire here and this wire are not connected. 495 00:36:55,609 --> 00:36:59,634 When this is open you expect a reflectivity of plus one, 496 00:36:59,634 --> 00:37:03,804 just like with the string, that the mountain comes back as 497 00:37:03,804 --> 00:37:07,554 a mountain. And, if that is the case, 498 00:37:07,554 --> 00:37:10,289 then here you get the maximum of the E field. 499 00:37:10,289 --> 00:37:13,583 You get an antinode here. But right here in the middle 500 00:37:13,583 --> 00:37:17,313 you get your nodal lines and right here in the middle you get 501 00:37:17,313 --> 00:37:19,675 your nodal lines, so you get a shift of 502 00:37:19,675 --> 00:37:22,721 one-quarter wavelength. From where you earlier had 503 00:37:22,721 --> 00:37:24,835 nodal lines, you get now antinodes. 504 00:37:24,835 --> 00:37:29,000 And where you had earlier antinodes you now get nodes. 505 00:37:29,000 --> 00:37:32,994 If this is open then these are the nodal lines. 506 00:37:32,994 --> 00:37:38,030 And so those add locations across the wire here where there 507 00:37:38,030 --> 00:37:42,285 is no potential difference at all moments in time. 508 00:37:42,285 --> 00:37:46,801 And so this is then what we would refer to as closed. 509 00:37:46,801 --> 00:37:51,316 And I want you to draw the parallel with our strings. 510 00:37:51,316 --> 00:37:56,439 And I even think that in my problem sets I give you a chance 511 00:37:56,439 --> 00:38:02,167 to work a little bit on this. Remember with strings, 512 00:38:02,167 --> 00:38:08,176 we can fix the lengths of the string in such a way that we get 513 00:38:08,176 --> 00:38:11,919 resonance. We can drive a string at one 514 00:38:11,919 --> 00:38:17,731 end, just moving it a teeny weenie little bit and the end is 515 00:38:17,731 --> 00:38:21,179 fixed. And we can set that system at 516 00:38:21,179 --> 00:38:24,922 resonance. The wave length that we then 517 00:38:24,922 --> 00:38:28,468 get, lambda of n, n being an integer, 518 00:38:28,468 --> 00:38:33,000 so n equals one, two, three, etc. 519 00:38:33,000 --> 00:38:36,671 That wavelength then is 2l divided by n. 520 00:38:36,671 --> 00:38:41,473 You can also wiggle it here and leave this end open. 521 00:38:41,473 --> 00:38:46,274 We even discussed that and I even demonstrated that. 522 00:38:46,274 --> 00:38:52,017 In which case you get resonance for lambda n equals 4l divided 523 00:38:52,017 --> 00:38:54,936 by 2n minus one. And, of course, 524 00:38:54,936 --> 00:39:00,302 the frequency in which you reach that resonance omega of n 525 00:39:00,302 --> 00:39:05,292 is always k of n times the speed of propagation which, 526 00:39:05,292 --> 00:39:10,000 in this case, is the speed of light. 527 00:39:10,000 --> 00:39:14,705 And k of n is then 2pi divided by the wavelength. 528 00:39:14,705 --> 00:39:20,392 And so the way I am going to do this demonstration for you, 529 00:39:20,392 --> 00:39:26,274 we have chosen the length of these wires such that the system 530 00:39:26,274 --> 00:39:33,823 is at resonance when it is open. We have a length which is 4.25 531 00:39:33,823 --> 00:39:41,215 meters, the wavelength is 3.4 meters, so the length is 1.25 532 00:39:41,215 --> 00:39:43,000 wavelength. 533 00:39:43,000 --> 00:39:48,000 534 00:39:48,000 --> 00:39:53,378 And so when it is open, which it is here, 535 00:39:53,378 --> 00:40:00,638 and this is one-quarter wavelength and this is one-half 536 00:40:00,638 --> 00:40:05,379 wavelength and so on. If I leave this open, 537 00:40:05,379 --> 00:40:09,452 and that is what I am going to do, and I have here a light that 538 00:40:09,452 --> 00:40:13,328 is connected between these two wires that I can slide along, 539 00:40:13,328 --> 00:40:15,824 so I have a light here, a light source, 540 00:40:15,824 --> 00:40:19,635 this is a very special light source, has an enormously high 541 00:40:19,635 --> 00:40:22,788 ohmic resistance. So that when I put it here that 542 00:40:22,788 --> 00:40:26,401 I am not disturbing the potential difference between the 543 00:40:26,401 --> 00:40:28,766 wires. I am not changing the boundary 544 00:40:28,766 --> 00:40:32,981 conditions, really. Because it has an enormously 545 00:40:32,981 --> 00:40:36,661 high ohmic resistance. And that light bulb lights up, 546 00:40:36,661 --> 00:40:39,067 even with a minutely small current. 547 00:40:39,067 --> 00:40:43,313 And so the first thing that I am going to show you is that if 548 00:40:43,313 --> 00:40:47,347 I turn on the transmitter, which I will use the same one I 549 00:40:47,347 --> 00:40:51,098 have here, and make sure I don't get electrocuted now. 550 00:40:51,098 --> 00:40:54,000 We are going to take this off. 551 00:40:54,000 --> 00:41:13,000 552 00:41:13,000 --> 00:41:16,563 And it may help, actually, if we give you a nice 553 00:41:16,563 --> 00:41:20,277 light situation so that it is not completely dark. 554 00:41:20,277 --> 00:41:24,296 Now I am going to move the light across the two wires, 555 00:41:24,296 --> 00:41:28,617 and you are going to see that there are locations were the 556 00:41:28,617 --> 00:41:32,483 potential difference is very large because we are at 557 00:41:32,483 --> 00:41:35,037 resonance. And, therefore, 558 00:41:35,037 --> 00:41:39,048 you will see a lot of lights. And there are locations where 559 00:41:39,048 --> 00:41:42,368 there are nodal lines, and you will see no light. 560 00:41:42,368 --> 00:41:45,204 Let's first go here. That is the open end. 561 00:41:45,204 --> 00:41:49,423 The open end is clearly where the electric field is very high. 562 00:41:49,423 --> 00:41:53,089 It is like the situation of a string with an open end. 563 00:41:53,089 --> 00:41:56,755 You have an antinode there. You have an antinode right 564 00:41:56,755 --> 00:41:59,696 here. But, look, when I move the 565 00:41:59,696 --> 00:42:02,176 light a quarter wavelength it goes out. 566 00:42:02,176 --> 00:42:05,896 I must be now near a nodal. Maybe not exactly on the nodal 567 00:42:05,896 --> 00:42:07,854 line. And now I move it another 568 00:42:07,854 --> 00:42:10,856 quarter wavelength and the light goes on again. 569 00:42:10,856 --> 00:42:13,857 Now I am going to move it back to the open end. 570 00:42:13,857 --> 00:42:17,577 And you see the light is on. I move it quarter wavelength, 571 00:42:17,577 --> 00:42:20,449 the light goes off. I move it another quarter 572 00:42:20,449 --> 00:42:23,777 wavelength, light goes on. I move it another quarter 573 00:42:23,777 --> 00:42:26,779 wavelength, light goes off. Isn't this amazing? 574 00:42:26,779 --> 00:42:32,000 Now I think I am going to move it another quarter wavelength. 575 00:42:32,000 --> 00:42:36,384 And the light goes on again. And so it is an amazing thing 576 00:42:36,384 --> 00:42:40,846 that you see in front of your eyes that there is a standing 577 00:42:40,846 --> 00:42:45,230 voltage wave between these two wires, and it is across the 578 00:42:45,230 --> 00:42:49,461 wires, not along the wires. Not what I am going to do is 579 00:42:49,461 --> 00:42:53,384 short this out here. When I put the light bulb right 580 00:42:53,384 --> 00:42:57,230 in the middle here, I put the light bulb at quarter 581 00:42:57,230 --> 00:43:01,214 wavelength. Now, when I short it out at the 582 00:43:01,214 --> 00:43:03,500 end, I am no longer at resonance. 583 00:43:03,500 --> 00:43:07,357 You no longer meet the resonance condition which I meet 584 00:43:07,357 --> 00:43:10,642 now at an open end, something that you can very 585 00:43:10,642 --> 00:43:13,642 easily verify. Now, the moment that I am no 586 00:43:13,642 --> 00:43:17,071 longer at resonance. I no longer have a very high 587 00:43:17,071 --> 00:43:21,071 potential difference at the antinodes, which I had before 588 00:43:21,071 --> 00:43:23,785 when I was at resonance. But, for sure, 589 00:43:23,785 --> 00:43:27,357 the moment that I short out these lines at the end, 590 00:43:27,357 --> 00:43:30,928 the locations which are now nodal lines must become 591 00:43:30,928 --> 00:43:35,101 antinodes. Now, maybe not super duper 592 00:43:35,101 --> 00:43:38,982 antinodes, which are a huge potential difference, 593 00:43:38,982 --> 00:43:42,862 but probably high enough for that light to go on. 594 00:43:42,862 --> 00:43:46,985 Are you ready for this? Now I am going to short this 595 00:43:46,985 --> 00:43:48,925 out. And, as you can see, 596 00:43:48,925 --> 00:43:52,886 the light goes on. Now, it may not be as bright as 597 00:43:52,886 --> 00:43:55,796 you have seen it before. But this is, 598 00:43:55,796 --> 00:44:00,000 again, an antinode and the light is on. 599 00:44:00,000 --> 00:44:04,224 And so you see a remarkable demonstration here where you see 600 00:44:04,224 --> 00:44:07,303 the standing waves, you see the nodal lines, 601 00:44:07,303 --> 00:44:11,742 you see the antinodal lines and you also see that it depends on 602 00:44:11,742 --> 00:44:15,608 the boundary conditions that if you make this open then 603 00:44:15,608 --> 00:44:19,546 one-quarter wavelength from the open end is an antinode. 604 00:44:19,546 --> 00:44:23,985 If you have it open one-quarter wavelength from the open end is 605 00:44:23,985 --> 00:44:27,136 a nodal line. And if you short cut it then it 606 00:44:27,136 --> 00:44:32,328 becomes an antinodal line. I have a second demonstration 607 00:44:32,328 --> 00:44:36,985 which is also very interesting. Let me turn this off before 608 00:44:36,985 --> 00:44:41,401 some of you touch those wires and we have some problems. 609 00:44:41,401 --> 00:44:44,452 I have another very nice demonstration, 610 00:44:44,452 --> 00:44:47,664 which is a cable that is 127 meters long. 611 00:44:47,664 --> 00:44:50,233 You see it here. It starts there, 612 00:44:50,233 --> 00:44:54,970 goes all the way here to make you see that it is really big, 613 00:44:54,970 --> 00:44:58,744 and then it goes back there. And, in this cable, 614 00:44:58,744 --> 00:45:03,000 I am going to send in a voltage pulse. 615 00:45:03,000 --> 00:45:07,429 It is a coaxial cable. And a coaxial cable has a wire 616 00:45:07,429 --> 00:45:11,432 which is at the core, metal, and then there is a 617 00:45:11,432 --> 00:45:15,094 cylinder around it. That is why it is called 618 00:45:15,094 --> 00:45:18,416 coaxial. This is a conductor and this is 619 00:45:18,416 --> 00:45:23,526 a conductor, and it is filled here with dialectic material to 620 00:45:23,526 --> 00:45:26,593 insulate the two conducting surfaces. 621 00:45:26,593 --> 00:45:32,763 And what are we going to do? We are going to send in here 622 00:45:32,763 --> 00:45:37,993 voltage pulses which have a length of 100 nanoseconds, 623 00:45:37,993 --> 00:45:43,717 100 times 10 to the 9 seconds, 100 nanosecond length of the 624 00:45:43,717 --> 00:45:45,789 pulse. In other words, 625 00:45:45,789 --> 00:45:51,513 if you look at the wire from the side, here is the wire and 626 00:45:51,513 --> 00:45:56,250 here is the shell, just take a cross-section here 627 00:45:56,250 --> 00:46:01,843 so you see only that shell. As the voltage pulse moves in 628 00:46:01,843 --> 00:46:05,529 this direction then you will see locally, between the outer 629 00:46:05,529 --> 00:46:08,516 conductor and the inner conductor, that E field. 630 00:46:08,516 --> 00:46:10,614 And it goes to zero here and here. 631 00:46:10,614 --> 00:46:14,237 And that marches then with a certain speed along the wire. 632 00:46:14,237 --> 00:46:17,923 And so, as that pulse passes, there will be an E field like 633 00:46:17,923 --> 00:46:19,957 this. But then there is no pulse. 634 00:46:19,957 --> 00:46:22,563 There is no E field between this and that. 635 00:46:22,563 --> 00:46:25,296 In that sense, it is sort of similar to this 636 00:46:25,296 --> 00:46:27,838 transmission line, except the geometry is 637 00:46:27,838 --> 00:46:33,025 different here. It is a coaxial cable. 638 00:46:33,025 --> 00:46:40,449 Now comes the punch line. I have an option to keep this 639 00:46:40,449 --> 00:46:48,699 end open so that the line at the very end is completely open, 640 00:46:48,699 --> 00:46:53,787 and I have the option to short cut it. 641 00:46:53,787 --> 00:47:00,800 And, when I short it out, a mountain comes back as a 642 00:47:00,800 --> 00:47:04,896 valley. When I leave it open, 643 00:47:04,896 --> 00:47:08,056 a mountain comes back as a mountain. 644 00:47:08,056 --> 00:47:12,933 We are going to show you this pulse at three locations. 645 00:47:12,933 --> 00:47:17,989 Where the pulse enters the system right at the beginning, 646 00:47:17,989 --> 00:47:22,053 which is here. Then we will show you the pulse 647 00:47:22,053 --> 00:47:26,929 at the end of the line here. And then we show it to you 648 00:47:26,929 --> 00:47:33,198 again as it comes back. You are going to see it three 649 00:47:33,198 --> 00:47:38,602 times, and you do. And the system now is closed at 650 00:47:38,602 --> 00:47:42,904 the end. Here is the pulse that goes in. 651 00:47:42,904 --> 00:47:49,301 And 1.3 microseconds later between this line and this line, 652 00:47:49,301 --> 00:47:54,816 there is a marker here which says 1.3 microseconds, 653 00:47:54,816 --> 00:48:00,000 the mountain has returned as a valley. 654 00:48:00,000 --> 00:48:05,099 This is the pulse that enters and then when it has come back. 655 00:48:05,099 --> 00:48:08,583 But now I am going to make it an open end. 656 00:48:08,583 --> 00:48:13,767 Here we see the incoming pulse. And you see the one that comes 657 00:48:13,767 --> 00:48:17,082 back is now 1.29 microseconds. Big deal. 658 00:48:17,082 --> 00:48:21,501 A mountain comes back as a mountain, but now there is 659 00:48:21,501 --> 00:48:25,835 something interesting. At the end of the cable where 660 00:48:25,835 --> 00:48:30,000 it is open you see twice the voltage. 661 00:48:30,000 --> 00:48:33,194 Do we understand that? I see someone say yes. 662 00:48:33,194 --> 00:48:36,461 Why do we see twice? Remember the open string? 663 00:48:36,461 --> 00:48:40,672 Now the reflecting wave and the incident wave have the same 664 00:48:40,672 --> 00:48:45,100 polarity, so at the end they add up together and they give you 665 00:48:45,100 --> 00:48:48,440 double the amplitude. That was exactly the same 666 00:48:48,440 --> 00:48:52,578 situation with the string. When we drove in a pulse on the 667 00:48:52,578 --> 00:48:56,789 string, when the end was open, remember the end moved up by 668 00:48:56,789 --> 00:49:01,000 twice the amplitude of the individual pulse. 669 00:49:01,000 --> 00:49:03,371 And you see that, too, here. 670 00:49:03,371 --> 00:49:07,937 It has twice the amplitude. This is right at the end. 671 00:49:07,937 --> 00:49:12,679 But now there is even more which I discovered purely by 672 00:49:12,679 --> 00:49:15,490 accident. Well, not so much of an 673 00:49:15,490 --> 00:49:18,739 accident. I always do some consistency 674 00:49:18,739 --> 00:49:21,637 checks to see where physics works. 675 00:49:21,637 --> 00:49:25,589 And so I said to myself, OK, 1.3 microseconds, 676 00:49:25,589 --> 00:49:32,000 that is what it takes to make a roundtrip through the wire. 677 00:49:32,000 --> 00:49:37,172 With all the confidence that I have in physics I said 2l 678 00:49:37,172 --> 00:49:40,181 divided by c, l being 127 meters, 679 00:49:40,181 --> 00:49:45,166 should therefore be 1.3 microseconds because that goes 680 00:49:45,166 --> 00:49:49,962 with the speed of light. But when I calculated this, 681 00:49:49,962 --> 00:49:54,006 I found that this is only 0.84 microseconds. 682 00:49:54,006 --> 00:49:59,460 But then I said to myself why is this speed slower than the 683 00:49:59,460 --> 00:50:04,464 speed of light? And I decided to ask you. 684 00:50:04,464 --> 00:50:08,014 Why is the speed substantially lower? 685 00:50:08,014 --> 00:50:11,661 Remember, we observe 1.3 microseconds. 686 00:50:11,661 --> 00:50:16,887 That is what we observe. This is only 65% of the speed 687 00:50:16,887 --> 00:50:19,845 of light. She knows the answer. 688 00:50:19,845 --> 00:50:24,084 Who else knows the answer? I don't blame it. 689 00:50:24,084 --> 00:50:30,000 It took me also two seconds before I realized it. 690 00:50:30,000 --> 00:50:36,000 691 00:50:36,000 --> 00:50:37,538 Nice try but it is the wrong answer. 692 00:50:37,538 --> 00:50:40,000 But it is better that you try than that you don't try. 693 00:50:40,000 --> 00:50:50,000 694 00:50:50,000 --> 00:50:54,425 I think you mean well. You may not express it in the 695 00:50:54,425 --> 00:50:57,462 most effective way. Do you remember, 696 00:50:57,462 --> 00:51:00,933 or do you not, that we discussed that the 697 00:51:00,933 --> 00:51:05,793 speed of propagation v is c divided by the square root of 698 00:51:05,793 --> 00:51:11,000 the index of refraction, of the dielectric constant? 699 00:51:11,000 --> 00:51:13,366 Remember that? I did at one point. 700 00:51:13,366 --> 00:51:17,452 I even showed you how this dielectric constant is a strong 701 00:51:17,452 --> 00:51:21,611 function of frequency for water. Well, when I looked up the 702 00:51:21,611 --> 00:51:25,267 dielectric constant of the material that is in here, 703 00:51:25,267 --> 00:51:30,000 I found that the dielectric constant kappa E is about 2.3. 704 00:51:30,000 --> 00:51:34,262 And if you calculate one divided by the square root of 705 00:51:34,262 --> 00:51:38,927 2.3 that happens to be 0.56. And so that's why the speed is 706 00:51:38,927 --> 00:51:43,351 65% of the speed of light. So you see also the effect of 707 00:51:43,351 --> 00:51:47,613 the dielectric constant here which is quite wonderful. 708 00:51:47,613 --> 00:51:51,957 Remember the square root of kappa E was also called the 709 00:51:51,957 --> 00:51:55,737 index of refraction. This is a nice moment for a 710 00:51:55,737 --> 00:51:58,230 break. We will reconvene in four 711 00:51:58,230 --> 00:52:00,000 minutes. 712 00:52:00,000 --> 00:52:05,000 713 00:52:05,000 --> 00:52:12,118 I would like to bring to a test now what we have learned, 714 00:52:12,118 --> 00:52:19,237 and so you get a chance to do some thinking for a change. 715 00:52:19,237 --> 00:52:27,118 I have here an electromagnetic wave linearly polarized E vector 716 00:52:27,118 --> 00:52:32,711 in this direction, going to propagate in this 717 00:52:32,711 --> 00:52:37,302 direction. And it is set up here 10 718 00:52:37,302 --> 00:52:42,195 gigahertz radiation, radar, so lambda is about three 719 00:52:42,195 --> 00:52:48,143 centimeters, and it is going to be polarized in this direction. 720 00:52:48,143 --> 00:52:51,118 You have to take my word for it. 721 00:52:51,118 --> 00:52:56,394 And, as it arrives here, this receiver is set so that it 722 00:52:56,394 --> 00:53:01,000 receives a radiation in this direction. 723 00:53:01,000 --> 00:53:05,896 If I rotate it 90 degrees then it doesn't receive it, 724 00:53:05,896 --> 00:53:10,415 but that is a different demonstration that we did 725 00:53:10,415 --> 00:53:13,993 before. Now I have a peculiar comb that 726 00:53:13,993 --> 00:53:17,759 is metal. These bars are not connected in 727 00:53:17,759 --> 00:53:20,584 here. They are just metal bars. 728 00:53:20,584 --> 00:53:26,516 And I am going to put this here at the end, and I am going to do 729 00:53:26,516 --> 00:53:32,276 it in two different ways. In one situation, 730 00:53:32,276 --> 00:53:39,241 the comb is like this. And in the other situation the 731 00:53:39,241 --> 00:53:45,133 comb is going to be like this. In both cases, 732 00:53:45,133 --> 00:53:53,035 will the electric field of the wave that comes in be in this 733 00:53:53,035 --> 00:53:57,455 direction? Here it will be in this 734 00:53:57,455 --> 00:54:01,954 direction. In one of those two cases, 735 00:54:01,954 --> 00:54:04,812 the radiation will go straight through. 736 00:54:04,812 --> 00:54:08,496 I will hold it in there and the receiver is happy. 737 00:54:08,496 --> 00:54:12,105 In the other situation, the radiation will not go 738 00:54:12,105 --> 00:54:14,285 through. It will be reflected. 739 00:54:14,285 --> 00:54:17,593 And now I am asking you to make a prediction. 740 00:54:17,593 --> 00:54:21,954 Do you think that if I hold the comb like this in the beam, 741 00:54:21,954 --> 00:54:25,488 that means like this, that the radiation will go 742 00:54:25,488 --> 00:54:27,969 through? Or, will the radiation be 743 00:54:27,969 --> 00:54:32,636 reflected? Or, do you think that in this 744 00:54:32,636 --> 00:54:37,909 case the radiation will go through or will it be reflected? 745 00:54:37,909 --> 00:54:43,000 Let's have a vote on that. First I want to ask the class, 746 00:54:43,000 --> 00:54:48,636 if I hold the comb in the beam like this, do you think that the 747 00:54:48,636 --> 00:54:53,363 wave will be reflected? Who thinks that the wave will 748 00:54:53,363 --> 00:54:57,181 be reflected? Who thinks that the wave will 749 00:54:57,181 --> 00:55:02,300 go straight through? Who thinks that in this case 750 00:55:02,300 --> 00:55:05,011 the wave will go straight through? 751 00:55:05,011 --> 00:55:09,036 Who thinks that it will be reflected in this case? 752 00:55:09,036 --> 00:55:12,733 I am actually quite impressed by your answers. 753 00:55:12,733 --> 00:55:17,087 Most of the time I get more wrong answers than correct 754 00:55:17,087 --> 00:55:20,291 answers. Indeed, it is in this case that 755 00:55:20,291 --> 00:55:23,413 the wave is practically 100% reflected. 756 00:55:23,413 --> 00:55:29,000 And, it is in this case, that the wave will go through. 757 00:55:29,000 --> 00:55:32,902 And, if you have a good understanding about the currents 758 00:55:32,902 --> 00:55:36,804 that are running at these surfaces to make sure you make 759 00:55:36,804 --> 00:55:40,494 the boundary conditions, you will be able to give the 760 00:55:40,494 --> 00:55:43,900 answer for yourself. If you cannot then see me or 761 00:55:43,900 --> 00:55:47,235 write me an email, but I want you to think about 762 00:55:47,235 --> 00:55:51,137 it a little bit for yourself. I will demonstrate it now. 763 00:55:51,137 --> 00:55:55,110 We have this 10 gigahertz signal modulated with 550 hertz 764 00:55:55,110 --> 00:55:59,013 sound so you can hear it. That is the only reason why we 765 00:55:59,013 --> 00:56:02,834 modulate it. So you can hear it. 766 00:56:02,834 --> 00:56:05,827 It is a nasty sound. Here it is. 767 00:56:05,827 --> 00:56:10,462 This is the transmitter and this is the receiver. 768 00:56:10,462 --> 00:56:15,096 Here is the comb. I am going to first stick it in 769 00:56:15,096 --> 00:56:18,379 like this, which is this situation. 770 00:56:18,379 --> 00:56:22,724 And I kill it. Now I rotate it and now it goes 771 00:56:22,724 --> 00:56:26,489 through. It is in this situation that it 772 00:56:26,489 --> 00:56:29,000 goes through. 773 00:56:29,000 --> 00:56:34,000 774 00:56:34,000 --> 00:56:39,890 My hand is not a bad conductor. All of your bodies are very 775 00:56:39,890 --> 00:56:43,445 good conductors. Look at my fingers. 776 00:56:43,445 --> 00:56:46,796 I kill it. Look now at my fingers. 777 00:56:46,796 --> 00:56:51,164 It goes through. You don't even need a comb, 778 00:56:51,164 --> 00:56:56,242 which is a conductor. You can just do it with blood 779 00:56:56,242 --> 00:57:00,000 and flesh. It goes through. 780 00:57:00,000 --> 00:57:05,979 This experiment should give you some insight into the secret 781 00:57:05,979 --> 00:57:10,743 behind Edwin Land's linear polarizers for light, 782 00:57:10,743 --> 00:57:16,418 which we never discussed. Imagine that the radiation that 783 00:57:16,418 --> 00:57:22,297 hits this plate is not linearly polarized but is completely 784 00:57:22,297 --> 00:57:26,148 unpolarized. Then the component in this 785 00:57:26,148 --> 00:57:32,073 direction will be reflected. But only the component in this 786 00:57:32,073 --> 00:57:34,354 direction will be allowed through. 787 00:57:34,354 --> 00:57:36,497 That becomes a linear polarizer. 788 00:57:36,497 --> 00:57:39,953 That is the whole idea behind the linear polarizer. 789 00:57:39,953 --> 00:57:42,649 So this, in a way, is for radar a linear 790 00:57:42,649 --> 00:57:45,483 polarizer. Shine on here unpolarized radar 791 00:57:45,483 --> 00:57:48,870 and all components in this direction will be gone, 792 00:57:48,870 --> 00:57:51,290 only this component will go through. 793 00:57:51,290 --> 00:57:53,640 Linearly radar light will come out. 794 00:57:53,640 --> 00:57:57,580 And so it may give you some thoughts about how the optical 795 00:57:57,580 --> 00:58:02,476 polarizers work. What they do is they align 796 00:58:02,476 --> 00:58:08,285 strings of molecules in such a way that you get a behavior not 797 00:58:08,285 --> 00:58:13,142 unlike these metal bars. Now I want to change gears, 798 00:58:13,142 --> 00:58:18,476 and I want to talk for the remaining time about radiation 799 00:58:18,476 --> 00:58:21,047 pressure. In modern physics, 800 00:58:21,047 --> 00:58:25,333 we don't think of electromagnetic radiation as 801 00:58:25,333 --> 00:58:29,714 plane parallel waves, which are infinite in all 802 00:58:29,714 --> 00:58:34,607 directions. But we think of them as 803 00:58:34,607 --> 00:58:40,388 bullets, as packages of waves. We call them photons. 804 00:58:40,388 --> 00:58:44,923 Photons can be produced by, for instance, 805 00:58:44,923 --> 00:58:50,591 atoms or molecules if they are in an excited state. 806 00:58:50,591 --> 00:58:55,238 And then they decay to a low energy state, 807 00:58:55,238 --> 00:59:00,000 they can radiate just one photon. 808 00:59:00,000 --> 00:59:03,044 But photons, in our modern physics, 809 00:59:03,044 --> 00:59:06,805 have momentum. If you throw a tomato at me, 810 00:59:06,805 --> 00:59:11,731 and it is a rotten tomato, it will go splat and I feel a 811 00:59:11,731 --> 00:59:13,522 push. Another tomato, 812 00:59:13,522 --> 00:59:16,029 another push. Another tomato, 813 00:59:16,029 --> 00:59:18,805 another push. So I feel a force. 814 00:59:18,805 --> 00:59:23,552 And, in that same way, the light, when it hits me also 815 00:59:23,552 --> 00:59:27,044 pushes on me. And that is what I want to 816 00:59:27,044 --> 00:59:31,093 discuss with you. First of all, 817 00:59:31,093 --> 00:59:36,394 the energy, and we have so many Es today, we even had an E of n, 818 00:59:36,394 --> 00:59:41,442 and we had an E which was the E vector, and now we have E for 819 00:59:41,442 --> 00:59:44,302 energy, so I will write the n here. 820 00:59:44,302 --> 00:59:49,014 This stands now for energy. The energy in a photon is Max 821 00:59:49,014 --> 00:59:53,305 Planck's constant times the frequency of the photon. 822 00:59:53,305 --> 00:59:56,838 And Max Planck's constant which, of course, 823 00:59:56,838 --> 1:00:01,802 plays a key role in quantum physics, is 6.63 times 10 to the 824 1:00:01,802 --> 1:00:06,517 minus 34 joule seconds. If you tell me what the 825 1:00:06,517 --> 1:00:09,419 frequency is, I will tell you what the energy 826 1:00:09,419 --> 1:00:13,575 is, which is very different from the classic idea that we always 827 1:00:13,575 --> 1:00:17,071 said, well, the energy is proportional to the electric 828 1:00:17,071 --> 1:00:20,369 field strength squared. And we had plane waves with 829 1:00:20,369 --> 1:00:24,129 infinity in this direction and infinity in this direction. 830 1:00:24,129 --> 1:00:27,691 So we always had infinite energy into a wave of a plane 831 1:00:27,691 --> 1:00:30,000 wave. Not for photons. 832 1:00:30,000 --> 1:00:35,550 Take the example radial waves. I will do only one example for 833 1:00:35,550 --> 1:00:37,863 you. We take 10 megahertz, 834 1:00:37,863 --> 1:00:42,859 10 to the 7 oscillations per second, so that would be a 835 1:00:42,859 --> 1:00:48,132 wavelength of about 30 meters. One photon contains energy, 836 1:00:48,132 --> 1:00:53,405 and each radial photon with this frequency has exactly the 837 1:00:53,405 --> 1:00:57,013 same energy. It is a quantized amount of 838 1:00:57,013 --> 1:01:00,436 energy. And if you substitute for this 839 1:01:00,436 --> 1:01:05,709 frequency, you multiply it by eight, then you get that the 840 1:01:05,709 --> 1:01:12,000 energy is about 6.6 times 10 to the minus 27 joules. 841 1:01:12,000 --> 1:01:16,736 In physics, we don't like to work with such very small 842 1:01:16,736 --> 1:01:22,365 numbers so we often convert them to what we call electron volts. 843 1:01:22,365 --> 1:01:27,370 One electron volt is 1.6 times 10 to the minus 19 joules. 844 1:01:27,370 --> 1:01:33,000 And so that energy then becomes a more digestible number. 845 1:01:33,000 --> 1:01:37,468 It is still small. It is about 4 times 10 to the 846 1:01:37,468 --> 1:01:42,127 minus electron volts. I did this exercise only for 847 1:01:42,127 --> 1:01:46,311 radial waves, but you can do it for any other 848 1:01:46,311 --> 1:01:51,731 frequency that you choose. And we call that then a photon. 849 1:01:51,731 --> 1:01:55,534 In modern physics, photons have momentum. 850 1:01:55,534 --> 1:02:02,000 And the momentum of a photon p is the energy divided by c. 851 1:02:02,000 --> 1:02:06,941 In other words, it is also hf divided by c. 852 1:02:06,941 --> 1:02:12,823 But you remember from 8.01 that the force is db/dt. 853 1:02:12,823 --> 1:02:18,705 Let's return to our tomatoes first so that you feel 854 1:02:18,705 --> 1:02:24,705 comfortable that you know what we are talking about. 855 1:02:24,705 --> 1:02:30,000 I am throwing rotten tomatoes at you. 856 1:02:30,000 --> 1:02:32,910 And those tomatoes come in like this. 857 1:02:32,910 --> 1:02:37,356 And then they lose all their momentum in this direction. 858 1:02:37,356 --> 1:02:40,994 Of course they go plop. Momentum is conserved. 859 1:02:40,994 --> 1:02:45,925 So the momentum that they lose must become my momentum because 860 1:02:45,925 --> 1:02:49,805 momentum is conserved. Suppose I throw at you ten 861 1:02:49,805 --> 1:02:53,928 tomatoes per second, that the mass of each tomato is 862 1:02:53,928 --> 1:02:59,374 one-quarter of a kilogram. And I throw them at you at 40 863 1:02:59,374 --> 1:03:03,387 meters per second, which I can do because I hire a 864 1:03:03,387 --> 1:03:07,071 baseball pitcher. And the baseball pitcher can 865 1:03:07,071 --> 1:03:12,148 throw them at 90 miles per hour, which is 40 meters per second. 866 1:03:12,148 --> 1:03:16,570 And now I can calculate what the average force you will 867 1:03:16,570 --> 1:03:20,255 experience is when all those tomatoes hit you. 868 1:03:20,255 --> 1:03:25,004 You feel a force because the momentum of a tomato is mv and 869 1:03:25,004 --> 1:03:30,000 the force that you will feel in the x direction. 870 1:03:30,000 --> 1:03:37,500 Momentum transfer is now going to be 10 times one-quarter times 871 1:03:37,500 --> 1:03:44,516 40 which is about 100 newtons. That is a substantial force. 872 1:03:44,516 --> 1:03:50,564 It may actually throw you over. It is a huge force. 873 1:03:50,564 --> 1:03:57,338 Let's now turn to photons. And so we have a beam of light 874 1:03:57,338 --> 1:04:02,200 that comes in. And let's give the beam of 875 1:04:02,200 --> 1:04:06,601 light cross-sectional area A. The light comes in this 876 1:04:06,601 --> 1:04:09,987 direction. And this light strikes you and 877 1:04:09,987 --> 1:04:13,457 you absorb it. It hits your face and it is 878 1:04:13,457 --> 1:04:16,504 absorbed. And so the momentum in this 879 1:04:16,504 --> 1:04:20,821 direction is destroyed. Momentum is conserved so you 880 1:04:20,821 --> 1:04:24,037 must have a momentum in this direction. 881 1:04:24,037 --> 1:04:30,597 You must feel a force. The pressure that you feel is 882 1:04:30,597 --> 1:04:37,917 obviously the force that you experience divided by the area. 883 1:04:37,917 --> 1:04:42,631 The force is db/dt divided by the area. 884 1:04:42,631 --> 1:04:48,338 But that is also, if this is the energy divided 885 1:04:48,338 --> 1:04:53,176 by c, I take db/dt, so I get one over c, 886 1:04:53,176 --> 1:05:00,000 then I get my one over A and then I get dE/dt. 887 1:05:00,000 --> 1:05:04,666 E being the energy now. I get here d of energy over dt. 888 1:05:04,666 --> 1:05:09,506 This is joules per second. How many joules per second hit 889 1:05:09,506 --> 1:05:11,839 me? That is what that means. 890 1:05:11,839 --> 1:05:15,814 And this is the area of the cross-section of B. 891 1:05:15,814 --> 1:05:20,308 And this is what we earlier called in this course the 892 1:05:20,308 --> 1:05:24,197 pointing vector. We called that the mean value 893 1:05:24,197 --> 1:05:28,000 of S. This is joules per second. 894 1:05:28,000 --> 1:05:31,404 So this is joules per second per square meter, 895 1:05:31,404 --> 1:05:34,808 which is exactly what the pointing vector was. 896 1:05:34,808 --> 1:05:39,120 And so we can write now that the average pressure that you 897 1:05:39,120 --> 1:05:42,676 will experience, if you absorb that light or the 898 1:05:42,676 --> 1:05:46,156 radial emissions or radars or whatever that is, 899 1:05:46,156 --> 1:05:50,770 is given by this relationship, namely that average pressure is 900 1:05:50,770 --> 1:05:53,796 the average pointing vector divided by c. 901 1:05:53,796 --> 1:05:58,411 You no longer have to think of S being E cross B divided by mu 902 1:05:58,411 --> 1:06:01,658 zero. In fact, the whole idea of 903 1:06:01,658 --> 1:06:04,401 individual photons, which is perfectly fine, 904 1:06:04,401 --> 1:06:08,228 that is where it all came from, can simply be replaced now by 905 1:06:08,228 --> 1:06:10,460 how much energy per second hits you. 906 1:06:10,460 --> 1:06:13,841 And whether this is uv or infrared or radial or radar, 907 1:06:13,841 --> 1:06:17,222 as long as you absorb it, if it goes straight through, 908 1:06:17,222 --> 1:06:19,199 of course, then nothing happens. 909 1:06:19,199 --> 1:06:22,899 But as long as you absorb it that is then the pressure that 910 1:06:22,899 --> 1:06:26,152 you will experience. Remember that we calculated the 911 1:06:26,152 --> 1:06:31,000 pointing vector at earth due to the radiation from the sun. 912 1:06:31,000 --> 1:06:34,624 And I still remember that number because it's the solar 913 1:06:34,624 --> 1:06:37,174 constant. It's the famous 1.4 kilowatts 914 1:06:37,174 --> 1:06:40,127 per square meter. That is the pointing vector 915 1:06:40,127 --> 1:06:43,080 from the solar radiation as it reaches earth, 916 1:06:43,080 --> 1:06:45,362 joules per second per square meter. 917 1:06:45,362 --> 1:06:48,516 I can calculate now, when I expose myself to the 918 1:06:48,516 --> 1:06:51,805 sun, what the pressure is, the radiation pressure, 919 1:06:51,805 --> 1:06:54,624 because I absorb it. You better believe it. 920 1:06:54,624 --> 1:06:57,174 None of that radiation goes through me. 921 1:06:57,174 --> 1:07:01,000 None of that radiation goes through me. 922 1:07:01,000 --> 1:07:03,697 I absorb it. And I feel a pressure. 923 1:07:03,697 --> 1:07:07,980 And I can calculate that pressure by taking this number 924 1:07:07,980 --> 1:07:11,153 and dividing by c. That pressure is very, 925 1:07:11,153 --> 1:07:12,978 very low. The c is huge. 926 1:07:12,978 --> 1:07:17,500 In fact, I can even calculate, if I hold my hand up in the 927 1:07:17,500 --> 1:07:21,387 direction of the sun, what the force will be on my 928 1:07:21,387 --> 1:07:24,242 hand. Well, the force is the pressure 929 1:07:24,242 --> 1:07:27,019 times the area. And, when I do that, 930 1:07:27,019 --> 1:07:31,620 then I find that the force on my hand is only 5 times 10 to 931 1:07:31,620 --> 1:07:35,703 the minus 8. So I hold my hand here. 932 1:07:35,703 --> 1:07:40,222 There is the sun and there is this force on my hand of 5 times 933 1:07:40,222 --> 1:07:43,407 10 to the minus. It is not going to knock me 934 1:07:43,407 --> 1:07:46,000 over. It is not going to be like the 935 1:07:46,000 --> 1:07:48,740 tomatoes. In fact, I don't even notice 936 1:07:48,740 --> 1:07:51,259 it. It is completely insignificant. 937 1:07:51,259 --> 1:07:55,037 The force, due to the radiation pressure of the sun, 938 1:07:55,037 --> 1:07:57,333 is for you and me insignificant. 939 1:07:57,333 --> 1:08:01,851 If I hold a mirror in my hands and I reflect the light back to 940 1:08:01,851 --> 1:08:05,037 the sun then, of course, the momentum change 941 1:08:05,037 --> 1:08:09,838 of the light doubles. It comes in like this and it 942 1:08:09,838 --> 1:08:12,925 goes back like this. You have twice the momentum 943 1:08:12,925 --> 1:08:15,814 transfer to you. And so then the force on you 944 1:08:15,814 --> 1:08:18,507 and the radiation pressure on you doubles. 945 1:08:18,507 --> 1:08:21,988 It is 100% reflection. The radiation pressure is twice 946 1:08:21,988 --> 1:08:26,059 what it is for 100% absorption. It is clear that you and I will 947 1:08:26,059 --> 1:08:30,000 never experience in our lives radiation pressure. 948 1:08:30,000 --> 1:08:34,972 For all the light sources that we deal with and the distances 949 1:08:34,972 --> 1:08:40,110 involved, the pointing vectors are always so small that it will 950 1:08:40,110 --> 1:08:42,762 not be noticeable for you and me. 951 1:08:42,762 --> 1:08:45,994 However, in astronomy that is different. 952 1:08:45,994 --> 1:08:50,055 There are stars 20, 30, 40 times more massive than 953 1:08:50,055 --> 1:08:55,027 the sun, which radiates so much energy that the atmosphere of 954 1:08:55,027 --> 1:09:00,000 the stars is floating on the radiation pressure. 955 1:09:00,000 --> 1:09:04,245 If the star would radiate more, it would blow its atmosphere 956 1:09:04,245 --> 1:09:06,332 off. And, if it radiates less, 957 1:09:06,332 --> 1:09:10,434 the atmosphere would come down. So it is very important in 958 1:09:10,434 --> 1:09:12,736 astronomy. Comets have two tails. 959 1:09:12,736 --> 1:09:16,047 I will show you shortly a picture of a comment. 960 1:09:16,047 --> 1:09:19,429 It has two tails. In general, you and I can only 961 1:09:19,429 --> 1:09:22,667 see one of the tails, which is the white tail, 962 1:09:22,667 --> 1:09:26,985 but there is also a blue tail. The blue tail is the result of 963 1:09:26,985 --> 1:09:31,361 solar winds. The sun, apart from emitting UV 964 1:09:31,361 --> 1:09:35,171 and electromagnetic radiation, also loses hydrogen. 965 1:09:35,171 --> 1:09:39,133 Protons and electrons. And, when they reach the earth 966 1:09:39,133 --> 1:09:42,942 in large quantities, we see Northern Light at night 967 1:09:42,942 --> 1:09:47,057 because they interact with the atmosphere of the earth. 968 1:09:47,057 --> 1:09:51,399 But they also interact with carbon monoxide and the carbon 969 1:09:51,399 --> 1:09:55,361 dioxide of the comet tail. And when they excite those 970 1:09:55,361 --> 1:10:01,000 molecules, and the molecules de-excite, they emit blue light. 971 1:10:01,000 --> 1:10:03,854 That is where the blue tail comes from. 972 1:10:03,854 --> 1:10:06,409 But independently a comet has dust. 973 1:10:06,409 --> 1:10:10,541 And it is the radiation pressure on these dust particles 974 1:10:10,541 --> 1:10:13,696 that blows the particles away from the sun, 975 1:10:13,696 --> 1:10:17,528 purely radiation pressure. These particles are a few 976 1:10:17,528 --> 1:10:20,909 microns in size. And so you get a huge tail of 977 1:10:20,909 --> 1:10:25,492 dust and now you get reflected sunlight off that tail of dust. 978 1:10:25,492 --> 1:10:30,000 The dust tail is formed by radiation pressure. 979 1:10:30,000 --> 1:10:33,564 And the reason why that light is not very bluish, 980 1:10:33,564 --> 1:10:37,352 which you might expect if you accept the scattering, 981 1:10:37,352 --> 1:10:41,214 is probably that those particles are a few microns in 982 1:10:41,214 --> 1:10:45,671 size whereby the preferential scattering for blue light is no 983 1:10:45,671 --> 1:10:49,087 longer very effective. That is why these tails, 984 1:10:49,087 --> 1:10:52,429 which can be 100 million kilometers in length, 985 1:10:52,429 --> 1:10:55,251 are white. Bright white ones are due to 986 1:10:55,251 --> 1:10:59,239 the radiation pressure. And the blue one, 987 1:10:59,239 --> 1:11:02,593 which is very faint, is due to the solar winds. 988 1:11:02,593 --> 1:11:06,312 And I would like to show you Hale-Bop which was very 989 1:11:06,312 --> 1:11:10,468 prominent in the sky in 1997. I hope that many of you have 990 1:11:10,468 --> 1:11:13,166 seen it. It was extremely easy to see. 991 1:11:13,166 --> 1:11:16,010 And I saw it 20, 30, 40 nights in a row. 992 1:11:16,010 --> 1:11:20,312 I couldn't get enough of it. Every night I was staring there 993 1:11:20,312 --> 1:11:23,885 for hours at the sky to see these gorgeous comets, 994 1:11:23,885 --> 1:11:27,093 but I only saw one tail. That was my problem, 995 1:11:27,093 --> 1:11:29,645 of course. Let's take a look at this 996 1:11:29,645 --> 1:11:34,872 beautiful comment Hale-Bop. Who had seen it in 1997? 997 1:11:34,872 --> 1:11:36,667 You were all born. Good. 998 1:11:36,667 --> 1:11:40,179 There you see Hale-Bop. You see the blue tail. 999 1:11:40,179 --> 1:11:44,939 That is the result of the solar winds, which is very faint and 1000 1:11:44,939 --> 1:11:48,606 not easy to see, unless you have a long exposure 1001 1:11:48,606 --> 1:11:53,367 like this picture and then you see the white tail which is the 1002 1:11:53,367 --> 1:11:57,736 result of radiation pressure. The sun is pushing onto the 1003 1:11:57,736 --> 1:12:02,564 small particles. And then you see the reflected 1004 1:12:02,564 --> 1:12:05,726 sunlight. You would expect that if the 1005 1:12:05,726 --> 1:12:11,025 dust particles are small enough, and if you see it at the right 1006 1:12:11,025 --> 1:12:16,239 angle that you would actually sometimes see this tail which is 1007 1:12:16,239 --> 1:12:19,914 mostly white, that you might see it actually 1008 1:12:19,914 --> 1:12:23,675 a little bluish, which we understand why that 1009 1:12:23,675 --> 1:12:28,034 would have to be bluish. Now I want you to test your 1010 1:12:28,034 --> 1:12:32,533 knowledge. Call it the brain teaser. 1011 1:12:32,533 --> 1:12:37,297 I have here a radiometer. Many of you may have a 1012 1:12:37,297 --> 1:12:41,858 radiometer in your rooms. They are very cheap. 1013 1:12:41,858 --> 1:12:46,925 It is a little glass bulb with low pressure inside. 1014 1:12:46,925 --> 1:12:50,878 It is not vacuum but it is low pressure. 1015 1:12:50,878 --> 1:12:56,858 And it has an arm here and an arm perpendicular to that like 1016 1:12:56,858 --> 1:13:02,236 windmills like this. And they can rotate with very 1017 1:13:02,236 --> 1:13:05,706 low friction in this direction. There are two, 1018 1:13:05,706 --> 1:13:08,406 one perpendicular and one like this. 1019 1:13:08,406 --> 1:13:13,264 And here is a little surface no larger than, which you will see, 1020 1:13:13,264 --> 1:13:17,583 a square centimeter or so. And this side is white and the 1021 1:13:17,583 --> 1:13:21,285 backside is black. And this side is black and the 1022 1:13:21,285 --> 1:13:24,910 backside is white. And then there is another one 1023 1:13:24,910 --> 1:13:27,763 like this. And we are going to radiate 1024 1:13:27,763 --> 1:13:31,980 light on this. The light will interact with 1025 1:13:31,980 --> 1:13:34,667 this whole thing. Now I am testing you. 1026 1:13:34,667 --> 1:13:38,557 Not very fair what I am doing because I know the answer. 1027 1:13:38,557 --> 1:13:40,962 Do you think, as the light strikes, 1028 1:13:40,962 --> 1:13:44,852 that this white surface will start to move in the board? 1029 1:13:44,852 --> 1:13:48,813 Because it is going to rotate. You are going to see that. 1030 1:13:48,813 --> 1:13:52,915 Will this go in the board and this come out of the board or 1031 1:13:52,915 --> 1:13:55,603 will it be reversed? Let's have a vote. 1032 1:13:55,603 --> 1:13:59,563 The first one to know who thinks that this white surface, 1033 1:13:59,563 --> 1:14:02,746 as it rotates, goes in the board and that this 1034 1:14:02,746 --> 1:14:07,454 will come out of the board? Who thinks that, 1035 1:14:07,454 --> 1:14:12,636 in fact, this white surface will come to you and this will 1036 1:14:12,636 --> 1:14:15,727 go in the board? It is about 50/50. 1037 1:14:15,727 --> 1:14:19,909 It is interesting. I can demonstrate it to you. 1038 1:14:19,909 --> 1:14:24,727 The one thing I can tell you, though, no, let me first 1039 1:14:24,727 --> 1:14:27,909 demonstrate it. Then we can test it. 1040 1:14:27,909 --> 1:14:31,144 It is nicer. There it is. 1041 1:14:31,144 --> 1:14:34,101 We will make it a little darker. 1042 1:14:34,101 --> 1:14:37,916 Try to remember what your prediction was. 1043 1:14:37,916 --> 1:14:43,163 Now I am shining onto it. You can clearly see the white. 1044 1:14:43,163 --> 1:14:46,596 This part is white and this is black. 1045 1:14:46,596 --> 1:14:51,747 And so the white surface is coming to you and the black 1046 1:14:51,747 --> 1:14:57,375 surface is going away from you. Have of you had it right and 1047 1:14:57,375 --> 1:15:02,818 half of you had it wrong. And the other half were not 1048 1:15:02,818 --> 1:15:05,175 even thinking so they had no answer. 1049 1:15:05,175 --> 1:15:08,138 How can it be that the white is coming to us? 1050 1:15:08,138 --> 1:15:11,977 Because radiation pressure would predict that the force on 1051 1:15:11,977 --> 1:15:16,085 the white surface would be about twice as high as the force on 1052 1:15:16,085 --> 1:15:20,059 the black surface because the white surface reflects and the 1053 1:15:20,059 --> 1:15:23,628 black surface absorbs. Well, the answer is very clear. 1054 1:15:23,628 --> 1:15:26,726 This has nothing to do with radiation pressure. 1055 1:15:26,726 --> 1:15:30,834 If you calculated the force on these surfaces due to radiation 1056 1:15:30,834 --> 1:15:34,892 pressure, it is insignificant. It is nothing. 1057 1:15:34,892 --> 1:15:38,333 It is even less than the force on my hand from the sun. 1058 1:15:38,333 --> 1:15:41,774 What makes this rotate has nothing to do with radiation 1059 1:15:41,774 --> 1:15:44,132 pressure. That is why it was not quite 1060 1:15:44,132 --> 1:15:46,617 fair when I said I am going to test you. 1061 1:15:46,617 --> 1:15:49,803 I like the idea that you may have sleepless nights, 1062 1:15:49,803 --> 1:15:52,544 you know that, and so I would leave you with 1063 1:15:52,544 --> 1:15:54,838 this. Think about why it is rotating. 1064 1:15:54,838 --> 1:15:58,215 And, if you cannot sleep, send me an email and we will 1065 1:15:58,215 --> 1:16:01,984 discuss it. Have a good weekend and see you 1066 1:16:01,984 next Tuesday.