WEBVTT 00:00:00.000 --> 00:00:00.960 align:middle line:90% 00:00:00.960 --> 00:00:02.400 align:middle line:90% [SQUEAKING] 00:00:02.400 --> 00:00:03.840 align:middle line:90% [RUSTLING] 00:00:03.840 --> 00:00:11.100 align:middle line:90% [CLICKING] 00:00:11.100 --> 00:00:13.440 align:middle line:84% JACK HARE: So today, we are starting a new topic, 00:00:13.440 --> 00:00:18.210 align:middle line:84% and we're going to be looking at radiation or self-emission 00:00:18.210 --> 00:00:20.640 align:middle line:90% from the plasmas. 00:00:20.640 --> 00:00:23.100 align:middle line:84% In a way, this is the natural language of the plasma. 00:00:23.100 --> 00:00:25.890 align:middle line:84% Previously, we have surrounded it with magnetic probes. 00:00:25.890 --> 00:00:28.080 align:middle line:84% We've plotted it with Langmuir probes. 00:00:28.080 --> 00:00:31.055 align:middle line:84% We've fired various beams of lasers and microwaves 00:00:31.055 --> 00:00:32.430 align:middle line:84% through it, but now we're looking 00:00:32.430 --> 00:00:34.710 align:middle line:84% at what the plasma produces itself 00:00:34.710 --> 00:00:39.810 align:middle line:84% as a way of trying to diagnose what's going on inside. 00:00:39.810 --> 00:00:41.580 align:middle line:84% And it's important to note, of course, 00:00:41.580 --> 00:00:43.420 align:middle line:84% plasma is not special in this sense. 00:00:43.420 --> 00:00:51.320 align:middle line:84% So all objects will omit electromagnetic radiation. 00:00:51.320 --> 00:00:54.590 align:middle line:90% 00:00:54.590 --> 00:00:59.701 align:middle line:84% Humans, of course, mostly in the IR. 00:00:59.701 --> 00:01:04.519 align:middle line:84% Pokers when they get red hot, famously, blow different colors. 00:01:04.519 --> 00:01:07.340 align:middle line:84% And of course, the universe, in the form of things 00:01:07.340 --> 00:01:09.260 align:middle line:90% like stars and black holes. 00:01:09.260 --> 00:01:12.470 align:middle line:84% But even in the completely empty parts of the universe, 00:01:12.470 --> 00:01:15.920 align:middle line:84% we still have the cosmic microwave background 00:01:15.920 --> 00:01:19.040 align:middle line:84% left over from the Big Bang rattling around. 00:01:19.040 --> 00:01:22.850 align:middle line:84% And so even something that is as cold as outer space 00:01:22.850 --> 00:01:26.360 align:middle line:84% still has some radiation associated with it. 00:01:26.360 --> 00:01:32.660 align:middle line:84% In general, if we have an object which is hotter, 00:01:32.660 --> 00:01:35.090 align:middle line:84% we get higher energy photons out. 00:01:35.090 --> 00:01:38.460 align:middle line:90% 00:01:38.460 --> 00:01:42.840 align:middle line:84% So we could say higher frequency or energy, which is just, 00:01:42.840 --> 00:01:44.700 align:middle line:84% of course, h bar times the frequency. 00:01:44.700 --> 00:01:48.470 align:middle line:90% 00:01:48.470 --> 00:01:49.650 align:middle line:90% Photon's out of it. 00:01:49.650 --> 00:01:51.710 align:middle line:84% And so we might expect in the case of plasmas, 00:01:51.710 --> 00:01:55.412 align:middle line:84% which are usually pretty hot, that we 00:01:55.412 --> 00:01:57.620 align:middle line:84% would be able to get some pretty high energy photons. 00:01:57.620 --> 00:02:00.170 align:middle line:84% And indeed, plasmas tend to span the gamut 00:02:00.170 --> 00:02:03.320 align:middle line:84% all the way from the radiofrequency 00:02:03.320 --> 00:02:08.720 align:middle line:84% up to X-rays and even gamma rays. 00:02:08.720 --> 00:02:12.090 align:middle line:90% 00:02:12.090 --> 00:02:14.480 align:middle line:84% So we're going to be trying to talk about radiation 00:02:14.480 --> 00:02:17.233 align:middle line:84% that spans different energy bands varying 00:02:17.233 --> 00:02:18.650 align:middle line:84% by many, many orders of magnitude. 00:02:18.650 --> 00:02:22.490 align:middle line:84% And just like when we talked about the different plasmas we 00:02:22.490 --> 00:02:24.290 align:middle line:84% worked on at the start of this class, 00:02:24.290 --> 00:02:26.120 align:middle line:84% we saw there was a huge diversity. 00:02:26.120 --> 00:02:28.470 align:middle line:84% Here, we're also dealing with a huge diversity, 00:02:28.470 --> 00:02:31.010 align:middle line:84% so we're going to try and use a framework that treats 00:02:31.010 --> 00:02:32.540 align:middle line:90% all of these in a similar way. 00:02:32.540 --> 00:02:35.157 align:middle line:84% But of course, there will be some subtle details. 00:02:35.157 --> 00:02:36.740 align:middle line:84% And at the end of the day, some things 00:02:36.740 --> 00:02:38.540 align:middle line:84% are going to look more like radio waves 00:02:38.540 --> 00:02:41.120 align:middle line:84% where maybe you need to think more about the wave equation 00:02:41.120 --> 00:02:43.940 align:middle line:84% because the wavelength of the radiation 00:02:43.940 --> 00:02:46.160 align:middle line:84% is on the order of the size of your plasma. 00:02:46.160 --> 00:02:48.440 align:middle line:84% And some things like X-rays and gamma rays 00:02:48.440 --> 00:02:50.978 align:middle line:84% are going to be up where the wavelength is so small compared 00:02:50.978 --> 00:02:52.520 align:middle line:84% to the size of our plasma that we can 00:02:52.520 --> 00:02:54.385 align:middle line:90% use a ray treatment instead. 00:02:54.385 --> 00:02:55.760 align:middle line:84% So just keep in mind that we have 00:02:55.760 --> 00:02:58.720 align:middle line:84% different conceptual frameworks for dealing with radiation 00:02:58.720 --> 00:03:00.470 align:middle line:84% and we may need to modify it a little bit, 00:03:00.470 --> 00:03:02.880 align:middle line:84% depending on what we're dealing with. 00:03:02.880 --> 00:03:05.280 align:middle line:84% Now, the fact that plasmas are hot and emit 00:03:05.280 --> 00:03:09.790 align:middle line:84% all of these different types of radiation has two consequences. 00:03:09.790 --> 00:03:17.790 align:middle line:84% First of all, this can be a significant ruling term. 00:03:17.790 --> 00:03:21.707 align:middle line:84% So for example, the radiation may cool down our plasma. 00:03:21.707 --> 00:03:24.290 align:middle line:84% You saw this particularly if you took the fusion energy class. 00:03:24.290 --> 00:03:27.740 align:middle line:84% And we remember that our Bremsstrahlung cooling there was 00:03:27.740 --> 00:03:30.710 align:middle line:84% a significant loss mechanism in our 0d power [INAUDIBLE] 00:03:30.710 --> 00:03:31.410 align:middle line:90% balance. 00:03:31.410 --> 00:03:34.610 align:middle line:84% But as well as being maybe pesky in the sense that they cool 00:03:34.610 --> 00:03:37.020 align:middle line:84% our nice, hot plasmas down, of course 00:03:37.020 --> 00:03:39.090 align:middle line:90% they also give us information. 00:03:39.090 --> 00:03:44.060 align:middle line:84% So if we can study the spectrum and the location 00:03:44.060 --> 00:03:48.180 align:middle line:84% and the temporal variation of this electromagnetic radiation, 00:03:48.180 --> 00:03:51.210 align:middle line:84% we can get information on our plasma. 00:03:51.210 --> 00:03:53.210 align:middle line:84% And this, of course, is our focus in this course 00:03:53.210 --> 00:03:55.640 align:middle line:84% because this is a diagnostic course. 00:03:55.640 --> 00:03:58.435 align:middle line:84% So in this course, we like radiation. 00:03:58.435 --> 00:03:59.810 align:middle line:84% Maybe in the previous course, you 00:03:59.810 --> 00:04:01.685 align:middle line:84% decided you didn't like radiation because you 00:04:01.685 --> 00:04:02.660 align:middle line:90% wanted to do fusion. 00:04:02.660 --> 00:04:05.690 align:middle line:90% But here, we like the radiation. 00:04:05.690 --> 00:04:08.060 align:middle line:84% Now in general, the way that radiation 00:04:08.060 --> 00:04:12.560 align:middle line:84% moves around inside the plasma, or indeed, inside any fluid 00:04:12.560 --> 00:04:13.820 align:middle line:90% is not trivial. 00:04:13.820 --> 00:04:17.029 align:middle line:84% And we want to think about how that radiation is transported. 00:04:17.029 --> 00:04:18.980 align:middle line:84% And this is a whole topic in itself 00:04:18.980 --> 00:04:21.079 align:middle line:84% that we'll only cover briefly here-- 00:04:21.079 --> 00:04:24.060 align:middle line:84% the topic of radiation transport. 00:04:24.060 --> 00:04:27.580 align:middle line:90% 00:04:27.580 --> 00:04:30.790 align:middle line:84% The basic idea in radiation transport 00:04:30.790 --> 00:04:34.150 align:middle line:84% is that you have some sort of plasma. 00:04:34.150 --> 00:04:40.200 align:middle line:84% There'll be radiation emitted in one place, 00:04:40.200 --> 00:04:42.240 align:middle line:84% and that will locally cool the plasma. 00:04:42.240 --> 00:04:46.920 align:middle line:84% And that radiation may be absorbed in another place, which 00:04:46.920 --> 00:04:48.240 align:middle line:90% will locally heat the plasma. 00:04:48.240 --> 00:04:54.220 align:middle line:84% But then that bit of plasma will also re-emit, and so the cycle 00:04:54.220 --> 00:04:55.130 align:middle line:90% continues. 00:04:55.130 --> 00:04:58.180 align:middle line:84% And you can see straight away that in three dimensions 00:04:58.180 --> 00:05:00.520 align:middle line:84% in a highly inhomogeneous system, 00:05:00.520 --> 00:05:02.960 align:middle line:84% this is going to be very, very complicated. 00:05:02.960 --> 00:05:05.965 align:middle line:84% So this is, effectively, the transport of energy. 00:05:05.965 --> 00:05:14.680 align:middle line:90% 00:05:14.680 --> 00:05:17.130 align:middle line:84% And this transport can be highly non-local. 00:05:17.130 --> 00:05:19.890 align:middle line:84% So you may be used to thinking about transport 00:05:19.890 --> 00:05:25.920 align:middle line:84% in a system where we've got some sort of heat, 00:05:25.920 --> 00:05:27.370 align:middle line:90% thermal conductivity. 00:05:27.370 --> 00:05:30.600 align:middle line:84% And we look at the diffusion of the heat through the material. 00:05:30.600 --> 00:05:31.813 align:middle line:90% That's a very local process. 00:05:31.813 --> 00:05:33.480 align:middle line:84% The amount of heat that's traveling just 00:05:33.480 --> 00:05:35.272 align:middle line:84% depends on the local temperature gradients. 00:05:35.272 --> 00:05:37.710 align:middle line:84% But here, we could have a region which is emitting, 00:05:37.710 --> 00:05:39.925 align:middle line:84% and it gets absorbed a very, very long way away. 00:05:39.925 --> 00:05:41.550 align:middle line:84% And because that radiation is traveling 00:05:41.550 --> 00:05:43.080 align:middle line:84% close to the speed of light, this 00:05:43.080 --> 00:05:44.500 align:middle line:84% can be a very, very fast process. 00:05:44.500 --> 00:05:47.340 align:middle line:84% So this non-locality makes solving the full radiation 00:05:47.340 --> 00:05:50.250 align:middle line:84% transport problem extremely difficult. 00:05:50.250 --> 00:05:53.220 align:middle line:84% In reality, if we want to solve radiation transport, 00:05:53.220 --> 00:05:57.630 align:middle line:84% we often assume that this is a diffusive process. 00:05:57.630 --> 00:06:02.010 align:middle line:84% And there are reasons why that assumption might be valid, 00:06:02.010 --> 00:06:04.050 align:middle line:84% but there are also good reasons why 00:06:04.050 --> 00:06:06.220 align:middle line:90% it may not be valid in general. 00:06:06.220 --> 00:06:09.610 align:middle line:84% So just to give you an overview that radiation transfer is 00:06:09.610 --> 00:06:10.110 align:middle line:90% complicated. 00:06:10.110 --> 00:06:12.870 align:middle line:84% And we're just doing a simplified version of it here. 00:06:12.870 --> 00:06:14.890 align:middle line:84% But radiation transport is extremely 00:06:14.890 --> 00:06:19.468 align:middle line:84% important for understanding how plasmas work-- 00:06:19.468 --> 00:06:21.010 align:middle line:84% for example, the significant cooling. 00:06:21.010 --> 00:06:22.750 align:middle line:84% But it's also very important for understanding 00:06:22.750 --> 00:06:24.460 align:middle line:84% how we get information from the plasmas 00:06:24.460 --> 00:06:29.920 align:middle line:84% because if you just have a camera sitting out here looking 00:06:29.920 --> 00:06:34.490 align:middle line:84% at the plasma and you just see some light coming out, 00:06:34.490 --> 00:06:37.010 align:middle line:84% you really want to have some model that tells you 00:06:37.010 --> 00:06:39.650 align:middle line:84% where that light came from inside your plasma. 00:06:39.650 --> 00:06:41.870 align:middle line:84% If you just think it's all coming from the surface, 00:06:41.870 --> 00:06:43.400 align:middle line:84% you'll get one result. If you think 00:06:43.400 --> 00:06:45.380 align:middle line:84% it's coming from modestly within the plasma, 00:06:45.380 --> 00:06:47.588 align:middle line:84% you'll get a different result. So it's very important 00:06:47.588 --> 00:06:49.950 align:middle line:84% to understand this radiation transport conception. 00:06:49.950 --> 00:06:52.280 align:middle line:84% So we're going to start by having a look at radiation 00:06:52.280 --> 00:06:54.740 align:middle line:84% transport before we even start to talk 00:06:54.740 --> 00:06:57.050 align:middle line:84% about what the radiation is or how it's produced. 00:06:57.050 --> 00:06:59.720 align:middle line:84% So even without knowing any of the details of what's 00:06:59.720 --> 00:07:01.970 align:middle line:84% making all these radio waves and X-rays and gamma rays 00:07:01.970 --> 00:07:03.140 align:middle line:84% and things like that, we're going 00:07:03.140 --> 00:07:05.557 align:middle line:84% to look at radiation transport framework, which we'll then 00:07:05.557 --> 00:07:07.340 align:middle line:84% apply for all of the different wavelengths 00:07:07.340 --> 00:07:09.260 align:middle line:90% that we're working with. 00:07:09.260 --> 00:07:09.812 align:middle line:90% OK. 00:07:09.812 --> 00:07:10.770 align:middle line:90% That's very high level. 00:07:10.770 --> 00:07:12.125 align:middle line:90% Any questions on that so far? 00:07:12.125 --> 00:07:17.917 align:middle line:90% 00:07:17.917 --> 00:07:18.750 align:middle line:90% AUDIENCE: Professor? 00:07:18.750 --> 00:07:20.490 align:middle line:90% JACK HARE: Yes. 00:07:20.490 --> 00:07:22.290 align:middle line:84% AUDIENCE: When you say assume diffuse, 00:07:22.290 --> 00:07:23.828 align:middle line:90% what does that exactly mean? 00:07:23.828 --> 00:07:25.870 align:middle line:84% JACK HARE: Sorry, I was talking at the same time. 00:07:25.870 --> 00:07:29.190 align:middle line:84% So the idea was that most of the time, when 00:07:29.190 --> 00:07:31.680 align:middle line:84% we're trying to solve radiation transport, 00:07:31.680 --> 00:07:34.110 align:middle line:84% we can't use this non-local model 00:07:34.110 --> 00:07:36.160 align:middle line:84% because it's very, very complicated. 00:07:36.160 --> 00:07:39.657 align:middle line:84% And so we often make assumptions that our transport is diffusive. 00:07:39.657 --> 00:07:42.240 align:middle line:84% I'm not really going into this in a great deal of detail here, 00:07:42.240 --> 00:07:42.960 align:middle line:90% but there are-- 00:07:42.960 --> 00:07:45.120 align:middle line:84% I can give some references later if you want. 00:07:45.120 --> 00:07:45.720 align:middle line:90% AUDIENCE: Yeah, that'd be great. 00:07:45.720 --> 00:07:46.650 align:middle line:90% Thank you. 00:07:46.650 --> 00:07:48.108 align:middle line:84% JACK HARE: The other thing I'll say 00:07:48.108 --> 00:07:50.520 align:middle line:84% is that a lot of this radiation transport stuff, 00:07:50.520 --> 00:07:52.410 align:middle line:84% as we'll find out, is important when 00:07:52.410 --> 00:07:56.250 align:middle line:84% you have some sort of opacity in your system for sufficiently 00:07:56.250 --> 00:07:58.380 align:middle line:84% high energy X-rays in a sufficiently 00:07:58.380 --> 00:08:00.390 align:middle line:90% sparse plasma like a tokamak. 00:08:00.390 --> 00:08:04.078 align:middle line:84% Then radiation transport is not necessarily very important. 00:08:04.078 --> 00:08:06.120 align:middle line:84% And so you might think, oh, I'm a tokamak person. 00:08:06.120 --> 00:08:07.110 align:middle line:90% I don't need to know this. 00:08:07.110 --> 00:08:09.750 align:middle line:84% But we'll find out very, very quickly that radiation transport 00:08:09.750 --> 00:08:12.970 align:middle line:84% is still incredibly important for the lower frequency waves, 00:08:12.970 --> 00:08:15.420 align:middle line:84% like the electron cyclotron emission in a tokamak. 00:08:15.420 --> 00:08:17.290 align:middle line:84% So people still need to pay attention 00:08:17.290 --> 00:08:19.790 align:middle line:84% even if they think they're too good for radiation transport. 00:08:19.790 --> 00:08:24.580 align:middle line:84% So let's have a look at what's going on here. 00:08:24.580 --> 00:08:26.770 align:middle line:84% Again, we're going to have some sort of plasma. 00:08:26.770 --> 00:08:30.620 align:middle line:90% 00:08:30.620 --> 00:08:33.110 align:middle line:84% And we're going to have some radiation coming 00:08:33.110 --> 00:08:34.610 align:middle line:90% into the plasma. 00:08:34.610 --> 00:08:38.120 align:middle line:84% Maybe this is radiation from another part of the plasma. 00:08:38.120 --> 00:08:41.809 align:middle line:84% Maybe we have generated a beam of X-rays 00:08:41.809 --> 00:08:44.630 align:middle line:84% or lasers or microwaves that we're using 00:08:44.630 --> 00:08:46.110 align:middle line:90% to shine through the plasma. 00:08:46.110 --> 00:08:47.360 align:middle line:90% It doesn't really matter here. 00:08:47.360 --> 00:08:50.900 align:middle line:84% This radiation is going to have an intensity, 00:08:50.900 --> 00:08:54.260 align:middle line:84% I, and we're going to parameterize 00:08:54.260 --> 00:08:57.380 align:middle line:84% the path of the radiation with this parameter, s. 00:08:57.380 --> 00:09:00.290 align:middle line:84% So we're going to call the point where the radiation enters 00:09:00.290 --> 00:09:03.070 align:middle line:90% the plasma s1. 00:09:03.070 --> 00:09:07.810 align:middle line:84% And then there is some path, s, through the plasma. 00:09:07.810 --> 00:09:09.910 align:middle line:84% And we're interested in the properties 00:09:09.910 --> 00:09:12.790 align:middle line:90% of the radiation at point s2. 00:09:12.790 --> 00:09:17.640 align:middle line:84% That's I at s2 when the radiation 00:09:17.640 --> 00:09:19.690 align:middle line:90% has exited the plasma here. 00:09:19.690 --> 00:09:22.390 align:middle line:84% So what we want to know is, how does I 00:09:22.390 --> 00:09:25.500 align:middle line:90% change along this path, s? 00:09:25.500 --> 00:09:27.720 align:middle line:84% Now, in general, this path, s, could 00:09:27.720 --> 00:09:30.840 align:middle line:84% be curved because we've talked extensively about the fact 00:09:30.840 --> 00:09:33.450 align:middle line:84% that when you have changes in the refractive index, 00:09:33.450 --> 00:09:36.180 align:middle line:84% you're going to have refraction of your red. 00:09:36.180 --> 00:09:38.522 align:middle line:84% Now, although I have drawn it curved here, 00:09:38.522 --> 00:09:40.980 align:middle line:84% a lot of the time I'm going to assume it's a straight line. 00:09:40.980 --> 00:09:44.100 align:middle line:90% 00:09:44.100 --> 00:09:45.480 align:middle line:90% So just watch out for that. 00:09:45.480 --> 00:09:47.270 align:middle line:84% But if you want to solve the full thing, 00:09:47.270 --> 00:09:50.000 align:middle line:84% you need to take this curvature into account. 00:09:50.000 --> 00:09:50.660 align:middle line:90% OK. 00:09:50.660 --> 00:09:53.460 align:middle line:84% And this quantity, I, that we're dealing with here, 00:09:53.460 --> 00:09:56.360 align:middle line:84% this is a quantity which we can formally define 00:09:56.360 --> 00:10:00.270 align:middle line:90% as the spectral radiance. 00:10:00.270 --> 00:10:07.830 align:middle line:90% 00:10:07.830 --> 00:10:09.990 align:middle line:84% I've used the symbol, I, which is often 00:10:09.990 --> 00:10:11.605 align:middle line:84% used for intensity because this is 00:10:11.605 --> 00:10:12.855 align:middle line:90% a symbol that's commonly used. 00:10:12.855 --> 00:10:16.330 align:middle line:90% 00:10:16.330 --> 00:10:18.970 align:middle line:84% Although people often refer to the spectral radiance 00:10:18.970 --> 00:10:21.280 align:middle line:84% as the intensity, they also use the word intensity 00:10:21.280 --> 00:10:22.910 align:middle line:84% to mean lots of different things. 00:10:22.910 --> 00:10:25.990 align:middle line:84% And so if you go on the radiosity Wikipedia page, 00:10:25.990 --> 00:10:27.790 align:middle line:84% there's an incredible table that has 00:10:27.790 --> 00:10:30.970 align:middle line:84% very, very niche words for all sorts of different quantities. 00:10:30.970 --> 00:10:32.943 align:middle line:84% And this is the table I always go 00:10:32.943 --> 00:10:35.360 align:middle line:84% to when I want to work out exactly what I'm talking about. 00:10:35.360 --> 00:10:37.660 align:middle line:84% And the reason is that this word, spectral radiance, 00:10:37.660 --> 00:10:42.670 align:middle line:84% here corresponds to one exact set of units, which 00:10:42.670 --> 00:10:49.270 align:middle line:84% is the watts, the joules per unit time being emitted 00:10:49.270 --> 00:10:53.020 align:middle line:84% through an area, meters squared here-- 00:10:53.020 --> 00:11:02.420 align:middle line:84% through a solid angle per some spectral unit here. 00:11:02.420 --> 00:11:10.410 align:middle line:84% So the spectral unit could be something like hertz. 00:11:10.410 --> 00:11:13.000 align:middle line:90% 00:11:13.000 --> 00:11:15.160 align:middle line:84% It could be something like joules. 00:11:15.160 --> 00:11:17.440 align:middle line:90% It could be electron volts. 00:11:17.440 --> 00:11:19.390 align:middle line:90% It could be meters. 00:11:19.390 --> 00:11:21.100 align:middle line:84% The difference in this being depends 00:11:21.100 --> 00:11:24.020 align:middle line:84% how you're resolving your radiation spectrum. 00:11:24.020 --> 00:11:25.930 align:middle line:84% So in terms of hertz, that's where 00:11:25.930 --> 00:11:28.900 align:middle line:90% we're using angular frequency. 00:11:28.900 --> 00:11:31.510 align:middle line:84% Joules and EV, that's where we're using energy. 00:11:31.510 --> 00:11:34.010 align:middle line:84% And meters, that's where we're measuring it in wavelength. 00:11:34.010 --> 00:11:37.960 align:middle line:84% So if we're talking about the spectral radiance in terms of, 00:11:37.960 --> 00:11:43.090 align:middle line:84% I don't know, gigahertz or per EV or per meters here. 00:11:43.090 --> 00:11:46.060 align:middle line:84% Of course, if you put meters in this one, 00:11:46.060 --> 00:11:47.568 align:middle line:84% then it starts to get complicated 00:11:47.568 --> 00:11:49.610 align:middle line:84% because this becomes a cubed and stuff like that. 00:11:49.610 --> 00:11:51.280 align:middle line:84% And so people will actually write this 00:11:51.280 --> 00:11:56.860 align:middle line:84% as watts per meter squared per steradian per meter, 00:11:56.860 --> 00:11:59.320 align:middle line:84% just to remind themselves that they haven't accidentally 00:11:59.320 --> 00:12:00.410 align:middle line:90% folded this in. 00:12:00.410 --> 00:12:05.530 align:middle line:84% So what this means is that there is some amount of radiation 00:12:05.530 --> 00:12:13.070 align:middle line:84% that's going through some unit area-- that's the meter squared. 00:12:13.070 --> 00:12:15.330 align:middle line:84% We've got watts going in this direction, 00:12:15.330 --> 00:12:20.490 align:middle line:84% and that watts is subtended by some solid angle that's 00:12:20.490 --> 00:12:21.810 align:middle line:90% measured in steradians. 00:12:21.810 --> 00:12:24.180 align:middle line:84% And we're also resolving it in terms 00:12:24.180 --> 00:12:26.400 align:middle line:90% of electronvolts or joules. 00:12:26.400 --> 00:12:28.950 align:middle line:90% This is a complicated quantity. 00:12:28.950 --> 00:12:30.930 align:middle line:84% But it will correspond to your intuition 00:12:30.930 --> 00:12:33.570 align:middle line:84% of what intensity of light is like if you 00:12:33.570 --> 00:12:35.670 align:middle line:90% think about it a little bit. 00:12:35.670 --> 00:12:37.230 align:middle line:90% OK. 00:12:37.230 --> 00:12:39.480 align:middle line:84% So let's just have an example here. 00:12:39.480 --> 00:12:41.665 align:middle line:84% Maybe our spectral radiance-- and I 00:12:41.665 --> 00:12:43.290 align:middle line:84% may start from this intensity every now 00:12:43.290 --> 00:12:46.305 align:middle line:84% and again-- initially has a spectrum. 00:12:46.305 --> 00:12:49.160 align:middle line:90% 00:12:49.160 --> 00:12:52.340 align:middle line:84% And I shall do this in hertz here. 00:12:52.340 --> 00:12:55.550 align:middle line:84% This is what Hutchinson uses in his book. 00:12:55.550 --> 00:13:00.860 align:middle line:84% Maybe our spectrum initially has some sort of complicated set 00:13:00.860 --> 00:13:03.470 align:middle line:84% of features-- spectral lines in the background, 00:13:03.470 --> 00:13:05.120 align:middle line:90% stuff like that-- 00:13:05.120 --> 00:13:06.300 align:middle line:90% like this. 00:13:06.300 --> 00:13:10.250 align:middle line:84% And by the time that it has traveled through the plasma, 00:13:10.250 --> 00:13:13.010 align:middle line:84% there will be some emission inside the plasma. 00:13:13.010 --> 00:13:16.250 align:middle line:84% The plasma will make more light which adds to our beam of light, 00:13:16.250 --> 00:13:18.920 align:middle line:84% and there'll be some absorption inside the plasma that 00:13:18.920 --> 00:13:19.980 align:middle line:90% gets rid of these. 00:13:19.980 --> 00:13:22.400 align:middle line:84% So for example, we might have absorption 00:13:22.400 --> 00:13:23.660 align:middle line:90% at these low frequencies. 00:13:23.660 --> 00:13:25.280 align:middle line:84% They've been completely wiped out. 00:13:25.280 --> 00:13:31.890 align:middle line:84% And then we will have some additional emission 00:13:31.890 --> 00:13:33.520 align:middle line:90% from inside your plasma. 00:13:33.520 --> 00:13:36.270 align:middle line:84% So here, we would have a region where there was absorption. 00:13:36.270 --> 00:13:45.310 align:middle line:90% 00:13:45.310 --> 00:13:47.710 align:middle line:84% And here we have a region where there was emission. 00:13:47.710 --> 00:13:54.840 align:middle line:90% 00:13:54.840 --> 00:13:56.270 align:middle line:84% So in general, we want to come up 00:13:56.270 --> 00:13:58.175 align:middle line:84% with some formulas that will tell us how, 00:13:58.175 --> 00:14:00.800 align:middle line:84% for a given amount of absorption and a given amount of emission 00:14:00.800 --> 00:14:02.217 align:middle line:84% in different parts of the plasma-- 00:14:02.217 --> 00:14:05.400 align:middle line:84% how the spectra will change from one point to the other. 00:14:05.400 --> 00:14:08.190 align:middle line:84% And this is the radiation transport equation, 00:14:08.190 --> 00:14:13.400 align:middle line:84% which is the very fundamental equation in this field. 00:14:13.400 --> 00:14:15.910 align:middle line:90% 00:14:15.910 --> 00:14:17.760 align:middle line:84% And the radiation transport equation 00:14:17.760 --> 00:14:25.660 align:middle line:84% says that the change in spectral radiance along our path 00:14:25.660 --> 00:14:29.650 align:middle line:84% that we've parametrized by s is simply 00:14:29.650 --> 00:14:35.880 align:middle line:84% equal to this quantity, j, which is 00:14:35.880 --> 00:14:52.240 align:middle line:84% the emissivity minus the intensity 00:14:52.240 --> 00:14:54.910 align:middle line:84% of the spectral radiance times by another quantity, 00:14:54.910 --> 00:14:58.300 align:middle line:90% alpha, which is the opacity. 00:14:58.300 --> 00:15:04.757 align:middle line:90% 00:15:04.757 --> 00:15:06.340 align:middle line:84% The important thing about the equation 00:15:06.340 --> 00:15:10.240 align:middle line:84% as we've written it here is that each of these quantities 00:15:10.240 --> 00:15:15.360 align:middle line:90% are functions of frequency. 00:15:15.360 --> 00:15:19.080 align:middle line:84% And effectively, this equation is linear in frequency, 00:15:19.080 --> 00:15:23.280 align:middle line:84% in the sense that we can solve the equations separately 00:15:23.280 --> 00:15:24.540 align:middle line:90% for each frequency. 00:15:24.540 --> 00:15:27.810 align:middle line:90% 00:15:27.810 --> 00:15:30.260 align:middle line:84% So whatever is happening down in the gigahertz range 00:15:30.260 --> 00:15:34.140 align:middle line:84% has nothing to do with what's happening up in the X-ray range. 00:15:34.140 --> 00:15:36.140 align:middle line:90% This is not necessarily true. 00:15:36.140 --> 00:15:37.590 align:middle line:84% In order to make this assumption, 00:15:37.590 --> 00:15:43.830 align:middle line:84% we have to neglect interesting processes like fluorescence, 00:15:43.830 --> 00:15:47.630 align:middle line:84% where we might excite part of the plasma at one frequency 00:15:47.630 --> 00:15:50.210 align:middle line:84% and get back lines at different frequencies. 00:15:50.210 --> 00:15:53.010 align:middle line:84% But in order to be able to solve this equation easily, 00:15:53.010 --> 00:15:54.600 align:middle line:90% we're going to neglect that. 00:15:54.600 --> 00:15:56.817 align:middle line:84% So this is an assumption that we've made. 00:15:56.817 --> 00:15:59.400 align:middle line:84% And I'll just tell you what the units of all these things are. 00:15:59.400 --> 00:16:02.040 align:middle line:84% You can work them out yourself, but the units of opacity-- 00:16:02.040 --> 00:16:07.000 align:middle line:84% very clearly, it has units of inverse length scale. 00:16:07.000 --> 00:16:11.080 align:middle line:84% And the units of emissivity are still 00:16:11.080 --> 00:16:17.960 align:middle line:90% watts per steradian per hertz. 00:16:17.960 --> 00:16:24.370 align:middle line:84% But now it's not per meter squared, but per meter cubed. 00:16:24.370 --> 00:16:27.480 align:middle line:84% And this reflects the fact that the emissivity 00:16:27.480 --> 00:16:30.720 align:middle line:84% is a measure of the light coming out 00:16:30.720 --> 00:16:36.610 align:middle line:84% from a small volume in every direction like this. 00:16:36.610 --> 00:16:38.680 align:middle line:84% The first region means that if you integrate over 00:16:38.680 --> 00:16:43.030 align:middle line:84% 4 pi steradians, you get the total radiated power 00:16:43.030 --> 00:16:45.670 align:middle line:90% per cubic meter per hertz. 00:16:45.670 --> 00:16:51.490 align:middle line:90% 00:16:51.490 --> 00:16:55.822 align:middle line:84% AUDIENCE: Emissivity and opacity are also functions of s, right? 00:16:55.822 --> 00:16:56.560 align:middle line:90% JACK HARE: Yes. 00:16:56.560 --> 00:17:00.370 align:middle line:84% I've suppressed the s in here, but you're quite right. 00:17:00.370 --> 00:17:03.485 align:middle line:84% As we go through the plasma, these will in general change. 00:17:03.485 --> 00:17:03.985 align:middle line:90% Yeah. 00:17:03.985 --> 00:17:06.740 align:middle line:90% 00:17:06.740 --> 00:17:08.720 align:middle line:84% These are properties in the material. 00:17:08.720 --> 00:17:11.300 align:middle line:84% It could be a plasma or it could be a lump of cloudy glass. 00:17:11.300 --> 00:17:14.240 align:middle line:84% We haven't actually decided or said anything about them. 00:17:14.240 --> 00:17:17.480 align:middle line:84% And we won't learn how to calculate these for some time. 00:17:17.480 --> 00:17:20.119 align:middle line:84% You just have to take my word for it that you can calculate 00:17:20.119 --> 00:17:21.428 align:middle line:90% the emissivity and the opacity. 00:17:21.428 --> 00:17:22.970 align:middle line:84% You've come across some of this stuff 00:17:22.970 --> 00:17:27.319 align:middle line:84% before if you've done blackbody radiation. 00:17:27.319 --> 00:17:30.060 align:middle line:84% But we're just going to leave it like that at the moment. 00:17:30.060 --> 00:17:30.560 align:middle line:90% OK. 00:17:30.560 --> 00:17:31.280 align:middle line:90% Any questions? 00:17:31.280 --> 00:17:32.690 align:middle line:90% Yes, Sean? 00:17:32.690 --> 00:17:34.778 align:middle line:90% AUDIENCE: Does this equation-- 00:17:34.778 --> 00:17:37.070 align:middle line:84% if you include temperature dependence of the emissivity 00:17:37.070 --> 00:17:40.070 align:middle line:84% and the opacity in this equation to be used to understand heating 00:17:40.070 --> 00:17:42.410 align:middle line:84% of the plasma, if you're thinking about maybe 00:17:42.410 --> 00:17:43.938 align:middle line:84% you're injecting waves or something. 00:17:43.938 --> 00:17:45.980 align:middle line:84% JACK HARE: So the question was, can this equation 00:17:45.980 --> 00:17:50.180 align:middle line:84% be used to understand heating of the plasma? 00:17:50.180 --> 00:17:52.460 align:middle line:84% Yeah, effectively this term is the amount 00:17:52.460 --> 00:17:57.030 align:middle line:84% of energy lost from the beam per unit length. 00:17:57.030 --> 00:18:00.140 align:middle line:84% And so you could add that energy into your plasma. 00:18:00.140 --> 00:18:03.390 align:middle line:84% So you could couple this to an energy equation in your plasma. 00:18:03.390 --> 00:18:06.470 align:middle line:84% But this itself is not an energy equation for the plasma. 00:18:06.470 --> 00:18:09.230 align:middle line:84% This is an energy equation effectively for the energy 00:18:09.230 --> 00:18:11.000 align:middle line:90% in the radiation. 00:18:11.000 --> 00:18:11.683 align:middle line:90% AUDIENCE: I see. 00:18:11.683 --> 00:18:12.350 align:middle line:90% JACK HARE: Yeah. 00:18:12.350 --> 00:18:15.290 align:middle line:84% But if you wanted to do full radiation transport, 00:18:15.290 --> 00:18:18.020 align:middle line:84% you would couple this to your energy equation, yeah. 00:18:18.020 --> 00:18:18.920 align:middle line:90% Other questions? 00:18:18.920 --> 00:18:25.950 align:middle line:90% 00:18:25.950 --> 00:18:26.450 align:middle line:90% OK. 00:18:26.450 --> 00:18:28.250 align:middle line:84% It's very convenient to introduce 00:18:28.250 --> 00:18:30.920 align:middle line:84% a slightly mysterious but important quantity called 00:18:30.920 --> 00:18:31.730 align:middle line:90% the optical depth. 00:18:31.730 --> 00:18:40.770 align:middle line:90% 00:18:40.770 --> 00:18:43.230 align:middle line:84% And the optical depth, which is still 00:18:43.230 --> 00:18:44.940 align:middle line:84% a function of frequency-- and I'm 00:18:44.940 --> 00:18:47.040 align:middle line:84% just going to keep trying to write these as long as possible 00:18:47.040 --> 00:18:48.900 align:middle line:84% to remind you of all functions of frequency. 00:18:48.900 --> 00:18:50.910 align:middle line:84% This is a quantity which is defined 00:18:50.910 --> 00:18:57.880 align:middle line:84% as the integral of the opacity along some section of path. 00:18:57.880 --> 00:19:01.210 align:middle line:84% And just written like this, this is an indefinite integral. 00:19:01.210 --> 00:19:04.850 align:middle line:84% We will come up with some limits to it later on. 00:19:04.850 --> 00:19:07.690 align:middle line:84% So at the moment, it doesn't have a huge meaning. 00:19:07.690 --> 00:19:09.790 align:middle line:84% But the important thing about it is 00:19:09.790 --> 00:19:19.760 align:middle line:84% that it depends on the actual path, s, and frequency. 00:19:19.760 --> 00:19:23.280 align:middle line:84% So the optical depth can be very different for X-rays-- 00:19:23.280 --> 00:19:26.340 align:middle line:84% good for microwaves-- so again, very important to realize that. 00:19:26.340 --> 00:19:29.130 align:middle line:84% The reason we introduced this is that if you stare 00:19:29.130 --> 00:19:31.450 align:middle line:84% at this definition and this equation, 00:19:31.450 --> 00:19:33.750 align:middle line:84% you might be able to convince yourself that we can then 00:19:33.750 --> 00:19:38.520 align:middle line:84% rewrite the equation as dI d tau is 00:19:38.520 --> 00:19:45.820 align:middle line:90% equal to j upon alpha minus I. 00:19:45.820 --> 00:19:47.290 align:middle line:84% And this is rather a nice equation 00:19:47.290 --> 00:19:49.680 align:middle line:84% because we use the equations of the form dI 00:19:49.680 --> 00:19:53.230 align:middle line:84% d tau equals minus I. These are exponential growth, 00:19:53.230 --> 00:19:54.820 align:middle line:90% exponential decay equations. 00:19:54.820 --> 00:19:58.040 align:middle line:84% And we're also used to adding an extra term in here, 00:19:58.040 --> 00:19:59.390 align:middle line:90% which is like a source term. 00:19:59.390 --> 00:20:02.080 align:middle line:84% So what this equation says is that the intensity 00:20:02.080 --> 00:20:06.490 align:middle line:84% is going to drop off along the path parametrized by s, 00:20:06.490 --> 00:20:08.020 align:middle line:90% which is now sitting inside tau. 00:20:08.020 --> 00:20:11.110 align:middle line:84% But the intensity is going to increase with a parameter that 00:20:11.110 --> 00:20:12.968 align:middle line:90% looks like J upon alpha. 00:20:12.968 --> 00:20:15.385 align:middle line:84% And I'll show you how we're going to use that in a second. 00:20:15.385 --> 00:20:25.580 align:middle line:90% 00:20:25.580 --> 00:20:26.650 align:middle line:90% Let's label explicitly. 00:20:26.650 --> 00:20:29.320 align:middle line:84% This, then, looks like a source function. 00:20:29.320 --> 00:20:42.790 align:middle line:90% 00:20:42.790 --> 00:20:44.520 align:middle line:84% So now we can go back to this problem 00:20:44.520 --> 00:20:46.390 align:middle line:90% and we can solve this equation. 00:20:46.390 --> 00:20:56.280 align:middle line:84% So we can solve for s1 to s2, like that. 00:20:56.280 --> 00:20:58.520 align:middle line:84% And we're going to solve it by integrating up 00:20:58.520 --> 00:21:00.500 align:middle line:90% with respect to tau here. 00:21:00.500 --> 00:21:07.060 align:middle line:84% And we're going to find that the intensity at s2 00:21:07.060 --> 00:21:09.130 align:middle line:84% on the other side of the plasma is 00:21:09.130 --> 00:21:14.740 align:middle line:84% going to be the intensity that we started out with at s1 times 00:21:14.740 --> 00:21:22.010 align:middle line:84% by a factor of tau 1 minus tau 2. 00:21:22.010 --> 00:21:24.980 align:middle line:84% So this is an attenuation that takes into account 00:21:24.980 --> 00:21:27.560 align:middle line:90% any absorption along the path. 00:21:27.560 --> 00:21:30.200 align:middle line:84% And then along with the initial radiation 00:21:30.200 --> 00:21:32.990 align:middle line:84% we started with, we're going to have a second term which 00:21:32.990 --> 00:21:35.420 align:middle line:84% corresponds to the radiation we've picked up 00:21:35.420 --> 00:21:37.970 align:middle line:84% through the plasma due to the emissivity. 00:21:37.970 --> 00:21:42.650 align:middle line:84% And that's going to be the integral from s1 to s2. 00:21:42.650 --> 00:21:45.370 align:middle line:90% 00:21:45.370 --> 00:21:49.390 align:middle line:84% Our emissivity, now explicitly parameterized as a function 00:21:49.390 --> 00:21:50.560 align:middle line:90% of s. 00:21:50.560 --> 00:21:52.900 align:middle line:84% As someone asked already, all of the J's and alphas 00:21:52.900 --> 00:21:53.890 align:middle line:90% are functions of s. 00:21:53.890 --> 00:22:02.710 align:middle line:84% But now I'm making explicit here e into the tau minus tau 2 ds. 00:22:02.710 --> 00:22:07.550 align:middle line:84% What this exponential here is doing is saying, yeah, OK. 00:22:07.550 --> 00:22:09.650 align:middle line:84% You're adding radiation, but that radiation 00:22:09.650 --> 00:22:11.150 align:middle line:90% may also be absorbed. 00:22:11.150 --> 00:22:14.970 align:middle line:84% If that radiation is emitted very early on, near s1, 00:22:14.970 --> 00:22:17.600 align:middle line:84% it's going to be absorbed a lot as it goes through the plasma. 00:22:17.600 --> 00:22:21.320 align:middle line:84% If the radiation is emitted late along the path, close to s2, 00:22:21.320 --> 00:22:23.000 align:middle line:84% it's not going to be absorbed very much. 00:22:23.000 --> 00:22:24.770 align:middle line:84% So it matters where along the path 00:22:24.770 --> 00:22:27.980 align:middle line:84% the radiation is actually emitted. 00:22:27.980 --> 00:22:30.060 align:middle line:84% So let me just label these terms here. 00:22:30.060 --> 00:22:33.980 align:middle line:90% So this is the original. 00:22:33.980 --> 00:22:36.590 align:middle line:90% 00:22:36.590 --> 00:22:40.280 align:middle line:84% I'm going to call it spectral radiance so I don't accidentally 00:22:40.280 --> 00:22:41.880 align:middle line:90% say intensity. 00:22:41.880 --> 00:22:44.330 align:middle line:84% So the original spectral radiance-- 00:22:44.330 --> 00:22:57.740 align:middle line:84% this is the optical depth from s1 to s2. 00:22:57.740 --> 00:23:10.270 align:middle line:84% This is the emission along the path, s. 00:23:10.270 --> 00:23:20.320 align:middle line:84% And this is the depth from s to s2, 00:23:20.320 --> 00:23:22.060 align:middle line:84% where s is wherever along the path 00:23:22.060 --> 00:23:25.080 align:middle line:90% this radiation has been emitted. 00:23:25.080 --> 00:23:27.090 align:middle line:84% This is a complicated equation, and I'm now 00:23:27.090 --> 00:23:29.580 align:middle line:84% going to give you two simple examples 00:23:29.580 --> 00:23:31.680 align:middle line:84% to try and build up your intuition for what 00:23:31.680 --> 00:23:33.010 align:middle line:90% this equation is saying. 00:23:33.010 --> 00:23:35.220 align:middle line:84% If you don't think it's complicated-- 00:23:35.220 --> 00:23:36.793 align:middle line:90% well, never mind. 00:23:36.793 --> 00:23:39.210 align:middle line:84% I think it's quite a complicated equation, especially what 00:23:39.210 --> 00:23:40.780 align:middle line:90% these taus are doing here. 00:23:40.780 --> 00:23:43.900 align:middle line:84% Any questions on it, though, before we keep going. 00:23:43.900 --> 00:23:46.470 align:middle line:90% Yeah? 00:23:46.470 --> 00:23:50.890 align:middle line:84% AUDIENCE: So for the emission along s term, 00:23:50.890 --> 00:23:55.920 align:middle line:84% if we're wondering about what's appearing at the point, s2, 00:23:55.920 --> 00:23:59.660 align:middle line:84% I'm just confused why we don't care about the e angle 00:23:59.660 --> 00:24:01.023 align:middle line:90% that it is emitted from, right? 00:24:01.023 --> 00:24:02.940 align:middle line:84% Because if it's emitted back along the pathway 00:24:02.940 --> 00:24:06.930 align:middle line:84% that you completed, it's not heading towards s2. 00:24:06.930 --> 00:24:08.922 align:middle line:84% I guess where is that dependence in there? 00:24:08.922 --> 00:24:09.630 align:middle line:90% JACK HARE: Right. 00:24:09.630 --> 00:24:13.770 align:middle line:84% So the question was, where is the direction of the emission 00:24:13.770 --> 00:24:15.240 align:middle line:90% featuring into this equation? 00:24:15.240 --> 00:24:17.268 align:middle line:84% And the question is because when we're 00:24:17.268 --> 00:24:18.810 align:middle line:84% talking about the emissivity, we said 00:24:18.810 --> 00:24:20.950 align:middle line:90% that it is in some solid angle. 00:24:20.950 --> 00:24:23.100 align:middle line:84% So we could imagine that this emissivity 00:24:23.100 --> 00:24:26.250 align:middle line:84% has an angle with respect to the magnetic field. 00:24:26.250 --> 00:24:29.040 align:middle line:84% And that would be a perfectly reasonable thing to think about. 00:24:29.040 --> 00:24:31.110 align:middle line:84% And indeed, this is where I've started 00:24:31.110 --> 00:24:34.200 align:middle line:84% slipping into a straight line kind of picture 00:24:34.200 --> 00:24:36.960 align:middle line:84% and also an isotropic emission kind of picture. 00:24:36.960 --> 00:24:38.940 align:middle line:84% If you want to do this properly, all 00:24:38.940 --> 00:24:40.680 align:middle line:84% of these things that are scalars start 00:24:40.680 --> 00:24:43.955 align:middle line:84% to become vector, or even worse, some sort of horrific tensor 00:24:43.955 --> 00:24:44.830 align:middle line:90% and things like that. 00:24:44.830 --> 00:24:46.872 align:middle line:84% And you have to account for all of that properly. 00:24:46.872 --> 00:24:49.250 align:middle line:84% So just to try and simplify this, I'm going to do that. 00:24:49.250 --> 00:24:49.690 align:middle line:90% But you're right. 00:24:49.690 --> 00:24:51.315 align:middle line:84% If you're dealing with an emission that 00:24:51.315 --> 00:24:53.980 align:middle line:84% is anisotropic like electron cyclotron emission, 00:24:53.980 --> 00:24:55.390 align:middle line:90% that might be very important. 00:24:55.390 --> 00:24:56.620 align:middle line:90% Yeah. 00:24:56.620 --> 00:24:58.460 align:middle line:90% Another question from Sean. 00:24:58.460 --> 00:25:00.993 align:middle line:84% AUDIENCE: Can you define tau an integral over the path? 00:25:00.993 --> 00:25:01.660 align:middle line:90% JACK HARE: Yeah. 00:25:01.660 --> 00:25:03.800 align:middle line:84% AUDIENCE: I'm a little confused how-- 00:25:03.800 --> 00:25:06.700 align:middle line:84% is tau inside the exponential inside the integral? 00:25:06.700 --> 00:25:09.310 align:middle line:84% Is this a sub integral to be evaluated before you 00:25:09.310 --> 00:25:10.450 align:middle line:90% evaluate the true integral? 00:25:10.450 --> 00:25:24.060 align:middle line:90% 00:25:24.060 --> 00:25:25.560 align:middle line:84% JACK HARE: I think there needs to be 00:25:25.560 --> 00:25:28.710 align:middle line:84% some primes and things like that in order to do this properly. 00:25:28.710 --> 00:25:31.170 align:middle line:84% You're certainly going to evaluate this tau, 00:25:31.170 --> 00:25:35.070 align:middle line:84% and it's going to be evaluated based on whereabouts 00:25:35.070 --> 00:25:37.390 align:middle line:90% in this integral, s, you are. 00:25:37.390 --> 00:25:39.570 align:middle line:84% So as you're incrementing s here, 00:25:39.570 --> 00:25:43.380 align:middle line:84% this tau is going to be evaluated from, for example, 00:25:43.380 --> 00:25:47.670 align:middle line:84% whatever is halfway between s1.5 to s of 2, 00:25:47.670 --> 00:25:51.552 align:middle line:84% where the endpoint of this tau is fixed already. 00:25:51.552 --> 00:25:53.760 align:middle line:84% I don't think I gave a very good explanation of that. 00:25:53.760 --> 00:25:56.302 align:middle line:84% I think probably if I went back and did this more rigorously, 00:25:56.302 --> 00:25:57.690 align:middle line:84% I'd put some more primes on this, 00:25:57.690 --> 00:25:59.398 align:middle line:84% and we'd have integrals within integrals. 00:25:59.398 --> 00:26:02.320 align:middle line:90% 00:26:02.320 --> 00:26:02.820 align:middle line:90% OK. 00:26:02.820 --> 00:26:03.487 align:middle line:90% Other questions? 00:26:03.487 --> 00:26:04.350 align:middle line:90% Yeah, I see Nicola. 00:26:04.350 --> 00:26:07.122 align:middle line:90% 00:26:07.122 --> 00:26:09.750 align:middle line:84% AUDIENCE: The path that the radiation takes 00:26:09.750 --> 00:26:14.390 align:middle line:84% through the plasma is going to depend on the radiation itself, 00:26:14.390 --> 00:26:14.890 align:middle line:90% right? 00:26:14.890 --> 00:26:19.367 align:middle line:84% So it's going to depend on what kind of radiation it is. 00:26:19.367 --> 00:26:20.450 align:middle line:90% JACK HARE: Yeah, could be. 00:26:20.450 --> 00:26:21.640 align:middle line:84% So if you're a different frequency, 00:26:21.640 --> 00:26:22.940 align:middle line:84% you might have a different refractive index, 00:26:22.940 --> 00:26:24.357 align:middle line:84% so you'd have a different bending. 00:26:24.357 --> 00:26:24.910 align:middle line:90% Yes, exactly. 00:26:24.910 --> 00:26:30.040 align:middle line:84% AUDIENCE: But in this equation, the path is predetermined. 00:26:30.040 --> 00:26:36.690 align:middle line:84% So it doesn't-- we're saying that if we know that the path is 00:26:36.690 --> 00:26:39.312 align:middle line:84% this, this is how we would calculate the-- 00:26:39.312 --> 00:26:40.020 align:middle line:90% JACK HARE: Right. 00:26:40.020 --> 00:26:43.770 align:middle line:84% But the path of the radiation, although it 00:26:43.770 --> 00:26:46.860 align:middle line:84% does depend on the frequency, doesn't depend on the intensity, 00:26:46.860 --> 00:26:48.070 align:middle line:90% necessarily. 00:26:48.070 --> 00:26:50.400 align:middle line:84% And so for a certain frequency, you 00:26:50.400 --> 00:26:52.943 align:middle line:84% can trace out a ray through your plasma. 00:26:52.943 --> 00:26:54.360 align:middle line:84% You'll know what trajectory it is. 00:26:54.360 --> 00:26:56.777 align:middle line:84% And then you can go back and solve the radiation transport 00:26:56.777 --> 00:26:59.020 align:middle line:90% equation along that path. 00:26:59.020 --> 00:27:01.740 align:middle line:84% AUDIENCE: And all the emissivity and opacity 00:27:01.740 --> 00:27:05.640 align:middle line:84% figures for that particular frequency 00:27:05.640 --> 00:27:06.720 align:middle line:90% that we are looking at. 00:27:06.720 --> 00:27:07.020 align:middle line:90% JACK HARE: Yes. 00:27:07.020 --> 00:27:08.100 align:middle line:84% At the moment, we're splitting it up 00:27:08.100 --> 00:27:10.800 align:middle line:84% so you could solve this for any different frequencies you want. 00:27:10.800 --> 00:27:14.040 align:middle line:84% And they don't interact in this simple model here. 00:27:14.040 --> 00:27:15.180 align:middle line:90% Yeah. 00:27:15.180 --> 00:27:15.725 align:middle line:90% OK. 00:27:15.725 --> 00:27:17.850 align:middle line:84% AUDIENCE: When you were answering Sean's question-- 00:27:17.850 --> 00:27:20.370 align:middle line:84% I'm a little confused about how the s 00:27:20.370 --> 00:27:23.215 align:middle line:84% 1.5 works into the bounds of the tau integral. 00:27:23.215 --> 00:27:26.100 align:middle line:90% 00:27:26.100 --> 00:27:28.290 align:middle line:84% If you're providing it with some estimate up to, 00:27:28.290 --> 00:27:30.748 align:middle line:84% what's the starting s, or if you're providing it, starting, 00:27:30.748 --> 00:27:32.717 align:middle line:84% that's with your ending s in that tau integral? 00:27:32.717 --> 00:27:35.050 align:middle line:84% AUDIENCE: So in this integral is an indefinite integral. 00:27:35.050 --> 00:27:37.230 align:middle line:84% So we're not providing any limits whatsoever 00:27:37.230 --> 00:27:37.980 align:middle line:90% at all on this. 00:27:37.980 --> 00:27:40.780 align:middle line:84% The limits only come when you start specifying the endpoint. 00:27:40.780 --> 00:27:42.480 align:middle line:84% So for example here, we completely 00:27:42.480 --> 00:27:46.110 align:middle line:84% specified both the start and the end, tau 1 and tau 2. 00:27:46.110 --> 00:27:48.285 align:middle line:84% And that integral is going to look like-- 00:27:48.285 --> 00:27:50.920 align:middle line:90% 00:27:50.920 --> 00:27:56.740 align:middle line:84% this is going to just look like e to the integral of alpha ds 00:27:56.740 --> 00:27:59.470 align:middle line:90% from s1 to s2. 00:27:59.470 --> 00:28:00.830 align:middle line:90% This is the equivalent. 00:28:00.830 --> 00:28:03.130 align:middle line:90% These two are equivalent here. 00:28:03.130 --> 00:28:07.180 align:middle line:84% In this case of tau 2, at the moment what we've got here-- 00:28:07.180 --> 00:28:10.130 align:middle line:84% we don't really have the lower bound. 00:28:10.130 --> 00:28:16.310 align:middle line:84% So we have e to the integral of s to s2 of alpha ds, 00:28:16.310 --> 00:28:19.720 align:middle line:84% like that, where the actual s that we put in the bottom here 00:28:19.720 --> 00:28:21.730 align:middle line:84% depends on whereabouts in evaluating 00:28:21.730 --> 00:28:23.500 align:middle line:90% this integral we actually are. 00:28:23.500 --> 00:28:26.890 align:middle line:84% Maybe that also might answer Sean's question. 00:28:26.890 --> 00:28:27.520 align:middle line:90% I don't know. 00:28:27.520 --> 00:28:28.450 align:middle line:90% OK, good. 00:28:28.450 --> 00:28:32.020 align:middle line:90% Thank you for teasing that out. 00:28:32.020 --> 00:28:32.740 align:middle line:90% OK. 00:28:32.740 --> 00:28:33.880 align:middle line:90% Just any questions online? 00:28:33.880 --> 00:28:39.540 align:middle line:90% 00:28:39.540 --> 00:28:40.800 align:middle line:90% OK, Nicola. 00:28:40.800 --> 00:28:43.410 align:middle line:84% AUDIENCE: Is it possible that radiation 00:28:43.410 --> 00:28:46.080 align:middle line:84% emitted from outside of that line 00:28:46.080 --> 00:28:48.960 align:middle line:84% would end up joining that same rate? 00:28:48.960 --> 00:28:50.375 align:middle line:90% JACK HARE: Yes, it could do. 00:28:50.375 --> 00:28:52.710 align:middle line:84% AUDIENCE: And they're not accounted for? 00:28:52.710 --> 00:28:55.560 align:middle line:90% JACK HARE: In this case, no. 00:28:55.560 --> 00:28:57.745 align:middle line:84% What we're going to do in an awful lot of this 00:28:57.745 --> 00:29:00.120 align:middle line:84% is assume that the radiation is moving in straight lines, 00:29:00.120 --> 00:29:02.070 align:middle line:90% in which case it won't cross. 00:29:02.070 --> 00:29:04.530 align:middle line:84% The reason is because this is already complicated enough 00:29:04.530 --> 00:29:05.580 align:middle line:90% to solve without that. 00:29:05.580 --> 00:29:10.463 align:middle line:84% If you need to solve it with that, then you do. 00:29:10.463 --> 00:29:12.630 align:middle line:84% If you're dealing with something like an X-ray going 00:29:12.630 --> 00:29:16.110 align:middle line:84% through a tokamak plasma, we are so far away 00:29:16.110 --> 00:29:17.880 align:middle line:84% from the critical density with an X-ray 00:29:17.880 --> 00:29:20.392 align:middle line:84% that the refractive index is unity, and it doesn't change. 00:29:20.392 --> 00:29:22.350 align:middle line:84% And so therefore, they do go in straight lines. 00:29:22.350 --> 00:29:23.850 align:middle line:90% But you're absolutely right. 00:29:23.850 --> 00:29:25.830 align:middle line:84% You could imagine all sorts of radiation 00:29:25.830 --> 00:29:27.773 align:middle line:84% coming through, crossing through this point, 00:29:27.773 --> 00:29:29.190 align:middle line:84% going back to an earlier question. 00:29:29.190 --> 00:29:31.440 align:middle line:84% That radiation could heat up a bit of the plasma, 00:29:31.440 --> 00:29:33.360 align:middle line:84% and then that plasma could then emit in a way 00:29:33.360 --> 00:29:35.580 align:middle line:84% that you didn't expect before by changing, obviously. 00:29:35.580 --> 00:29:40.820 align:middle line:84% So radiation transport is extremely complicated, even 00:29:40.820 --> 00:29:41.820 align:middle line:90% without magnetic fields. 00:29:41.820 --> 00:29:43.695 align:middle line:84% And then when you put the magnetic fields in, 00:29:43.695 --> 00:29:45.300 align:middle line:84% the whole thing becomes much worse. 00:29:45.300 --> 00:29:46.207 align:middle line:90% OK. 00:29:46.207 --> 00:29:47.040 align:middle line:90% Any other questions? 00:29:47.040 --> 00:29:49.240 align:middle line:84% I will give some examples of this equation, 00:29:49.240 --> 00:29:51.880 align:middle line:84% so hopefully it will begin to make more sense. 00:29:51.880 --> 00:29:52.380 align:middle line:90% OK. 00:29:52.380 --> 00:29:53.860 align:middle line:90% Let's keep going onto that. 00:29:53.860 --> 00:29:54.360 align:middle line:90% Good. 00:29:54.360 --> 00:29:58.290 align:middle line:84% So let's consider a really simple system where we have 00:29:58.290 --> 00:30:02.110 align:middle line:90% some radiation coming in here. 00:30:02.110 --> 00:30:09.290 align:middle line:84% We have a plasma, which I'm just going to split up 00:30:09.290 --> 00:30:10.878 align:middle line:90% into two regions here. 00:30:10.878 --> 00:30:12.920 align:middle line:84% These regions are actually going to be identical, 00:30:12.920 --> 00:30:15.860 align:middle line:84% but I want to show you how the intensity changes 00:30:15.860 --> 00:30:17.210 align:middle line:90% in each of these regions. 00:30:17.210 --> 00:30:20.840 align:middle line:84% And then we've got some radiation coming back out here. 00:30:20.840 --> 00:30:24.240 align:middle line:90% 00:30:24.240 --> 00:30:34.450 align:middle line:84% So in this plasma, we're going to have an opacity, which 00:30:34.450 --> 00:30:38.230 align:middle line:84% is just simply some amount of opacity 00:30:38.230 --> 00:30:41.290 align:middle line:90% as a single frequency, nu 1. 00:30:41.290 --> 00:30:47.340 align:middle line:84% And we're going to have some emissivity, which is simply 00:30:47.340 --> 00:30:53.430 align:middle line:84% some amount of emission at a different frequency, nu 2, 00:30:53.430 --> 00:30:55.260 align:middle line:90% like this. 00:30:55.260 --> 00:31:03.060 align:middle line:84% And my initial radiation that I'm going to put through here 00:31:03.060 --> 00:31:08.890 align:middle line:84% is just going to have some amount of emission at nu 1 here. 00:31:08.890 --> 00:31:11.250 align:middle line:84% So I've really simplified this, where effectively we 00:31:11.250 --> 00:31:13.583 align:middle line:84% are solving the radiation transport for all frequencies. 00:31:13.583 --> 00:31:15.708 align:middle line:84% It's just that I've come up with this problem where 00:31:15.708 --> 00:31:16.830 align:middle line:90% there are two frequencies. 00:31:16.830 --> 00:31:18.160 align:middle line:90% So this is much easier. 00:31:18.160 --> 00:31:19.910 align:middle line:84% You can come up with something arbitrarily 00:31:19.910 --> 00:31:22.950 align:middle line:84% more complicated than this if you want to. 00:31:22.950 --> 00:31:25.950 align:middle line:84% I just want to point out, you're going to find out soon 00:31:25.950 --> 00:31:30.870 align:middle line:84% that these two, emissivity and opacities, are not compatible. 00:31:30.870 --> 00:31:33.150 align:middle line:84% In fact, there's a really strong thermodynamic link 00:31:33.150 --> 00:31:34.075 align:middle line:90% between the two. 00:31:34.075 --> 00:31:35.200 align:middle line:90% So don't shout at me there. 00:31:35.200 --> 00:31:37.335 align:middle line:84% This is just to make life easier for you right now. 00:31:37.335 --> 00:31:39.210 align:middle line:84% But it turns out that you couldn't physically 00:31:39.210 --> 00:31:40.930 align:middle line:90% have a pattern like this. 00:31:40.930 --> 00:31:42.660 align:middle line:84% We'll talk about that in a little bit. 00:31:42.660 --> 00:31:43.260 align:middle line:90% OK. 00:31:43.260 --> 00:31:46.080 align:middle line:84% So what does this look like as we go through the plasma? 00:31:46.080 --> 00:31:51.290 align:middle line:84% So at this first step here, we're 00:31:51.290 --> 00:31:55.010 align:middle line:84% going to absorb some amounts of the initial radiation, which 00:31:55.010 --> 00:31:56.120 align:middle line:90% was at nu 1. 00:31:56.120 --> 00:31:57.990 align:middle line:84% So our radiation is going to go down. 00:31:57.990 --> 00:32:02.510 align:middle line:84% But we're going to pick up some amount of radiation at nu 2 00:32:02.510 --> 00:32:04.840 align:middle line:90% because the plasma is emitting. 00:32:04.840 --> 00:32:08.900 align:middle line:84% And this next step here, this process continues. 00:32:08.900 --> 00:32:12.140 align:middle line:90% Nu 1 goes down, nu 2 goes up. 00:32:12.140 --> 00:32:16.070 align:middle line:84% And then finally, as we exit the plasma, 00:32:16.070 --> 00:32:20.780 align:middle line:84% we might end up with a system where nu 1 is very, very small 00:32:20.780 --> 00:32:23.148 align:middle line:90% and nu 2 is very, very large. 00:32:23.148 --> 00:32:25.190 align:middle line:84% So effectively, as we've gone through the plasma, 00:32:25.190 --> 00:32:28.520 align:middle line:84% we have been absorbing that nu 1, the size of that line 00:32:28.520 --> 00:32:31.910 align:middle line:84% has gone down, and we've been emitting that nu 2. 00:32:31.910 --> 00:32:36.130 align:middle line:84% The size of that line has gone up here. 00:32:36.130 --> 00:32:37.590 align:middle line:90% So that's a cartoon example. 00:32:37.590 --> 00:32:39.960 align:middle line:84% It turns out that for this approximation, 00:32:39.960 --> 00:32:43.020 align:middle line:84% that the plasma is homogeneous, then 00:32:43.020 --> 00:32:45.660 align:middle line:84% we can solve this equation analytically as well. 00:32:45.660 --> 00:32:57.537 align:middle line:90% 00:32:57.537 --> 00:32:58.370 align:middle line:90% AUDIENCE: Professor? 00:32:58.370 --> 00:32:59.070 align:middle line:90% JACK HARE: Yeah. 00:32:59.070 --> 00:33:03.420 align:middle line:84% AUDIENCE: So just to clarify, the absorption 00:33:03.420 --> 00:33:06.480 align:middle line:84% is throughout both sections of the plasma. 00:33:06.480 --> 00:33:08.218 align:middle line:90% JACK HARE: Yeah. 00:33:08.218 --> 00:33:10.260 align:middle line:84% I split it into two sections so I could just draw 00:33:10.260 --> 00:33:11.580 align:middle line:90% the intensity in two places. 00:33:11.580 --> 00:33:14.700 align:middle line:84% But the plasma has uniform properties throughout. 00:33:14.700 --> 00:33:17.358 align:middle line:90% 00:33:17.358 --> 00:33:21.030 align:middle line:84% This applies to both sections, and this applies 00:33:21.030 --> 00:33:23.670 align:middle line:90% to both sections like that. 00:33:23.670 --> 00:33:25.410 align:middle line:90% Yeah, good question. 00:33:25.410 --> 00:33:26.587 align:middle line:90% AUDIENCE: Thank you. 00:33:26.587 --> 00:33:27.170 align:middle line:90% JACK HARE: OK. 00:33:27.170 --> 00:33:30.440 align:middle line:90% Any other questions? 00:33:30.440 --> 00:33:30.940 align:middle line:90% OK. 00:33:30.940 --> 00:33:38.075 align:middle line:90% 00:33:38.075 --> 00:33:39.700 align:middle line:84% At these times, I wish I'd more boards. 00:33:39.700 --> 00:33:46.303 align:middle line:90% 00:33:46.303 --> 00:33:47.470 align:middle line:90% I really want that equation. 00:33:47.470 --> 00:34:06.430 align:middle line:90% 00:34:06.430 --> 00:34:07.140 align:middle line:90% Yeah? 00:34:07.140 --> 00:34:11.550 align:middle line:84% AUDIENCE: Can we assume anything about energy conservation 00:34:11.550 --> 00:34:12.900 align:middle line:90% in this equation? 00:34:12.900 --> 00:34:15.120 align:middle line:84% Can your additions have more energy 00:34:15.120 --> 00:34:16.650 align:middle line:84% out than what caused the emission? 00:34:16.650 --> 00:34:18.810 align:middle line:90% Or I guess-- 00:34:18.810 --> 00:34:21.090 align:middle line:84% JACK HARE: So the question was, are we conserving 00:34:21.090 --> 00:34:22.090 align:middle line:90% energy in this equation? 00:34:22.090 --> 00:34:24.120 align:middle line:84% So explicitly, we are not conserving 00:34:24.120 --> 00:34:25.230 align:middle line:90% energy in this equation. 00:34:25.230 --> 00:34:28.469 align:middle line:84% The plasma is the source and the sink of the energy. 00:34:28.469 --> 00:34:31.409 align:middle line:84% Any radiation we lose is heating the plasma, 00:34:31.409 --> 00:34:33.900 align:middle line:84% any radiation we gain goes to cooling the plasma. 00:34:33.900 --> 00:34:36.150 align:middle line:84% That's why you would need to couple an energy equation 00:34:36.150 --> 00:34:37.558 align:middle line:90% into this to do it properly. 00:34:37.558 --> 00:34:38.100 align:middle line:90% AUDIENCE: OK. 00:34:38.100 --> 00:34:38.767 align:middle line:90% JACK HARE: Yeah. 00:34:38.767 --> 00:34:45.076 align:middle line:90% 00:34:45.076 --> 00:34:46.590 align:middle line:90% I'll put that equation there. 00:34:46.590 --> 00:34:49.210 align:middle line:90% 00:34:49.210 --> 00:34:55.650 align:middle line:84% So if we have a system, which is now homogeneous so 00:34:55.650 --> 00:35:09.480 align:middle line:84% that along our path, s, this quantity, J upon alpha 00:35:09.480 --> 00:35:11.260 align:middle line:90% doesn't change. 00:35:11.260 --> 00:35:14.910 align:middle line:84% So again, this is just the homogeneous condition. 00:35:14.910 --> 00:35:19.360 align:middle line:90% 00:35:19.360 --> 00:35:21.550 align:middle line:84% We can solve this analytically, and we 00:35:21.550 --> 00:35:27.140 align:middle line:84% get that the intensity at point s2 00:35:27.140 --> 00:35:32.150 align:middle line:84% is simply equal to the intensity of point s1, 00:35:32.150 --> 00:35:34.940 align:middle line:84% attenuated by a factor e to the minus 00:35:34.940 --> 00:35:42.780 align:middle line:84% Cal 2, 1, which is, as we said before, the integral from l 2, 1 00:35:42.780 --> 00:35:48.120 align:middle line:84% is equal to the integral from s1 to s2 of alpha ds. 00:35:48.120 --> 00:35:52.690 align:middle line:84% So this is just an exponential damping factor on the intensity. 00:35:52.690 --> 00:35:55.770 align:middle line:84% This is what you would expect for some sort 00:35:55.770 --> 00:35:57.570 align:middle line:84% of constant opacity through your system. 00:35:57.570 --> 00:36:00.930 align:middle line:84% The further you go, the more that initial signal 00:36:00.930 --> 00:36:02.100 align:middle line:90% is going to be damped. 00:36:02.100 --> 00:36:04.350 align:middle line:84% And then we're also going to have a term, 00:36:04.350 --> 00:36:14.410 align:middle line:84% J upon alpha 1 minus e to the minus tau 2, 1-- 00:36:14.410 --> 00:36:15.490 align:middle line:90% same again here. 00:36:15.490 --> 00:36:18.105 align:middle line:90% 00:36:18.105 --> 00:36:18.605 align:middle line:90% OK. 00:36:18.605 --> 00:36:21.613 align:middle line:90% 00:36:21.613 --> 00:36:23.030 align:middle line:84% And when we look at this equation, 00:36:23.030 --> 00:36:25.270 align:middle line:84% we can see that there's a strong dependence on tau. 00:36:25.270 --> 00:36:27.970 align:middle line:84% And so we want to identify two limits which 00:36:27.970 --> 00:36:33.130 align:middle line:84% have names, one of which is this tau 2, 1 the optical depth 00:36:33.130 --> 00:36:35.920 align:middle line:90% is much, much less than 1. 00:36:35.920 --> 00:36:40.660 align:middle line:84% And the other one is tau 2, 1 is much, much more than 1. 00:36:40.660 --> 00:36:51.600 align:middle line:84% And these are called optically thin and optically thick. 00:36:51.600 --> 00:36:55.030 align:middle line:90% 00:36:55.030 --> 00:36:57.880 align:middle line:84% In these limits, this equation reduces 00:36:57.880 --> 00:37:16.940 align:middle line:84% to either Is2 is equal to Is1 or Is2 is equal to J on alpha here. 00:37:16.940 --> 00:37:20.270 align:middle line:84% Optically thin corresponds to the radiation 00:37:20.270 --> 00:37:22.520 align:middle line:84% streaming through without being absorbed. 00:37:22.520 --> 00:37:25.940 align:middle line:84% Optically thick corresponds to the radiation being 00:37:25.940 --> 00:37:27.700 align:middle line:90% very, very strongly absorbed. 00:37:27.700 --> 00:37:30.800 align:middle line:84% So the optically thick case, we have no information 00:37:30.800 --> 00:37:33.470 align:middle line:90% about the initial intensity. 00:37:33.470 --> 00:37:36.203 align:middle line:84% The optically thin case, we only have information 00:37:36.203 --> 00:37:37.370 align:middle line:90% about the initial intensity. 00:37:37.370 --> 00:37:39.890 align:middle line:84% We don't have any added emission here. 00:37:39.890 --> 00:37:49.620 align:middle line:84% So people may also call these transparent and opaque. 00:37:49.620 --> 00:37:52.910 align:middle line:90% 00:37:52.910 --> 00:37:56.250 align:middle line:84% Now, remember, this is a strong function of frequency. 00:37:56.250 --> 00:37:59.600 align:middle line:84% And so you can be optically thin or transparent to X-rays 00:37:59.600 --> 00:38:02.990 align:middle line:84% and optically thick or opaque to the first and second harmonics 00:38:02.990 --> 00:38:05.660 align:middle line:84% of the electron cyclotron emission and tokamak. 00:38:05.660 --> 00:38:07.520 align:middle line:90% And these are not contradictory. 00:38:07.520 --> 00:38:09.410 align:middle line:84% We have solved these equations separately 00:38:09.410 --> 00:38:11.570 align:middle line:84% for every different frequency that we're 00:38:11.570 --> 00:38:14.710 align:middle line:90% interested in in our system. 00:38:14.710 --> 00:38:17.740 align:middle line:84% Now, the interesting thing about this optically thick case 00:38:17.740 --> 00:38:24.760 align:middle line:84% is that this corresponds to a black body, which 00:38:24.760 --> 00:38:26.920 align:middle line:84% is a thermodynamic system that I'm 00:38:26.920 --> 00:38:30.020 align:middle line:84% sure you've studied in which, again, 00:38:30.020 --> 00:38:34.600 align:middle line:84% the radiation is in equilibrium with the temperature 00:38:34.600 --> 00:38:35.680 align:middle line:90% inside the plasma-- 00:38:35.680 --> 00:38:37.780 align:middle line:84% perfect thermodynamic equilibrium. 00:38:37.780 --> 00:38:42.190 align:middle line:84% And for a black body, we know a second expression 00:38:42.190 --> 00:38:43.480 align:middle line:90% for the intensity here. 00:38:43.480 --> 00:38:47.055 align:middle line:90% For a black body, I black body-- 00:38:47.055 --> 00:38:49.720 align:middle line:90% 00:38:49.720 --> 00:38:52.110 align:middle line:90% or, yeah. 00:38:52.110 --> 00:38:55.560 align:middle line:84% I as a function of frequency is equal to a function that's 00:38:55.560 --> 00:38:59.280 align:middle line:84% often called B. I'm guessing the black body. 00:38:59.280 --> 00:39:05.010 align:middle line:84% And this is equal to u squared upon C squared, Planck's 00:39:05.010 --> 00:39:16.840 align:middle line:84% constant Hu over the exponential of h nu upon T minus 1. 00:39:16.840 --> 00:39:22.270 align:middle line:84% Or in many cases, it suffices to use the approximation 00:39:22.270 --> 00:39:25.150 align:middle line:90% nu squared t upon C squared. 00:39:25.150 --> 00:39:30.460 align:middle line:84% And this is valid for h nu much, much less than t. 00:39:30.460 --> 00:39:32.100 align:middle line:90% And this is the classical limit. 00:39:32.100 --> 00:39:37.250 align:middle line:84% So this is the limit we had in classical physics for some time. 00:39:37.250 --> 00:39:38.720 align:middle line:84% And it was the violation of this, 00:39:38.720 --> 00:39:40.970 align:middle line:84% the ultraviolet catastrophe implied by the fact 00:39:40.970 --> 00:39:42.650 align:middle line:84% that, as nu keeps getting larger, 00:39:42.650 --> 00:39:44.840 align:middle line:84% you keep emitting more radiation. 00:39:44.840 --> 00:39:46.610 align:middle line:84% That catastrophe is well known too. 00:39:46.610 --> 00:39:49.370 align:middle line:84% In some part, the development of quantum mechanics-- and this 00:39:49.370 --> 00:39:51.480 align:middle line:84% quantum mechanically correct correction. 00:39:51.480 --> 00:39:53.810 align:middle line:84% It just turns out that this is pretty 00:39:53.810 --> 00:39:55.340 align:middle line:90% good with [INAUDIBLE] law. 00:39:55.340 --> 00:39:58.490 align:middle line:84% It's a pretty good approximation for low frequencies, 00:39:58.490 --> 00:40:01.551 align:middle line:84% where the frequency is low compared to the temperature. 00:40:01.551 --> 00:40:04.260 align:middle line:90% 00:40:04.260 --> 00:40:06.240 align:middle line:84% What is interesting, then, is we now 00:40:06.240 --> 00:40:10.420 align:middle line:84% have two expressions for the optically thick case. 00:40:10.420 --> 00:40:12.810 align:middle line:84% We have an expression for the black body radiation, 00:40:12.810 --> 00:40:15.540 align:middle line:84% and we have an expression to with J upon alpha. 00:40:15.540 --> 00:40:18.490 align:middle line:84% And these expressions must be equal to each other. 00:40:18.490 --> 00:40:22.020 align:middle line:84% And so this means that the emissivity of the opacity 00:40:22.020 --> 00:40:24.290 align:middle line:90% is equal to-- 00:40:24.290 --> 00:40:26.210 align:middle line:84% I'm just going to use the classical limit-- 00:40:26.210 --> 00:40:30.000 align:middle line:84% nu squared upon C squared times the temperature. 00:40:30.000 --> 00:40:32.300 align:middle line:84% And this is called Kirchhoff's law, 00:40:32.300 --> 00:40:34.998 align:middle line:84% along with all the other things that people cotransport. 00:40:34.998 --> 00:40:44.610 align:middle line:90% 00:40:44.610 --> 00:40:46.020 align:middle line:84% Kirchhoff's law is very profound. 00:40:46.020 --> 00:40:50.860 align:middle line:90% 00:40:50.860 --> 00:40:54.940 align:middle line:84% Although we derived it in the limits of an optically thick 00:40:54.940 --> 00:40:59.470 align:middle line:84% body, it still has to apply into an optically thin body. 00:40:59.470 --> 00:41:04.240 align:middle line:84% And effectively, what this says is if you know the emissivity, 00:41:04.240 --> 00:41:07.480 align:middle line:84% then you automatically know the opacity. 00:41:07.480 --> 00:41:10.000 align:middle line:84% These two quantities are intimately related. 00:41:10.000 --> 00:41:14.170 align:middle line:84% This is why I said that this wasn't 00:41:14.170 --> 00:41:17.830 align:middle line:84% actually valid because this does not obey Kirchhoff's law. 00:41:17.830 --> 00:41:20.602 align:middle line:90% 00:41:20.602 --> 00:41:22.060 align:middle line:84% So the nice thing about that is you 00:41:22.060 --> 00:41:26.080 align:middle line:84% only have to calculate the emissivity for a system, 00:41:26.080 --> 00:41:29.150 align:middle line:84% and then you automatically get the opacity. 00:41:29.150 --> 00:41:36.440 align:middle line:84% So if you calculate J, you get alpha for free. 00:41:36.440 --> 00:41:39.280 align:middle line:90% That's rather convenient. 00:41:39.280 --> 00:41:41.950 align:middle line:84% That's nice, but maybe the more profound thing 00:41:41.950 --> 00:41:44.660 align:middle line:84% about Kirchhoff's law is the fact 00:41:44.660 --> 00:41:49.630 align:middle line:84% that where you have high emissivity, 00:41:49.630 --> 00:41:55.130 align:middle line:90% we also have high opacity. 00:41:55.130 --> 00:41:59.360 align:middle line:84% What this means is that regions which strongly emit also 00:41:59.360 --> 00:42:01.190 align:middle line:90% strongly absorb. 00:42:01.190 --> 00:42:06.270 align:middle line:84% And that also means that regions which weakly emit do not-- 00:42:06.270 --> 00:42:06.770 align:middle line:90% sorry. 00:42:06.770 --> 00:42:10.460 align:middle line:84% Regions which weakly absorb do not strongly emit. 00:42:10.460 --> 00:42:11.510 align:middle line:90% They also weakly emit. 00:42:11.510 --> 00:42:12.840 align:middle line:90% It's confusing. 00:42:12.840 --> 00:42:14.150 align:middle line:90% This is where this comes in. 00:42:14.150 --> 00:42:15.660 align:middle line:84% Some of you been staring at this being like, 00:42:15.660 --> 00:42:16.670 align:middle line:90% wait, is this obvious? 00:42:16.670 --> 00:42:19.730 align:middle line:84% The point is if you have very, very low absorption, 00:42:19.730 --> 00:42:21.950 align:middle line:84% you are also going to have very, very low emission. 00:42:21.950 --> 00:42:24.740 align:middle line:84% And so you're not going to see significant self-emission 00:42:24.740 --> 00:42:28.410 align:middle line:84% from the plasma being added to your initial beam of radiation. 00:42:28.410 --> 00:42:30.080 align:middle line:84% And so you might think, oh, perhaps 00:42:30.080 --> 00:42:32.533 align:middle line:84% I would get the initial beam plus some extra term. 00:42:32.533 --> 00:42:33.950 align:middle line:84% Well, you would get an extra term, 00:42:33.950 --> 00:42:36.380 align:middle line:84% but that term will be proportional to tau to 1, 00:42:36.380 --> 00:42:38.990 align:middle line:84% and we've already said the brightness is much less than 1. 00:42:38.990 --> 00:42:41.000 align:middle line:84% So that extra radiation wouldn't be 00:42:41.000 --> 00:42:43.400 align:middle line:84% very significant compared to the initial beam 00:42:43.400 --> 00:42:45.230 align:middle line:90% that you put through. 00:42:45.230 --> 00:42:46.400 align:middle line:90% OK. 00:42:46.400 --> 00:42:48.430 align:middle line:90% Questions? 00:42:48.430 --> 00:42:50.962 align:middle line:84% AUDIENCE: Does the built-in thermodynamic equilibrium 00:42:50.962 --> 00:42:52.670 align:middle line:84% assumption or something-- like you always 00:42:52.670 --> 00:42:54.830 align:middle line:84% have the same temperature everywhere along your path 00:42:54.830 --> 00:42:55.788 align:middle line:90% or something like that? 00:42:55.788 --> 00:42:56.570 align:middle line:90% Or is It just-- 00:42:56.570 --> 00:42:57.153 align:middle line:90% JACK HARE: No. 00:42:57.153 --> 00:42:59.000 align:middle line:84% I think the power of Kirchhoff's law 00:42:59.000 --> 00:43:01.325 align:middle line:84% is although you derive it using all these black body 00:43:01.325 --> 00:43:03.530 align:middle line:84% assumptions, it then works everywhere else as well. 00:43:03.530 --> 00:43:08.300 align:middle line:84% Because these two have to obey this principle for a black body, 00:43:08.300 --> 00:43:10.070 align:middle line:84% but it means it also pins it even when 00:43:10.070 --> 00:43:15.350 align:middle line:84% you're not in non-local thermodynamic equilibrium. 00:43:15.350 --> 00:43:15.950 align:middle line:90% Yeah. 00:43:15.950 --> 00:43:17.825 align:middle line:84% So I think that's why it's a powerful result. 00:43:17.825 --> 00:43:20.690 align:middle line:90% It doesn't just apply to bodies. 00:43:20.690 --> 00:43:21.860 align:middle line:90% Yeah, another question. 00:43:21.860 --> 00:43:24.800 align:middle line:84% AUDIENCE: You said that you can calculate the emissivity 00:43:24.800 --> 00:43:26.540 align:middle line:90% and get absorption. 00:43:26.540 --> 00:43:29.580 align:middle line:84% That also requires you knowing the temperature, though, right? 00:43:29.580 --> 00:43:31.260 align:middle line:84% JACK HARE: Well, if you're going to calculate the emissivity, 00:43:31.260 --> 00:43:32.490 align:middle line:84% you need to know the temperature of your plasma. 00:43:32.490 --> 00:43:34.020 align:middle line:84% We haven't got there yet, but the emissivity 00:43:34.020 --> 00:43:35.730 align:middle line:84% is always going to be a very strong function 00:43:35.730 --> 00:43:36.550 align:middle line:90% of the temperature. 00:43:36.550 --> 00:43:36.870 align:middle line:90% AUDIENCE: OK. 00:43:36.870 --> 00:43:37.537 align:middle line:90% JACK HARE: Yeah. 00:43:37.537 --> 00:43:39.270 align:middle line:84% So to calculate the emissivity, you're 00:43:39.270 --> 00:43:40.860 align:middle line:84% going to need to know the density and the temperature 00:43:40.860 --> 00:43:41.890 align:middle line:90% of your plasma. 00:43:41.890 --> 00:43:44.370 align:middle line:84% And if you want to do it anisotropically, 00:43:44.370 --> 00:43:46.710 align:middle line:84% you'll need to know magnetic field direction, all sorts 00:43:46.710 --> 00:43:47.710 align:middle line:90% of fun things like that. 00:43:47.710 --> 00:43:49.950 align:middle line:84% But at the very basic, we're going 00:43:49.950 --> 00:43:51.720 align:middle line:84% to do Bremsstrahlung in a little bit-- 00:43:51.720 --> 00:43:54.975 align:middle line:84% Bremsstrahlung, density squared, temperature to the half. 00:43:54.975 --> 00:43:56.850 align:middle line:84% So you need to know those two things in order 00:43:56.850 --> 00:43:59.940 align:middle line:84% to be able to get the Bremsstrahlung out. 00:43:59.940 --> 00:44:02.608 align:middle line:84% AUDIENCE: So we assume that you divide ds of J 00:44:02.608 --> 00:44:03.900 align:middle line:90% over alpha on this [INAUDIBLE]. 00:44:03.900 --> 00:44:07.095 align:middle line:84% Are we-- do we also have to assume alpha is constant? 00:44:07.095 --> 00:44:13.437 align:middle line:90% 00:44:13.437 --> 00:44:15.020 align:middle line:84% JACK HARE: Maybe this is more powerful 00:44:15.020 --> 00:44:16.603 align:middle line:84% than just simply a homogeneous plasma. 00:44:16.603 --> 00:44:19.820 align:middle line:84% Yeah, it looks like they both go up and down together. 00:44:19.820 --> 00:44:21.290 align:middle line:90% Yeah. 00:44:21.290 --> 00:44:23.720 align:middle line:84% I don't think we need to assume that they're actually 00:44:23.720 --> 00:44:26.540 align:middle line:84% homogeneous in order to get this result. So maybe you're right. 00:44:26.540 --> 00:44:27.740 align:middle line:90% Yeah, maybe. 00:44:27.740 --> 00:44:30.500 align:middle line:84% As long as J and alpha increase and decrease together. 00:44:30.500 --> 00:44:33.465 align:middle line:84% I'm trying to work out, is that effectively a statement 00:44:33.465 --> 00:44:34.840 align:middle line:84% that the temperature is constant? 00:44:34.840 --> 00:44:36.840 align:middle line:90% I think it is. 00:44:36.840 --> 00:44:40.140 align:middle line:84% So the temperature is constant because of this result. 00:44:40.140 --> 00:44:43.590 align:middle line:84% But I think that means the density could change. 00:44:43.590 --> 00:44:45.945 align:middle line:84% So it's homogeneous in temperature only. 00:44:45.945 --> 00:44:53.460 align:middle line:90% 00:44:53.460 --> 00:44:53.960 align:middle line:90% OK. 00:44:53.960 --> 00:44:54.835 align:middle line:90% Any questions online? 00:44:54.835 --> 00:45:00.640 align:middle line:90% 00:45:00.640 --> 00:45:01.600 align:middle line:90% Yeah? 00:45:01.600 --> 00:45:04.550 align:middle line:84% AUDIENCE: Maybe I just missed this in the second [INAUDIBLE] 00:45:04.550 --> 00:45:05.050 align:middle line:90% question. 00:45:05.050 --> 00:45:10.810 align:middle line:84% But this differential equation in gamma alpha equals zero. 00:45:10.810 --> 00:45:16.690 align:middle line:84% Why do we need Kirchhoff's law to conclude that IJ and I alpha 00:45:16.690 --> 00:45:19.512 align:middle line:84% are the same, that they increase and decrease in lockstep? 00:45:19.512 --> 00:45:21.220 align:middle line:84% And what that differential equation says. 00:45:21.220 --> 00:45:25.650 align:middle line:90% 00:45:25.650 --> 00:45:27.090 align:middle line:84% JACK HARE: This is an assumption. 00:45:27.090 --> 00:45:28.860 align:middle line:84% This equation here is an assumption 00:45:28.860 --> 00:45:33.490 align:middle line:84% that we've made in order to derive this equation, which 00:45:33.490 --> 00:45:35.740 align:middle line:84% is particularly simple to understand the optically 00:45:35.740 --> 00:45:39.880 align:middle line:84% thin and optically thick limits because tau just 00:45:39.880 --> 00:45:43.960 align:middle line:84% linearly increases with distance rather than changing rapidly 00:45:43.960 --> 00:45:45.550 align:middle line:90% over different bits of plasma. 00:45:45.550 --> 00:45:49.600 align:middle line:84% Once we've done that, we then find out 00:45:49.600 --> 00:45:51.340 align:middle line:84% for the optical case that this is true. 00:45:51.340 --> 00:45:52.362 align:middle line:90% And then, I don't know. 00:45:52.362 --> 00:45:54.070 align:middle line:84% I don't think it's obvious from the start 00:45:54.070 --> 00:45:56.380 align:middle line:84% that we were going to end up with this result. 00:45:56.380 --> 00:45:58.630 align:middle line:84% And this result is still true for inhomogeneous 00:45:58.630 --> 00:45:59.830 align:middle line:90% plasmas as well. 00:45:59.830 --> 00:46:01.450 align:middle line:84% It just happens to be true for-- we've 00:46:01.450 --> 00:46:04.040 align:middle line:84% proved it in the case of a homogeneous plasma. 00:46:04.040 --> 00:46:04.540 align:middle line:90% Yeah. 00:46:04.540 --> 00:46:05.800 align:middle line:90% I think I see your question. 00:46:05.800 --> 00:46:10.270 align:middle line:84% But I don't think we've baked in our result in our assumptions, 00:46:10.270 --> 00:46:11.800 align:middle line:90% if that's what you're asking. 00:46:11.800 --> 00:46:12.940 align:middle line:90% AUDIENCE: Yeah. 00:46:12.940 --> 00:46:15.190 align:middle line:84% JACK HARE: I don't think so, but yeah, OK. 00:46:15.190 --> 00:46:16.127 align:middle line:90% OK, other questions? 00:46:16.127 --> 00:46:17.710 align:middle line:84% Anything online, anything in the room? 00:46:17.710 --> 00:46:22.820 align:middle line:90% 00:46:22.820 --> 00:46:23.320 align:middle line:90% All right. 00:46:23.320 --> 00:46:25.300 align:middle line:84% So now what we probably want to do 00:46:25.300 --> 00:46:34.250 align:middle line:84% is calculate J for a variety of different cases. 00:46:34.250 --> 00:46:38.560 align:middle line:84% So we could calculate J, the emissivity from free electrons. 00:46:38.560 --> 00:46:42.070 align:middle line:84% What sort of radiation do we get from free electrons? 00:46:42.070 --> 00:46:45.395 align:middle line:90% 00:46:45.395 --> 00:46:46.830 align:middle line:90% Brems, OK. 00:46:46.830 --> 00:46:49.680 align:middle line:90% 00:46:49.680 --> 00:46:51.993 align:middle line:90% Anything else? 00:46:51.993 --> 00:46:52.910 align:middle line:90% AUDIENCE: Synchrotron? 00:46:52.910 --> 00:46:53.930 align:middle line:90% JACK HARE: Synchrotron? 00:46:53.930 --> 00:46:56.780 align:middle line:84% You're not going to do synchrotron. 00:46:56.780 --> 00:46:59.940 align:middle line:90% AUDIENCE: Thomson scattering? 00:46:59.940 --> 00:47:01.860 align:middle line:90% JACK HARE: Thomson scattering. 00:47:01.860 --> 00:47:03.420 align:middle line:90% AUDIENCE: Oh, no, Compton. 00:47:03.420 --> 00:47:05.190 align:middle line:90% JACK HARE: Compton scattering? 00:47:05.190 --> 00:47:09.210 align:middle line:84% Compton scattering is just relativistic Thomson scattering. 00:47:09.210 --> 00:47:12.510 align:middle line:84% So we're not going to consider that because that 00:47:12.510 --> 00:47:17.550 align:middle line:84% is scattering of radiation from external radiation 00:47:17.550 --> 00:47:19.200 align:middle line:90% by the plasma. 00:47:19.200 --> 00:47:21.990 align:middle line:84% So we're talking about radiation being produced by the plasma 00:47:21.990 --> 00:47:22.490 align:middle line:90% here. 00:47:22.490 --> 00:47:23.830 align:middle line:90% But this is a good point. 00:47:23.830 --> 00:47:26.490 align:middle line:84% We will talk about scattering extensively later on, 00:47:26.490 --> 00:47:29.580 align:middle line:90% but not in this case. 00:47:29.580 --> 00:47:31.040 align:middle line:90% Anything else? 00:47:31.040 --> 00:47:32.040 align:middle line:90% AUDIENCE: Recombination. 00:47:32.040 --> 00:47:35.420 align:middle line:90% JACK HARE: Recombination, yeah. 00:47:35.420 --> 00:47:37.650 align:middle line:84% It starts with three electronic beats, so it counts. 00:47:37.650 --> 00:47:40.812 align:middle line:90% 00:47:40.812 --> 00:47:41.395 align:middle line:90% Anything else? 00:47:41.395 --> 00:47:45.400 align:middle line:90% 00:47:45.400 --> 00:47:46.570 align:middle line:90% AUDIENCE: Lamar radiation? 00:47:46.570 --> 00:47:47.768 align:middle line:90% Or is that just-- 00:47:47.768 --> 00:47:49.810 align:middle line:84% JACK HARE: Yeah, that's just cyclotron radiation. 00:47:49.810 --> 00:47:53.097 align:middle line:84% To be honest, if that's the same as synchrotron, 00:47:53.097 --> 00:47:53.930 align:middle line:90% that explains a lot. 00:47:53.930 --> 00:47:55.070 align:middle line:84% So I've always been confused about that. 00:47:55.070 --> 00:47:56.050 align:middle line:84% But we're definitely doing cyclotron. 00:47:56.050 --> 00:47:57.640 align:middle line:90% Is that the same as synchrotron? 00:47:57.640 --> 00:47:58.990 align:middle line:84% AUDIENCE: I think they're slightly different. 00:47:58.990 --> 00:48:00.865 align:middle line:84% JACK HARE: Isn't one the relativistic version 00:48:00.865 --> 00:48:01.630 align:middle line:90% of the other? 00:48:01.630 --> 00:48:03.640 align:middle line:84% AUDIENCE: They're definitely related. 00:48:03.640 --> 00:48:04.705 align:middle line:90% JACK HARE: Anyone? 00:48:04.705 --> 00:48:06.370 align:middle line:84% AUDIENCE: With any three positrons? 00:48:06.370 --> 00:48:07.180 align:middle line:90% Do you think that? 00:48:07.180 --> 00:48:09.935 align:middle line:84% JACK HARE: What would we get if we had three positrons? 00:48:09.935 --> 00:48:10.810 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:48:10.810 --> 00:48:11.643 align:middle line:90% JACK HARE: Yeah, OK. 00:48:11.643 --> 00:48:13.840 align:middle line:84% So we're not going to do that, but that'll be great. 00:48:13.840 --> 00:48:14.740 align:middle line:90% No, no, no. 00:48:14.740 --> 00:48:16.028 align:middle line:90% That would be neat. 00:48:16.028 --> 00:48:17.320 align:middle line:90% We're not going to derive that. 00:48:17.320 --> 00:48:20.890 align:middle line:84% It's a little bit terminal to the [INAUDIBLE], isn't it? 00:48:20.890 --> 00:48:22.420 align:middle line:90% Yeah. 00:48:22.420 --> 00:48:24.460 align:middle line:90% That's all I've got here. 00:48:24.460 --> 00:48:27.758 align:middle line:84% And then we'll also do this for bound electrons. 00:48:27.758 --> 00:48:29.300 align:middle line:84% And really, for bound electrons, this 00:48:29.300 --> 00:48:31.445 align:middle line:84% is the whole zoo of light emission. 00:48:31.445 --> 00:48:37.610 align:middle line:90% 00:48:37.610 --> 00:48:39.510 align:middle line:84% I think I'd be very inconsistent whether I've 00:48:39.510 --> 00:48:41.860 align:middle line:84% put two m's or one m in emission throughout this lecture. 00:48:41.860 --> 00:48:43.527 align:middle line:84% So you can tell how good my spelling is. 00:48:43.527 --> 00:48:44.340 align:middle line:90% OK. 00:48:44.340 --> 00:48:48.070 align:middle line:84% Now, in a previous version of this course, 00:48:48.070 --> 00:48:50.507 align:middle line:84% we went through all of these without talking 00:48:50.507 --> 00:48:52.590 align:middle line:84% at any point about what you'd actually do with it. 00:48:52.590 --> 00:48:55.327 align:middle line:84% And I found this very difficult to teach. 00:48:55.327 --> 00:48:56.910 align:middle line:84% So what we're going to do is I'm going 00:48:56.910 --> 00:48:59.550 align:middle line:84% to cover each of these topics in turn, 00:48:59.550 --> 00:49:03.150 align:middle line:84% and then interject with here's how a bolometer works or here's 00:49:03.150 --> 00:49:06.600 align:middle line:84% how a pinhole camera works or here's how a spectrometer works. 00:49:06.600 --> 00:49:09.330 align:middle line:84% But the point is that all of the techniques I'll discuss 00:49:09.330 --> 00:49:11.915 align:middle line:84% are applicable to all of these different types of radiation. 00:49:11.915 --> 00:49:13.290 align:middle line:84% I'm just trying to break it up so 00:49:13.290 --> 00:49:15.000 align:middle line:84% it's not learn every type of radiation 00:49:15.000 --> 00:49:18.090 align:middle line:84% and then learn every type of diagnosing radiation. 00:49:18.090 --> 00:49:23.870 align:middle line:84% But we will start, as Hutchinson does, with cyclotron. 00:49:23.870 --> 00:49:28.160 align:middle line:90% 00:49:28.160 --> 00:49:32.090 align:middle line:84% But before that, I have a little question for you about something 00:49:32.090 --> 00:49:34.550 align:middle line:84% that I thought about in the shower the other day, 00:49:34.550 --> 00:49:37.920 align:middle line:84% and thought, I wonder if this is obvious or not. 00:49:37.920 --> 00:49:40.010 align:middle line:84% And you can tell me whether or not it's obvious. 00:49:40.010 --> 00:49:50.970 align:middle line:84% And my question for you is, we've 00:49:50.970 --> 00:49:54.060 align:middle line:84% encountered two places in which radiation 00:49:54.060 --> 00:49:57.900 align:middle line:90% ceases to go through a plasma. 00:49:57.900 --> 00:50:02.290 align:middle line:84% We have found a limit where the optical depth is much, 00:50:02.290 --> 00:50:04.870 align:middle line:90% much greater than 1. 00:50:04.870 --> 00:50:06.190 align:middle line:90% So that's particularly thick. 00:50:06.190 --> 00:50:10.860 align:middle line:90% 00:50:10.860 --> 00:50:13.320 align:middle line:84% And we've also talked about a cut-off. 00:50:13.320 --> 00:50:16.010 align:middle line:84% That's where the refractive index is less than 0. 00:50:16.010 --> 00:50:20.050 align:middle line:90% 00:50:20.050 --> 00:50:22.180 align:middle line:84% In the context of a plasma, remember, we 00:50:22.180 --> 00:50:27.400 align:middle line:84% might have n is 1 minus ne over 2 and C. 00:50:27.400 --> 00:50:32.630 align:middle line:84% And so when we get close to n critical, the wave is cut off. 00:50:32.630 --> 00:50:37.230 align:middle line:84% And that also appeared to stop the radiation going. 00:50:37.230 --> 00:50:42.712 align:middle line:84% So what I want to ask you is, are these two things the same? 00:50:42.712 --> 00:50:43.920 align:middle line:90% And if not, why are they not? 00:50:43.920 --> 00:50:50.290 align:middle line:90% 00:50:50.290 --> 00:50:53.360 align:middle line:84% AUDIENCE: But they're not the same, the reason 00:50:53.360 --> 00:50:57.260 align:middle line:84% being that the optically thick case includes absorption, 00:50:57.260 --> 00:51:01.590 align:middle line:84% but not reflection, whereas the in less than 0 case, [INAUDIBLE] 00:51:01.590 --> 00:51:02.797 align:middle line:90% reflection. 00:51:02.797 --> 00:51:03.380 align:middle line:90% JACK HARE: OK. 00:51:03.380 --> 00:51:05.250 align:middle line:84% So let's start writing down some differences. 00:51:05.250 --> 00:51:05.760 align:middle line:90% I agree with you. 00:51:05.760 --> 00:51:07.170 align:middle line:90% I don't think they are the same. 00:51:07.170 --> 00:51:09.830 align:middle line:90% So here, we have reflection. 00:51:09.830 --> 00:51:12.574 align:middle line:84% There's no energy absorbed, at least in our WKB 00:51:12.574 --> 00:51:13.790 align:middle line:90% picture of this. 00:51:13.790 --> 00:51:15.470 align:middle line:90% There might be reality. 00:51:15.470 --> 00:51:17.780 align:middle line:90% Here, we just have absorption. 00:51:17.780 --> 00:51:21.670 align:middle line:90% 00:51:21.670 --> 00:51:25.345 align:middle line:84% What happens to the wave in these two cases? 00:51:25.345 --> 00:51:27.595 align:middle line:84% What happens to the electric field in these two cases? 00:51:27.595 --> 00:51:33.730 align:middle line:90% 00:51:33.730 --> 00:51:34.230 align:middle line:90% OK. 00:51:34.230 --> 00:51:37.660 align:middle line:84% So in both cases, what happens to the electric field? 00:51:37.660 --> 00:51:42.100 align:middle line:84% I guess we have a critical surface here. 00:51:42.100 --> 00:51:44.890 align:middle line:90% Our wave is coming in. 00:51:44.890 --> 00:51:47.380 align:middle line:84% What happens to the electric field 00:51:47.380 --> 00:51:50.620 align:middle line:84% past a critical surface in the case of a cut-off? 00:51:50.620 --> 00:51:51.940 align:middle line:90% AUDIENCE: It can't propagate? 00:51:51.940 --> 00:51:53.190 align:middle line:90% JACK HARE: It can't propagate. 00:51:53.190 --> 00:51:54.400 align:middle line:90% So what does it do instead? 00:51:54.400 --> 00:51:56.200 align:middle line:90% AUDIENCE: Reflects? 00:51:56.200 --> 00:51:57.880 align:middle line:84% JACK HARE: Is there no electric field? 00:51:57.880 --> 00:51:58.840 align:middle line:90% AUDIENCE: Evanescence. 00:51:58.840 --> 00:51:59.798 align:middle line:90% JACK HARE: Evanescence. 00:51:59.798 --> 00:52:00.610 align:middle line:90% So what does it do? 00:52:00.610 --> 00:52:01.480 align:middle line:90% AUDIENCE: Decays exponentially? 00:52:01.480 --> 00:52:02.540 align:middle line:90% JACK HARE: Decays exponentially. 00:52:02.540 --> 00:52:04.780 align:middle line:84% AUDIENCE: It's like an Airy function somewhere in there. 00:52:04.780 --> 00:52:05.363 align:middle line:90% JACK HARE: OK. 00:52:05.363 --> 00:52:10.060 align:middle line:84% So this is because our refractive index is less than 0. 00:52:10.060 --> 00:52:17.180 align:middle line:84% Our refractive index is defined as k squared over omega squared 00:52:17.180 --> 00:52:19.790 align:middle line:90% C squared, like this. 00:52:19.790 --> 00:52:22.220 align:middle line:84% Omega and C are positive, which means 00:52:22.220 --> 00:52:25.460 align:middle line:84% that k squared is less than 0, which means 00:52:25.460 --> 00:52:27.500 align:middle line:90% we can define some quantity. 00:52:27.500 --> 00:52:30.650 align:middle line:84% I'm trying to write this very exaggerated kappa to try to make 00:52:30.650 --> 00:52:32.460 align:middle line:90% it look different from my k's-- 00:52:32.460 --> 00:52:36.800 align:middle line:90% which is equal to Ik, like this. 00:52:36.800 --> 00:52:45.290 align:middle line:84% And then our electric field now goes as exponential to the minus 00:52:45.290 --> 00:52:47.930 align:middle line:90% kappa x, like this. 00:52:47.930 --> 00:52:50.280 align:middle line:84% The electric field itself decays. 00:52:50.280 --> 00:52:51.590 align:middle line:90% It does not oscillate. 00:52:51.590 --> 00:52:54.500 align:middle line:90% 00:52:54.500 --> 00:52:56.495 align:middle line:84% What about in the case where optically thick? 00:52:56.495 --> 00:52:59.490 align:middle line:90% 00:52:59.490 --> 00:53:01.862 align:middle line:90% Does the wave oscillate or not? 00:53:01.862 --> 00:53:03.970 align:middle line:84% AUDIENCE: Can you just attenuate the amplitude? 00:53:03.970 --> 00:53:04.690 align:middle line:90% JACK HARE: Yeah. 00:53:04.690 --> 00:53:06.760 align:middle line:84% So in the case where we're optically thick, 00:53:06.760 --> 00:53:08.210 align:middle line:90% the amplitude just goes down. 00:53:08.210 --> 00:53:14.600 align:middle line:84% So we still have a wave which is oscillating. 00:53:14.600 --> 00:53:23.450 align:middle line:84% So e is going as exponential of Ikx minus omega t. 00:53:23.450 --> 00:53:29.210 align:middle line:84% It's just that this envelope, I, which 00:53:29.210 --> 00:53:32.703 align:middle line:84% is proportional to e times its complex 00:53:32.703 --> 00:53:34.120 align:middle line:84% conjugate-- it's not proportional. 00:53:34.120 --> 00:53:36.800 align:middle line:84% This is e times its complex conjugate. 00:53:36.800 --> 00:53:42.800 align:middle line:84% That is dropping like exponential of minus tau. 00:53:42.800 --> 00:53:45.128 align:middle line:84% But the wave is still oscillating. 00:53:45.128 --> 00:53:54.600 align:middle line:90% 00:53:54.600 --> 00:53:57.300 align:middle line:84% Any other differences about this? 00:53:57.300 --> 00:54:01.140 align:middle line:90% 00:54:01.140 --> 00:54:02.850 align:middle line:84% AUDIENCE: Here's a clarifying question. 00:54:02.850 --> 00:54:04.830 align:middle line:84% These cases are actually the same, right? 00:54:04.830 --> 00:54:06.980 align:middle line:84% It's just that in one case, you've entirely 00:54:06.980 --> 00:54:09.830 align:middle line:84% gone to an imaginary k, and in the other case, 00:54:09.830 --> 00:54:12.715 align:middle line:84% you put, in principle, a complex k or something. 00:54:12.715 --> 00:54:14.840 align:middle line:84% JACK HARE: That actually feeds into my final point. 00:54:14.840 --> 00:54:17.615 align:middle line:84% So yeah, keep going with that thought. 00:54:17.615 --> 00:54:20.990 align:middle line:84% AUDIENCE: I guess just to say one case could become the other, 00:54:20.990 --> 00:54:23.773 align:middle line:84% assuming you lost all of your real part. 00:54:23.773 --> 00:54:24.440 align:middle line:90% JACK HARE: Yeah. 00:54:24.440 --> 00:54:26.510 align:middle line:84% So people online, the question was these 00:54:26.510 --> 00:54:28.590 align:middle line:84% may actually be the same, in some sense. 00:54:28.590 --> 00:54:29.850 align:middle line:90% This is a slow decay. 00:54:29.850 --> 00:54:31.160 align:middle line:90% This is a very rapid decay. 00:54:31.160 --> 00:54:33.080 align:middle line:84% The difference I see here is that you 00:54:33.080 --> 00:54:36.410 align:middle line:84% can have this slow decay for many different values 00:54:36.410 --> 00:54:38.180 align:middle line:90% of the optical depth. 00:54:38.180 --> 00:54:40.548 align:middle line:84% This happens suddenly and only when 00:54:40.548 --> 00:54:41.840 align:middle line:90% we get to the critical density. 00:54:41.840 --> 00:54:44.180 align:middle line:84% Before that, the wave knows nothing about it. 00:54:44.180 --> 00:54:45.980 align:middle line:84% In reality, if the density is ramping up, 00:54:45.980 --> 00:54:47.272 align:middle line:90% the wavelength will get longer. 00:54:47.272 --> 00:54:50.270 align:middle line:84% But the amplitude of the wave packet in our [? wkp ?] 00:54:50.270 --> 00:54:53.000 align:middle line:84% approximation that we use in the derivable equations-- 00:54:53.000 --> 00:54:54.990 align:middle line:90% that amplitude doesn't change. 00:54:54.990 --> 00:54:58.190 align:middle line:84% So I guess the way I summarize it in my notes 00:54:58.190 --> 00:55:00.365 align:middle line:84% is that this is a gradual process. 00:55:00.365 --> 00:55:02.902 align:middle line:90% 00:55:02.902 --> 00:55:04.110 align:middle line:90% And this is a sudden process. 00:55:04.110 --> 00:55:07.050 align:middle line:90% 00:55:07.050 --> 00:55:09.630 align:middle line:84% In a sense, this is an overdamped oscillator. 00:55:09.630 --> 00:55:12.450 align:middle line:84% The amplitude just drops without oscillating. 00:55:12.450 --> 00:55:15.240 align:middle line:84% And this is an underdamped or maybe critically damped 00:55:15.240 --> 00:55:17.280 align:middle line:84% oscillator, where the wave keeps oscillating, 00:55:17.280 --> 00:55:18.540 align:middle line:90% but it goes down slowly. 00:55:18.540 --> 00:55:21.560 align:middle line:90% 00:55:21.560 --> 00:55:23.250 align:middle line:90% And this will happen. 00:55:23.250 --> 00:55:25.730 align:middle line:84% There will always be some absorption in some plasma. 00:55:25.730 --> 00:55:27.917 align:middle line:84% So you will always have very gradual decrease, 00:55:27.917 --> 00:55:29.750 align:middle line:84% whereas you don't actually have to have this 00:55:29.750 --> 00:55:30.922 align:middle line:90% happen in any plasma at all. 00:55:30.922 --> 00:55:32.630 align:middle line:84% It doesn't have to be a critical density. 00:55:32.630 --> 00:55:35.270 align:middle line:84% But there will always be some finite opacity, 00:55:35.270 --> 00:55:37.160 align:middle line:90% even if it's very, very small. 00:55:37.160 --> 00:55:39.175 align:middle line:84% Anyway, I don't know if that's profound or not. 00:55:39.175 --> 00:55:40.550 align:middle line:84% I just thought it was interesting 00:55:40.550 --> 00:55:42.500 align:middle line:84% that these two phenomena look quite similar, 00:55:42.500 --> 00:55:44.000 align:middle line:84% but they're actually very different. 00:55:44.000 --> 00:55:46.430 align:middle line:90% 00:55:46.430 --> 00:55:47.030 align:middle line:90% OK. 00:55:47.030 --> 00:55:48.240 align:middle line:90% More questions, yes. 00:55:48.240 --> 00:55:49.460 align:middle line:90% AUDIENCE: Weird follow-up. 00:55:49.460 --> 00:55:51.043 align:middle line:84% We talked a little bit about not being 00:55:51.043 --> 00:55:53.337 align:middle line:84% able to see [INAUDIBLE] for some things. 00:55:53.337 --> 00:55:54.920 align:middle line:84% JACK HARE: Usually reflectometry, yes. 00:55:54.920 --> 00:55:55.545 align:middle line:90% AUDIENCE: Yeah. 00:55:55.545 --> 00:55:59.870 align:middle line:84% So if you have a case where your decay is longer than the length 00:55:59.870 --> 00:56:02.133 align:middle line:84% scale over which your critical density is maintained, 00:56:02.133 --> 00:56:03.800 align:middle line:84% do you get weird things where it becomes 00:56:03.800 --> 00:56:05.127 align:middle line:90% a wave on the other side? 00:56:05.127 --> 00:56:06.710 align:middle line:84% JACK HARE: So the question was, if you 00:56:06.710 --> 00:56:08.720 align:middle line:84% have a region where the density drops 00:56:08.720 --> 00:56:10.130 align:middle line:90% below the critical density-- 00:56:10.130 --> 00:56:11.070 align:middle line:90% for example here. 00:56:11.070 --> 00:56:12.928 align:middle line:84% So any less than critical density, 00:56:12.928 --> 00:56:13.970 align:middle line:90% what happens to the wave? 00:56:13.970 --> 00:56:15.350 align:middle line:90% Has anyone done this experiment? 00:56:15.350 --> 00:56:16.267 align:middle line:90% I did it in undergrad. 00:56:16.267 --> 00:56:18.200 align:middle line:84% It was incredible, life changing. 00:56:18.200 --> 00:56:20.880 align:middle line:90% 00:56:20.880 --> 00:56:21.390 align:middle line:90% Yeah? 00:56:21.390 --> 00:56:22.515 align:middle line:90% AUDIENCE: Keep oscillating? 00:56:22.515 --> 00:56:24.390 align:middle line:84% JACK HARE: It will start oscillating again. 00:56:24.390 --> 00:56:27.030 align:middle line:84% So you will couple an evanescent wave. 00:56:27.030 --> 00:56:28.483 align:middle line:84% You can couple an oscillating wave 00:56:28.483 --> 00:56:30.150 align:middle line:84% through an evanescent gap where the wave 00:56:30.150 --> 00:56:32.220 align:middle line:84% itself does not propagate, and energy will start 00:56:32.220 --> 00:56:33.360 align:middle line:90% coming out the other side. 00:56:33.360 --> 00:56:35.722 align:middle line:84% We did it with wax blocks and microwaves 00:56:35.722 --> 00:56:37.680 align:middle line:84% in an underground lab-- these big blocks of wax 00:56:37.680 --> 00:56:38.940 align:middle line:90% and a microwave generator. 00:56:38.940 --> 00:56:41.790 align:middle line:84% And as you move the wax blocks apart, 00:56:41.790 --> 00:56:45.570 align:middle line:84% you can generate a wave with increasingly small amplitude. 00:56:45.570 --> 00:56:48.750 align:middle line:84% But the wave is still there, and it bridged the gap. 00:56:48.750 --> 00:56:51.390 align:middle line:84% And then the remaining energy-- because now your wave is just 00:56:51.390 --> 00:56:53.700 align:middle line:84% oscillating on the other side, much smaller. 00:56:53.700 --> 00:56:56.910 align:middle line:84% That remaining energy is reflected instead. 00:56:56.910 --> 00:57:02.600 align:middle line:84% But this is a really cool example of electromagnetism. 00:57:02.600 --> 00:57:08.090 align:middle line:84% I assume the same thing happens in a tokamak or in any plasma. 00:57:08.090 --> 00:57:10.430 align:middle line:90% Does it, people who do waves? 00:57:10.430 --> 00:57:13.250 align:middle line:84% AUDIENCE: [INAUDIBLE] lower hybrid. 00:57:13.250 --> 00:57:13.980 align:middle line:90% JACK HARE: Right. 00:57:13.980 --> 00:57:14.480 align:middle line:90% OK. 00:57:14.480 --> 00:57:16.910 align:middle line:84% So you can get it evanescently crossing a bit of the plasma 00:57:16.910 --> 00:57:18.680 align:middle line:84% and then coming back to life on the other side. 00:57:18.680 --> 00:57:18.920 align:middle line:90% AUDIENCE: Yeah. 00:57:18.920 --> 00:57:20.628 align:middle line:84% That's what you want to put your launcher 00:57:20.628 --> 00:57:23.180 align:middle line:84% as close to the plasma [INAUDIBLE] as you can. 00:57:23.180 --> 00:57:26.855 align:middle line:84% Obviously, that gives you plasma surface problems. 00:57:26.855 --> 00:57:28.480 align:middle line:84% JACK HARE: Just for the people on line, 00:57:28.480 --> 00:57:30.950 align:middle line:84% Grant is saying that you want to put your launcher very, very 00:57:30.950 --> 00:57:32.240 align:middle line:84% close to the last closed flux surface 00:57:32.240 --> 00:57:33.698 align:middle line:84% because otherwise it's evanescently 00:57:33.698 --> 00:57:34.656 align:middle line:90% decaying in free space. 00:57:34.656 --> 00:57:37.156 align:middle line:84% So you want it very close to the plasma, where it's actually 00:57:37.156 --> 00:57:38.630 align:middle line:90% in oscillating mode instead. 00:57:38.630 --> 00:57:39.890 align:middle line:90% OK. 00:57:39.890 --> 00:57:41.510 align:middle line:90% Any other questions on this? 00:57:41.510 --> 00:57:45.020 align:middle line:84% And then we will go on to deriving radiation 00:57:45.020 --> 00:57:48.820 align:middle line:90% from free charges. 00:57:48.820 --> 00:57:49.430 align:middle line:90% No screams. 00:57:49.430 --> 00:57:50.888 align:middle line:84% None of you've done Jackson before. 00:57:50.888 --> 00:57:58.088 align:middle line:90% 00:57:58.088 --> 00:58:00.380 align:middle line:84% The good news is we're not going to do the full Jackson 00:58:00.380 --> 00:58:01.370 align:middle line:90% treatment on this. 00:58:01.370 --> 00:58:04.783 align:middle line:84% It's extremely boring, and you've hopefully got it before. 00:58:04.783 --> 00:58:06.200 align:middle line:84% And if you haven't, there's no way 00:58:06.200 --> 00:58:09.233 align:middle line:84% I'm going to teach you it in a couple of lectures. 00:58:09.233 --> 00:58:11.150 align:middle line:84% We're going to quote some of the main results. 00:58:11.150 --> 00:58:13.100 align:middle line:84% If they look completely perplexing, 00:58:13.100 --> 00:58:16.310 align:middle line:84% then it might be worth going to have a look at something that 00:58:16.310 --> 00:58:18.400 align:middle line:84% deals with radiation from free chapters. 00:58:18.400 --> 00:58:20.150 align:middle line:84% But we're not going to do the whole thing. 00:58:20.150 --> 00:58:22.040 align:middle line:84% If you want to see it in very rigorous detail, 00:58:22.040 --> 00:58:23.582 align:middle line:84% you should look at Hutchinson's book. 00:58:23.582 --> 00:58:26.420 align:middle line:84% It really does go into this with a lot of rigor. 00:58:26.420 --> 00:58:29.080 align:middle line:84% But what I want to start with is I 00:58:29.080 --> 00:58:35.950 align:middle line:84% want to start with a very simple picture of why charges radiate. 00:58:35.950 --> 00:58:41.898 align:middle line:84% And I have not been able to find this in any textbook. 00:58:41.898 --> 00:58:43.440 align:middle line:84% But it was taught to me in undergrad, 00:58:43.440 --> 00:58:45.660 align:middle line:84% and I thought it was a rather nice physical picture, 00:58:45.660 --> 00:58:47.560 align:middle line:90% so I will teach it to you. 00:58:47.560 --> 00:58:51.330 align:middle line:84% And it may or may not be helpful to you. 00:58:51.330 --> 00:58:56.370 align:middle line:84% Radiation is measured by the Poynting flux. 00:58:56.370 --> 00:59:00.960 align:middle line:84% The Poynting flux is equal to the electric field crossed 00:59:00.960 --> 00:59:06.360 align:middle line:84% with the magnetic field over a factor of mu 0, or probably C, 00:59:06.360 --> 00:59:07.890 align:middle line:90% if you're using CGS. 00:59:07.890 --> 00:59:09.910 align:middle line:90% I don't know. 00:59:09.910 --> 00:59:13.380 align:middle line:84% The point is that radiation moves in a direction which 00:59:13.380 --> 00:59:16.110 align:middle line:84% is perpendicular to the electric and the magnetic fields 00:59:16.110 --> 00:59:16.770 align:middle line:90% in the system. 00:59:16.770 --> 00:59:18.780 align:middle line:84% And we're going to use this simple formula 00:59:18.780 --> 00:59:21.010 align:middle line:84% and sketch a few different moving charges. 00:59:21.010 --> 00:59:23.010 align:middle line:84% And we're going to use that to see whether those 00:59:23.010 --> 00:59:25.480 align:middle line:90% charges radiate or not. 00:59:25.480 --> 00:59:29.880 align:middle line:84% So let us start with a very simple system-- 00:59:29.880 --> 00:59:34.860 align:middle line:90% an electron at rest. 00:59:34.860 --> 00:59:36.960 align:middle line:84% What are the electric and magnetic fields in this? 00:59:36.960 --> 00:59:41.440 align:middle line:90% 00:59:41.440 --> 00:59:42.243 align:middle line:90% Yes? 00:59:42.243 --> 00:59:43.660 align:middle line:84% AUDIENCE: A radial electric field? 00:59:43.660 --> 00:59:44.980 align:middle line:90% It's a known [INAUDIBLE]. 00:59:44.980 --> 00:59:47.610 align:middle line:84% JACK HARE: So it's a radial electric field like this. 00:59:47.610 --> 00:59:54.950 align:middle line:90% 00:59:54.950 --> 01:00:00.170 align:middle line:84% Electric field drops off as 1 upon R squared, Gauss's law 01:00:00.170 --> 01:00:02.150 align:middle line:90% in the R half direction. 01:00:02.150 --> 01:00:06.470 align:middle line:84% The magnetic field is 0 because there are no moving charges, 01:00:06.470 --> 01:00:07.760 align:middle line:90% so there's no currents. 01:00:07.760 --> 01:00:13.000 align:middle line:84% And so therefore, the Poynting vector is 0. 01:00:13.000 --> 01:00:15.580 align:middle line:84% Our stationary charge is not radiated. 01:00:15.580 --> 01:00:18.760 align:middle line:90% OK, next one. 01:00:18.760 --> 01:00:23.280 align:middle line:84% Now we've got a particle, and it's traveling 01:00:23.280 --> 01:00:26.060 align:middle line:90% at a constant velocity. 01:00:26.060 --> 01:00:30.960 align:middle line:84% For example, it's traveling in this direction. 01:00:30.960 --> 01:00:35.870 align:middle line:84% At a snapshot in time, I'm looking at this particle 01:00:35.870 --> 01:00:37.890 align:middle line:90% from my lab frame. 01:00:37.890 --> 01:00:40.290 align:middle line:84% What are the electric and the magnetic fields here? 01:00:40.290 --> 01:00:44.510 align:middle line:90% 01:00:44.510 --> 01:00:45.140 align:middle line:90% Not you again. 01:00:45.140 --> 01:00:47.120 align:middle line:90% We'll have someone else. 01:00:47.120 --> 01:00:49.885 align:middle line:90% But I'm glad you know. 01:00:49.885 --> 01:00:51.670 align:middle line:84% AUDIENCE: The magnetic field, you 01:00:51.670 --> 01:00:57.013 align:middle line:84% can do by the right-hand rule so it's a moving charge. 01:00:57.013 --> 01:00:57.680 align:middle line:90% JACK HARE: Yeah. 01:00:57.680 --> 01:00:59.920 align:middle line:84% So there'll be some sort of-- there's current 01:00:59.920 --> 01:01:02.650 align:middle line:84% in this direction, so there'll be a magnetic field surrounding 01:01:02.650 --> 01:01:03.150 align:middle line:90% it. 01:01:03.150 --> 01:01:05.415 align:middle line:84% I've drawn this tilted, out of the page. 01:01:05.415 --> 01:01:07.790 align:middle line:84% Otherwise, I'd just have to draw it straight up and down, 01:01:07.790 --> 01:01:08.740 align:middle line:90% which would be hard to do. 01:01:08.740 --> 01:01:09.340 align:middle line:90% OK. 01:01:09.340 --> 01:01:11.470 align:middle line:90% This is the theta. 01:01:11.470 --> 01:01:13.330 align:middle line:90% What about the electric fields? 01:01:13.330 --> 01:01:19.190 align:middle line:90% 01:01:19.190 --> 01:01:22.450 align:middle line:90% Not a trick question. 01:01:22.450 --> 01:01:23.355 align:middle line:90% AUDIENCE: Radially. 01:01:23.355 --> 01:01:23.980 align:middle line:90% JACK HARE: Yes. 01:01:23.980 --> 01:01:29.770 align:middle line:84% There was still the same electric fields radially in. 01:01:29.770 --> 01:01:31.590 align:middle line:84% AUDIENCE: Can I ask the annoying question? 01:01:31.590 --> 01:01:33.520 align:middle line:84% We're observing from close enough 01:01:33.520 --> 01:01:37.060 align:middle line:84% that we're able to see what's moved and all that good stuff? 01:01:37.060 --> 01:01:39.850 align:middle line:90% JACK HARE: Yes. 01:01:39.850 --> 01:01:41.950 align:middle line:84% This is definitely not rigorous, but it 01:01:41.950 --> 01:01:43.250 align:middle line:90% does get the right answer. 01:01:43.250 --> 01:01:47.390 align:middle line:84% So please bear with me, even if you're like, oh, no. 01:01:47.390 --> 01:01:47.890 align:middle line:90% OK. 01:01:47.890 --> 01:01:50.560 align:middle line:84% So then we've got electric fields, which 01:01:50.560 --> 01:01:54.880 align:middle line:90% go as 1 upon r squared, r hat. 01:01:54.880 --> 01:02:03.550 align:middle line:84% Magnetic fields, which are going to go as 1 upon r theta hat. 01:02:03.550 --> 01:02:05.740 align:middle line:84% This is assuming that we've got this electron 01:02:05.740 --> 01:02:06.990 align:middle line:90% as a current carrying wire. 01:02:06.990 --> 01:02:08.740 align:middle line:84% Clearly, it's not a current carrying wire. 01:02:08.740 --> 01:02:10.750 align:middle line:84% That's a result for a current carrying wire. 01:02:10.750 --> 01:02:14.120 align:middle line:84% But it's dropping off in this fashion. 01:02:14.120 --> 01:02:17.170 align:middle line:90% And so that means what is s? 01:02:17.170 --> 01:02:24.360 align:middle line:90% 01:02:24.360 --> 01:02:27.795 align:middle line:84% How does it scale and what direction is it pointed? 01:02:27.795 --> 01:02:30.260 align:middle line:84% AUDIENCE: It's pointing in the bi-direction 01:02:30.260 --> 01:02:32.833 align:middle line:84% or whatever depends on your unit. 01:02:32.833 --> 01:02:33.500 align:middle line:90% JACK HARE: Yeah. 01:02:33.500 --> 01:02:35.570 align:middle line:84% Well, the system's a bit screwy here. 01:02:35.570 --> 01:02:38.240 align:middle line:84% Let's say I'm using cylindrical coordinates, where 01:02:38.240 --> 01:02:43.130 align:middle line:84% I've got z direction here, a radial direction here, 01:02:43.130 --> 01:02:46.250 align:middle line:84% and I've got some theta angle like this. 01:02:46.250 --> 01:02:48.768 align:middle line:90% 01:02:48.768 --> 01:02:51.060 align:middle line:84% I know that doesn't make sense with the definition of r 01:02:51.060 --> 01:02:54.820 align:middle line:90% that we had earlier, but again. 01:02:54.820 --> 01:02:55.390 align:middle line:90% Yeah. 01:02:55.390 --> 01:02:58.113 align:middle line:84% AUDIENCE: That'd be 1 over r cubed mu 0. 01:02:58.113 --> 01:02:58.780 align:middle line:90% JACK HARE: Yeah. 01:02:58.780 --> 01:03:01.000 align:middle line:90% I'm going to drop the mu 0. 01:03:01.000 --> 01:03:02.800 align:middle line:90% Treat myself. 01:03:02.800 --> 01:03:04.610 align:middle line:84% And what direction is it pointed? 01:03:04.610 --> 01:03:05.290 align:middle line:90% AUDIENCE: Z. 01:03:05.290 --> 01:03:06.748 align:middle line:84% JACK HARE: In the z direction, yes. 01:03:06.748 --> 01:03:09.790 align:middle line:90% 01:03:09.790 --> 01:03:11.350 align:middle line:84% What is this Poynting flux doing? 01:03:11.350 --> 01:03:14.290 align:middle line:90% 01:03:14.290 --> 01:03:17.040 align:middle line:84% The Poynting flux is the transport 01:03:17.040 --> 01:03:18.870 align:middle line:90% of electromagnetic energy. 01:03:18.870 --> 01:03:19.710 align:middle line:90% What's it doing? 01:03:19.710 --> 01:03:22.080 align:middle line:84% Why is it pointing in the z direction? 01:03:22.080 --> 01:03:23.987 align:middle line:90% Is it radiated? 01:03:23.987 --> 01:03:25.570 align:middle line:84% AUDIENCE: It's following the particle. 01:03:25.570 --> 01:03:27.403 align:middle line:84% JACK HARE: It's just following the particle. 01:03:27.403 --> 01:03:28.960 align:middle line:84% This Poynting flux is simply moving 01:03:28.960 --> 01:03:31.390 align:middle line:84% the electric and magnetic fields that the particle 01:03:31.390 --> 01:03:33.940 align:middle line:90% has with the particle. 01:03:33.940 --> 01:03:36.100 align:middle line:84% And it drops off as 1 over r cubed. 01:03:36.100 --> 01:03:40.900 align:middle line:84% It drops off very, very quickly, away from the particle. 01:03:40.900 --> 01:03:45.610 align:middle line:84% If I draw a surface with surface area r squared around it, 01:03:45.610 --> 01:03:48.207 align:middle line:84% and I ask, how much total power do I 01:03:48.207 --> 01:03:49.540 align:middle line:90% have going through that surface? 01:03:49.540 --> 01:03:51.733 align:middle line:84% That amount total power will drop off 01:03:51.733 --> 01:03:52.900 align:middle line:90% as I make my surface bigger. 01:03:52.900 --> 01:03:54.460 align:middle line:90% This is not a propagating wave. 01:03:54.460 --> 01:03:56.540 align:middle line:90% You can't observe this. 01:03:56.540 --> 01:03:58.360 align:middle line:84% Again, you need to be close enough to see 01:03:58.360 --> 01:03:59.568 align:middle line:90% the electric magnetic fields. 01:03:59.568 --> 01:04:02.920 align:middle line:84% But this is not a radiation you can see from far away. 01:04:02.920 --> 01:04:04.940 align:middle line:90% OK. 01:04:04.940 --> 01:04:20.230 align:middle line:84% So this moves the n energy with the electron like that. 01:04:20.230 --> 01:04:21.620 align:middle line:90% This is the point of it. 01:04:21.620 --> 01:04:24.520 align:middle line:84% I can add-- sorry, now that we've decided 01:04:24.520 --> 01:04:26.835 align:middle line:90% it's in the z direction. 01:04:26.835 --> 01:04:27.335 align:middle line:90% OK. 01:04:27.335 --> 01:04:29.930 align:middle line:90% 01:04:29.930 --> 01:04:33.410 align:middle line:84% Now finally, we're going to look at a system 01:04:33.410 --> 01:04:36.090 align:middle line:84% where the velocity is not constant. 01:04:36.090 --> 01:04:38.960 align:middle line:84% And I'm going to look at a very, very specific velocity profile 01:04:38.960 --> 01:04:39.460 align:middle line:90% here. 01:04:39.460 --> 01:04:41.920 align:middle line:84% And you'll see why I've chosen this in a moment. 01:04:41.920 --> 01:04:45.900 align:middle line:84% This is a profile in which the velocity of our particle 01:04:45.900 --> 01:04:50.160 align:middle line:84% in this sort of z direction is initially at some value, 01:04:50.160 --> 01:04:55.910 align:middle line:84% and then suddenly, instantaneously drops to 0. 01:04:55.910 --> 01:05:03.480 align:middle line:84% A function of time, which means that the x-coordinate is going 01:05:03.480 --> 01:05:06.270 align:middle line:90% to go up and then flatten off. 01:05:06.270 --> 01:05:08.250 align:middle line:84% The particle will then be at rest, 01:05:08.250 --> 01:05:13.196 align:middle line:84% and we'll define this to be e equals 0 here. 01:05:13.196 --> 01:05:26.330 align:middle line:90% 01:05:26.330 --> 01:05:28.113 align:middle line:90% What am I going to get? 01:05:28.113 --> 01:05:30.030 align:middle line:84% I've changed something just before the lecture 01:05:30.030 --> 01:05:31.200 align:middle line:90% to make it clearer. 01:05:31.200 --> 01:05:32.700 align:middle line:84% Now I'm not sure it's consistent. 01:05:32.700 --> 01:05:33.283 align:middle line:90% Give it to me. 01:05:33.283 --> 01:05:42.180 align:middle line:90% 01:05:42.180 --> 01:05:44.840 align:middle line:84% I'm doing it the other way around. 01:05:44.840 --> 01:05:45.340 align:middle line:90% Let's see. 01:05:45.340 --> 01:05:47.170 align:middle line:84% The particle is initially at rest, 01:05:47.170 --> 01:05:50.348 align:middle line:84% and then it suddenly accelerates to some velocity. 01:05:50.348 --> 01:05:51.640 align:middle line:90% I think this will make it work. 01:05:51.640 --> 01:06:02.640 align:middle line:90% 01:06:02.640 --> 01:06:06.110 align:middle line:84% So the particle is initially at rest. 01:06:06.110 --> 01:06:08.870 align:middle line:90% Let's say it's at rest here. 01:06:08.870 --> 01:06:11.900 align:middle line:84% And then all of a sudden, it's over, 01:06:11.900 --> 01:06:15.830 align:middle line:84% moving in this direction like this. 01:06:15.830 --> 01:06:19.190 align:middle line:84% Now, information about the electric and magnetic fields 01:06:19.190 --> 01:06:22.440 align:middle line:84% of this particle can only propagate at the speed of light. 01:06:22.440 --> 01:06:26.700 align:middle line:84% And so that means if I'm an observer some distance away-- 01:06:26.700 --> 01:06:34.890 align:middle line:84% and let's say that distance is this circle, like that. 01:06:34.890 --> 01:06:38.280 align:middle line:84% The information I have about the electric and magnetic fields 01:06:38.280 --> 01:06:41.622 align:middle line:90% is the same information I had-- 01:06:41.622 --> 01:06:43.080 align:middle line:84% from my point of view, the particle 01:06:43.080 --> 01:06:47.490 align:middle line:84% is still at rest, which means that outside of this circle, all 01:06:47.490 --> 01:06:53.600 align:middle line:84% I can say is that the electric fields are pointing inwards 01:06:53.600 --> 01:06:54.590 align:middle line:90% here. 01:06:54.590 --> 01:06:57.470 align:middle line:84% Inside the circle, I now know that the electric fields 01:06:57.470 --> 01:06:59.370 align:middle line:84% are doing something quite different. 01:06:59.370 --> 01:07:04.690 align:middle line:90% They're now looking like this. 01:07:04.690 --> 01:07:11.870 align:middle line:84% And we have some sort of magnetic field like this. 01:07:11.870 --> 01:07:14.120 align:middle line:84% Now, the electric field lines cannot be broken. 01:07:14.120 --> 01:07:16.850 align:middle line:84% These electric field lines therefore, 01:07:16.850 --> 01:07:19.550 align:middle line:84% in this very fictitious scenario, 01:07:19.550 --> 01:07:22.230 align:middle line:84% have to be joined up in this way. 01:07:22.230 --> 01:07:26.480 align:middle line:84% So I'm also going to draw some electric field lines like that, 01:07:26.480 --> 01:07:28.260 align:middle line:90% electric field lines like that. 01:07:28.260 --> 01:07:39.010 align:middle line:84% And these ones are outside like this. 01:07:39.010 --> 01:07:40.710 align:middle line:84% And now again on the inside, I've 01:07:40.710 --> 01:07:46.630 align:middle line:84% got some magnetic field lines, which is still 01:07:46.630 --> 01:07:48.730 align:middle line:90% in this poloidal direction. 01:07:48.730 --> 01:07:52.410 align:middle line:90% 01:07:52.410 --> 01:07:56.472 align:middle line:90% So what's happening now? 01:07:56.472 --> 01:07:58.180 align:middle line:84% So while pointing flux or electric fields 01:07:58.180 --> 01:08:01.620 align:middle line:84% or magnetic fields, let's go through the magnetic field 01:08:01.620 --> 01:08:05.240 align:middle line:84% and say that at this interface here-- 01:08:05.240 --> 01:08:13.690 align:middle line:84% so at this interface, we still have a magnetic field, which 01:08:13.690 --> 01:08:17.640 align:middle line:90% opposes upon our theta hat. 01:08:17.640 --> 01:08:19.154 align:middle line:90% We've got an electric field. 01:08:19.154 --> 01:08:21.960 align:middle line:90% 01:08:21.960 --> 01:08:24.000 align:middle line:84% OK, I think this is the bit where I just 01:08:24.000 --> 01:08:26.730 align:middle line:90% have to tell you to believe me. 01:08:26.730 --> 01:08:29.070 align:middle line:84% If I'm right, I can prove this straight away. 01:08:29.070 --> 01:08:33.170 align:middle line:90% 01:08:33.170 --> 01:08:37.850 align:middle line:84% This electric field turns out to have a scaling which goes as 1 01:08:37.850 --> 01:08:38.680 align:middle line:90% upon r. 01:08:38.680 --> 01:08:41.029 align:middle line:84% I can't remember where that comes from. 01:08:41.029 --> 01:08:42.680 align:middle line:84% Not 1 upon r squared, but it's very 01:08:42.680 --> 01:08:44.359 align:middle line:90% important for the conclusion. 01:08:44.359 --> 01:08:48.722 align:middle line:84% And also here, if we think about this direction, 01:08:48.722 --> 01:08:50.180 align:middle line:84% we're going to have something which 01:08:50.180 --> 01:08:52.770 align:middle line:84% is tangential to this theta direction. 01:08:52.770 --> 01:08:56.090 align:middle line:84% So I'm going to call it some sort of [INAUDIBLE] where 01:08:56.090 --> 01:09:02.029 align:middle line:84% I'm subtly switching into a spherical coordinate system. 01:09:02.029 --> 01:09:07.010 align:middle line:84% And all of this together means that our Poynting vector now 01:09:07.010 --> 01:09:09.350 align:middle line:90% goes as 1 upon r squared. 01:09:09.350 --> 01:09:13.830 align:middle line:90% And it has a direction, r hat. 01:09:13.830 --> 01:09:15.450 align:middle line:84% And this means that finally, we've 01:09:15.450 --> 01:09:17.279 align:middle line:90% got a system which is radiated. 01:09:17.279 --> 01:09:22.979 align:middle line:84% So this is moving radiation radially outwards 01:09:22.979 --> 01:09:24.220 align:middle line:90% in every direction. 01:09:24.220 --> 01:09:27.149 align:middle line:84% And this is the result that you probably 01:09:27.149 --> 01:09:31.710 align:middle line:84% remember is that accelerated charged particles radiate. 01:09:31.710 --> 01:09:52.470 align:middle line:90% 01:09:52.470 --> 01:09:55.620 align:middle line:84% These disappointing lectures are not simply shuffling the energy 01:09:55.620 --> 01:09:57.000 align:middle line:90% along with the particle. 01:09:57.000 --> 01:09:58.910 align:middle line:84% It's actually taking energy from the particle 01:09:58.910 --> 01:10:00.660 align:middle line:84% and putting it into electromagnetic waves. 01:10:00.660 --> 01:10:02.340 align:middle line:84% And like I said, it'd be much more convincing 01:10:02.340 --> 01:10:04.260 align:middle line:84% if I can remember the argument for this fact. 01:10:04.260 --> 01:10:06.270 align:middle line:84% And I decided not to write it down in my notes, 01:10:06.270 --> 01:10:08.130 align:middle line:90% so I can't remember it. 01:10:08.130 --> 01:10:10.140 align:middle line:84% But it's clearly a very different circumstance 01:10:10.140 --> 01:10:11.700 align:middle line:90% than what we had here. 01:10:11.700 --> 01:10:14.430 align:middle line:84% The reason we've derived in this very hand-wavy way 01:10:14.430 --> 01:10:16.740 align:middle line:84% is I'm about to write down the actual correct equation. 01:10:16.740 --> 01:10:18.810 align:middle line:84% And you'll see that it has almost no intuition 01:10:18.810 --> 01:10:20.310 align:middle line:84% involved in it whatsoever, whereas I 01:10:20.310 --> 01:10:22.890 align:middle line:84% like to think that this gives you a little bit of intuition 01:10:22.890 --> 01:10:25.560 align:middle line:84% about what's going on and why, moving charges right here. 01:10:25.560 --> 01:10:26.820 align:middle line:90% But questions? 01:10:26.820 --> 01:10:35.540 align:middle line:90% 01:10:35.540 --> 01:10:36.440 align:middle line:90% Yeah. 01:10:36.440 --> 01:10:38.300 align:middle line:84% AUDIENCE: So in this situation, you've 01:10:38.300 --> 01:10:41.900 align:middle line:84% drawn in your third situation with the non-constant B 01:10:41.900 --> 01:10:46.670 align:middle line:84% and C lightcone of the particle as it begins to move. 01:10:46.670 --> 01:10:48.410 align:middle line:84% That's where the energy is located? 01:10:48.410 --> 01:10:49.077 align:middle line:90% JACK HARE: Yeah. 01:10:49.077 --> 01:10:51.980 align:middle line:84% So this distance here, this is where 01:10:51.980 --> 01:10:54.320 align:middle line:84% an observer outside of this would still believe 01:10:54.320 --> 01:10:56.870 align:middle line:90% the particle is located here. 01:10:56.870 --> 01:10:58.640 align:middle line:84% Inside this lightcone, the observer 01:10:58.640 --> 01:11:00.567 align:middle line:84% can see that the particle has moved, 01:11:00.567 --> 01:11:02.150 align:middle line:84% so of course, the particle is actually 01:11:02.150 --> 01:11:05.347 align:middle line:84% going to be moving in several discrete-- 01:11:05.347 --> 01:11:07.430 align:middle line:84% I've drawn these electric field lines as straight. 01:11:07.430 --> 01:11:09.222 align:middle line:84% They should actually be curved inside here. 01:11:09.222 --> 01:11:11.780 align:middle line:84% I've just drawn them as straight from the current position 01:11:11.780 --> 01:11:14.920 align:middle line:90% of the particle at this point. 01:11:14.920 --> 01:11:17.460 align:middle line:84% And so there must be a tangential discontinuity 01:11:17.460 --> 01:11:19.800 align:middle line:84% of the electric field at this surface 01:11:19.800 --> 01:11:23.010 align:middle line:84% to go from one radial vector pointing at one point 01:11:23.010 --> 01:11:25.350 align:middle line:84% to a new radial vector pointing at a new point. 01:11:25.350 --> 01:11:27.430 align:middle line:84% And it's that tangential discontinuity 01:11:27.430 --> 01:11:31.800 align:middle line:84% which generates this electric field perpendicular 01:11:31.800 --> 01:11:35.760 align:middle line:84% to the magnetic field that gives us Rossby Poynting vector. 01:11:35.760 --> 01:11:37.583 align:middle line:90% And that goes radially outwards. 01:11:37.583 --> 01:11:39.000 align:middle line:84% AUDIENCE: And so for the situation 01:11:39.000 --> 01:11:42.540 align:middle line:84% where we have basically a step function in the velocity, 01:11:42.540 --> 01:11:46.133 align:middle line:84% the energy is infinitely localized to this shell? 01:11:46.133 --> 01:11:46.800 align:middle line:90% JACK HARE: Yeah. 01:11:46.800 --> 01:11:51.450 align:middle line:84% In this case, the particle is only accelerating at this time. 01:11:51.450 --> 01:11:56.970 align:middle line:84% The acceleration just looks like a delta function. 01:11:56.970 --> 01:12:00.930 align:middle line:84% And so therefore, there's only radiation here, 01:12:00.930 --> 01:12:03.280 align:middle line:84% propagating outwards as a spherical wave. 01:12:03.280 --> 01:12:05.880 align:middle line:84% And from then onwards, the particle 01:12:05.880 --> 01:12:07.950 align:middle line:84% is just in this situation again where it's 01:12:07.950 --> 01:12:09.160 align:middle line:90% shuffling its fields along. 01:12:09.160 --> 01:12:11.580 align:middle line:90% And so this is acceleration. 01:12:11.580 --> 01:12:16.240 align:middle line:90% 01:12:16.240 --> 01:12:16.950 align:middle line:90% Yeah. 01:12:16.950 --> 01:12:20.200 align:middle line:84% AUDIENCE: So does that mean like if you have somehow 01:12:20.200 --> 01:12:22.300 align:middle line:84% the spatial distribution of energy 01:12:22.300 --> 01:12:25.283 align:middle line:84% as a function of your acceleration, [INAUDIBLE]. 01:12:25.283 --> 01:12:26.200 align:middle line:90% JACK HARE: Absolutely. 01:12:26.200 --> 01:12:28.720 align:middle line:84% And we will see that in the long and complicated formula 01:12:28.720 --> 01:12:30.430 align:middle line:90% in a moment, yes. 01:12:30.430 --> 01:12:33.497 align:middle line:84% You can do this stuff rigorously as well. 01:12:33.497 --> 01:12:34.330 align:middle line:90% Any other questions? 01:12:34.330 --> 01:12:35.590 align:middle line:90% Yeah. 01:12:35.590 --> 01:12:39.430 align:middle line:84% AUDIENCE: Just my vague memory of how the 1 over r thing is 01:12:39.430 --> 01:12:41.400 align:middle line:90% because you're-- 01:12:41.400 --> 01:12:47.080 align:middle line:84% I think it's like to have your voltage make some big nonsense, 01:12:47.080 --> 01:12:49.750 align:middle line:84% you have to integrate along that path length 01:12:49.750 --> 01:12:51.627 align:middle line:84% that it's getting extended across. 01:12:51.627 --> 01:12:52.210 align:middle line:90% JACK HARE: OK. 01:12:52.210 --> 01:12:53.860 align:middle line:84% So there's a suggestion this 1 over r 01:12:53.860 --> 01:12:55.690 align:middle line:84% comes from trying to keep the voltage 01:12:55.690 --> 01:12:58.540 align:middle line:84% to a reasonable value and your integration along the path. 01:12:58.540 --> 01:12:59.200 align:middle line:90% AUDIENCE: Yeah. 01:12:59.200 --> 01:13:00.140 align:middle line:90% There's some weird-- 01:13:00.140 --> 01:13:01.210 align:middle line:84% JACK HARE: Have you seen this before, then? 01:13:01.210 --> 01:13:01.720 align:middle line:90% AUDIENCE: Yeah. 01:13:01.720 --> 01:13:02.620 align:middle line:90% JACK HARE: OK, good. 01:13:02.620 --> 01:13:04.720 align:middle line:90% I'm glad I'm not just crazy. 01:13:04.720 --> 01:13:06.100 align:middle line:90% I didn't come up with this. 01:13:06.100 --> 01:13:09.560 align:middle line:84% It was underground, and I'll try to reconstruct it. 01:13:09.560 --> 01:13:10.060 align:middle line:90% OK. 01:13:10.060 --> 01:13:13.250 align:middle line:84% Any other questions before we move on? 01:13:13.250 --> 01:13:16.440 align:middle line:90% Any questions online? 01:13:16.440 --> 01:13:17.370 align:middle line:90% I see Matt's hand. 01:13:17.370 --> 01:13:18.600 align:middle line:90% Yeah. 01:13:18.600 --> 01:13:19.307 align:middle line:90% AUDIENCE: Yeah. 01:13:19.307 --> 01:13:21.390 align:middle line:84% Maybe you said this, but I just want to make sure. 01:13:21.390 --> 01:13:27.180 align:middle line:84% The Poynting flux direction in the second picture. 01:13:27.180 --> 01:13:28.800 align:middle line:90% Yeah. 01:13:28.800 --> 01:13:33.150 align:middle line:84% That's only valid in the plane perpendicular to the motion 01:13:33.150 --> 01:13:34.290 align:middle line:90% of the particle, right? 01:13:34.290 --> 01:13:40.470 align:middle line:84% Because the electric field direction would be rho, not-- 01:13:40.470 --> 01:13:42.683 align:middle line:84% it would be in cylindrical coordinates, right? 01:13:42.683 --> 01:13:44.850 align:middle line:84% JACK HARE: I was saying that the fact that I decided 01:13:44.850 --> 01:13:46.590 align:middle line:84% to use cylindrical coordinates here 01:13:46.590 --> 01:13:49.440 align:middle line:84% is not actually very useful because the electric field isn't 01:13:49.440 --> 01:13:52.350 align:middle line:84% just in that direction or in this direction. 01:13:52.350 --> 01:13:53.920 align:middle line:90% It is in every direction. 01:13:53.920 --> 01:13:58.200 align:middle line:84% It's just the only place where e cross B is significant 01:13:58.200 --> 01:13:59.908 align:middle line:90% is really in this plane here. 01:13:59.908 --> 01:14:00.450 align:middle line:90% AUDIENCE: OK. 01:14:00.450 --> 01:14:01.425 align:middle line:90% Sure, sure. 01:14:01.425 --> 01:14:02.383 align:middle line:90% JACK HARE: [INAUDIBLE]. 01:14:02.383 --> 01:14:03.660 align:middle line:90% And it will drop off. 01:14:03.660 --> 01:14:05.077 align:middle line:84% If you go back to this point here, 01:14:05.077 --> 01:14:06.785 align:middle line:84% e cross B-- because they're not very well 01:14:06.785 --> 01:14:08.110 align:middle line:90% anti-aligned, it will drop off. 01:14:08.110 --> 01:14:12.960 align:middle line:84% And so you have a ring moving the electric fields 01:14:12.960 --> 01:14:14.543 align:middle line:84% and magnetic fields with the particle. 01:14:14.543 --> 01:14:15.918 align:middle line:84% AUDIENCE: Yeah, that makes sense. 01:14:15.918 --> 01:14:16.540 align:middle line:90% OK, thanks. 01:14:16.540 --> 01:14:17.123 align:middle line:90% JACK HARE: OK. 01:14:17.123 --> 01:14:18.920 align:middle line:90% Any other questions? 01:14:18.920 --> 01:14:19.420 align:middle line:90% OK. 01:14:19.420 --> 01:14:22.105 align:middle line:90% Let's see the full version. 01:14:22.105 --> 01:14:39.990 align:middle line:90% 01:14:39.990 --> 01:14:43.400 align:middle line:84% So if you do this properly, you get out 01:14:43.400 --> 01:15:00.205 align:middle line:84% of the equation for the radiation from a single moving 01:15:00.205 --> 01:15:00.705 align:middle line:90% charge. 01:15:00.705 --> 01:15:08.580 align:middle line:90% 01:15:08.580 --> 01:15:11.700 align:middle line:84% Like I said, if you haven't seen this 01:15:11.700 --> 01:15:13.112 align:middle line:84% or can't remember how to do this, 01:15:13.112 --> 01:15:14.820 align:middle line:84% I recommend you go take a look at Jackson 01:15:14.820 --> 01:15:19.300 align:middle line:84% or some other sufficiently advanced text. 01:15:19.300 --> 01:15:22.750 align:middle line:84% And this equation is the electric field 01:15:22.750 --> 01:15:27.310 align:middle line:84% seen by the observer who is some distance away from the charge. 01:15:27.310 --> 01:15:33.740 align:middle line:84% And this electric field has got the standard q over 4 pi epsilon 01:15:33.740 --> 01:15:35.960 align:middle line:84% 0 that we know and love from Gauss's law. 01:15:35.960 --> 01:15:39.170 align:middle line:84% And then it has two terms corresponding 01:15:39.170 --> 01:15:42.720 align:middle line:84% to radiation, which we call the near field and the far field. 01:15:42.720 --> 01:15:46.940 align:middle line:84% So the first term, which is part of the near field, is a 1 01:15:46.940 --> 01:15:49.220 align:middle line:90% over k squared. 01:15:49.220 --> 01:15:52.070 align:middle line:84% This k is not the normal k that we've 01:15:52.070 --> 01:15:54.080 align:middle line:90% been using before the moment. 01:15:54.080 --> 01:15:55.865 align:middle line:84% And I've just realized this is a cube. 01:15:55.865 --> 01:16:00.010 align:middle line:90% k cubed r squared. 01:16:00.010 --> 01:16:01.850 align:middle line:84% And then you have a term that looks 01:16:01.850 --> 01:16:08.330 align:middle line:84% like r hat minus the particle velocity on C. 1 01:16:08.330 --> 01:16:11.812 align:middle line:84% minus the particle velocity squared on C squared. 01:16:11.812 --> 01:16:14.270 align:middle line:84% We're going to see lots of velocities showing up normalized 01:16:14.270 --> 01:16:16.610 align:middle line:84% to C because obviously the speed of light 01:16:16.610 --> 01:16:19.470 align:middle line:84% is pretty important for this sort of stuff. 01:16:19.470 --> 01:16:22.380 align:middle line:84% Here, I've defined two new terms. 01:16:22.380 --> 01:16:26.370 align:middle line:84% This k takes account of the fact that we're looking 01:16:26.370 --> 01:16:28.350 align:middle line:90% at the particle at some time. 01:16:28.350 --> 01:16:29.975 align:middle line:84% We're looking at the particle and we're 01:16:29.975 --> 01:16:31.933 align:middle line:84% seeing where that particle is, but the particle 01:16:31.933 --> 01:16:32.730 align:middle line:90% has already moved. 01:16:32.730 --> 01:16:34.860 align:middle line:84% So this is helping us with our retarded 01:16:34.860 --> 01:16:37.860 align:middle line:84% time, which is looking at the earlier time that we're at. 01:16:37.860 --> 01:16:50.710 align:middle line:84% So this k is defined as 1 minus r 0 b upon r C. 01:16:50.710 --> 01:16:55.190 align:middle line:84% And I haven't defined r either, so I should do that. 01:16:55.190 --> 01:16:57.830 align:middle line:84% This r is a vector that is effectively 01:16:57.830 --> 01:17:00.750 align:middle line:84% the vector that joins the observer to the particle. 01:17:00.750 --> 01:17:03.530 align:middle line:84% So this is defined as the position 01:17:03.530 --> 01:17:07.790 align:middle line:84% of the observer minus the position of the particle 01:17:07.790 --> 01:17:08.810 align:middle line:90% at time, t. 01:17:08.810 --> 01:17:10.940 align:middle line:84% So if I draw a little diagram, we've 01:17:10.940 --> 01:17:13.700 align:middle line:84% got some origin to our coordinate system. 01:17:13.700 --> 01:17:19.310 align:middle line:84% There's some vector, r of t, which is our electron. 01:17:19.310 --> 01:17:24.120 align:middle line:84% And then there's some vector, x, which defines our observer. 01:17:24.120 --> 01:17:28.460 align:middle line:84% And so this is the vector r here. 01:17:28.460 --> 01:17:30.215 align:middle line:84% We could get rid of this by just assuming 01:17:30.215 --> 01:17:32.180 align:middle line:84% either the observer or the particle 01:17:32.180 --> 01:17:34.220 align:middle line:84% is at the origin of the coordinate system. 01:17:34.220 --> 01:17:36.050 align:middle line:84% But as the particle is moving, that's 01:17:36.050 --> 01:17:37.710 align:middle line:84% not necessarily particularly useful. 01:17:37.710 --> 01:17:40.860 align:middle line:84% And so we're doing it in this more generalized way here. 01:17:40.860 --> 01:17:44.880 align:middle line:84% So we'll talk about what happens to this term in a moment. 01:17:44.880 --> 01:17:56.050 align:middle line:84% The second term inside here is 1 upon C squared k cubed R r hat. 01:17:56.050 --> 01:17:59.370 align:middle line:84% And we're just going to find r hat 01:17:59.370 --> 01:18:05.210 align:middle line:84% is equal to the R vector over the size of the R vector-- 01:18:05.210 --> 01:18:07.570 align:middle line:90% so just a normalized vector. 01:18:07.570 --> 01:18:17.600 align:middle line:84% And that r hat is crossed with r minus the C, which is, 01:18:17.600 --> 01:18:22.710 align:middle line:84% in turn, crossed with v dot upon C. 01:18:22.710 --> 01:18:26.070 align:middle line:84% And v dot is going to be extremely important in a moment. 01:18:26.070 --> 01:18:27.750 align:middle line:84% And I'm going to close my brackets. 01:18:27.750 --> 01:18:33.610 align:middle line:90% 01:18:33.610 --> 01:18:35.320 align:middle line:90% So we have two terms here. 01:18:35.320 --> 01:18:41.200 align:middle line:84% We have a term which we call the near field and a term 01:18:41.200 --> 01:18:43.385 align:middle line:90% that we call the far field. 01:18:43.385 --> 01:18:49.210 align:middle line:90% 01:18:49.210 --> 01:18:52.960 align:middle line:84% We're not going to explicitly derive the magnetic field here 01:18:52.960 --> 01:18:55.090 align:middle line:84% because fortunately, these are properties 01:18:55.090 --> 01:18:57.520 align:middle line:84% of electromagnetic waves in a vacuum. 01:18:57.520 --> 01:18:59.560 align:middle line:84% And we're still talking about vacuum waves here. 01:18:59.560 --> 01:19:08.520 align:middle line:84% B is simply equal to 1 upon C e crossed with r hat, like that. 01:19:08.520 --> 01:19:10.920 align:middle line:84% So the magnetic field is perpendicular 01:19:10.920 --> 01:19:14.310 align:middle line:84% to the electric field, as we expect. 01:19:14.310 --> 01:19:27.200 align:middle line:84% And this means that for the near-field case, 01:19:27.200 --> 01:19:29.740 align:middle line:84% we have a Poynting vector, s, which 01:19:29.740 --> 01:19:33.920 align:middle line:84% again is equal to e cross B with some constants. 01:19:33.920 --> 01:19:39.280 align:middle line:84% And that s is going to go as 1 over r to the 4th. 01:19:39.280 --> 01:19:40.610 align:middle line:90% And so the far field case. 01:19:40.610 --> 01:19:48.150 align:middle line:90% 01:19:48.150 --> 01:19:55.000 align:middle line:84% That s is going to go like R. And s is going to go as 1 upon R 01:19:55.000 --> 01:19:56.530 align:middle line:90% squared. 01:19:56.530 --> 01:19:59.280 align:middle line:90% Here we had r squared of this. 01:19:59.280 --> 01:20:03.170 align:middle line:90% 01:20:03.170 --> 01:20:06.740 align:middle line:84% So this straight away tells us that if we consider the power 01:20:06.740 --> 01:20:11.750 align:middle line:84% through a sphere of some radius, r, the total power, 01:20:11.750 --> 01:20:13.730 align:middle line:84% s, integrated over the surface of the sphere 01:20:13.730 --> 01:20:16.670 align:middle line:84% will drop off as R squared for the near-field. 01:20:16.670 --> 01:20:18.690 align:middle line:84% But it will stay constant for the far field. 01:20:18.690 --> 01:20:24.710 align:middle line:84% So only the far field is actual propagating radiation. 01:20:24.710 --> 01:20:26.995 align:middle line:84% That's the only thing we're actually going to observe. 01:20:26.995 --> 01:20:29.850 align:middle line:90% 01:20:29.850 --> 01:20:33.300 align:middle line:84% So we don't actually need to have 01:20:33.300 --> 01:20:36.180 align:middle line:84% this near-field term in the equations we're looking at. 01:20:36.180 --> 01:20:38.870 align:middle line:84% It just drops out when you solve this equation. 01:20:38.870 --> 01:20:40.830 align:middle line:90% We figured [INAUDIBLE]. 01:20:40.830 --> 01:20:52.900 align:middle line:84% They're propagating e and m, e [INAUDIBLE] 01:20:52.900 --> 01:20:57.340 align:middle line:84% The only reason to show you this is, first of all, to show-- 01:20:57.340 --> 01:20:59.050 align:middle line:84% as many of you know, with the radiation 01:20:59.050 --> 01:21:01.682 align:middle line:84% even from a single moving charge is very complicated. 01:21:01.682 --> 01:21:03.140 align:middle line:84% And then of course, we have to ask, 01:21:03.140 --> 01:21:05.455 align:middle line:84% what is it that we're trying to achieve in a plasma? 01:21:05.455 --> 01:21:10.190 align:middle line:90% 01:21:10.190 --> 01:21:19.640 align:middle line:84% But the first thing we want to do in a plasma 01:21:19.640 --> 01:21:23.450 align:middle line:84% is we notice that our near field far field term 01:21:23.450 --> 01:21:27.260 align:middle line:84% has lots of things that we might know, like where we're sitting 01:21:27.260 --> 01:21:28.650 align:middle line:90% and how long it's been. 01:21:28.650 --> 01:21:33.230 align:middle line:84% But there are several things inside here that we don't know, 01:21:33.230 --> 01:21:36.890 align:middle line:90% such as v and v dot. 01:21:36.890 --> 01:21:38.480 align:middle line:84% So that tells us that we're going 01:21:38.480 --> 01:21:45.290 align:middle line:84% to need to solve the equation of motion for a particle, something 01:21:45.290 --> 01:21:47.360 align:middle line:84% that you've done several times in a plasma 01:21:47.360 --> 01:21:50.615 align:middle line:84% already without a magnetic field, with a magnetic field. 01:21:50.615 --> 01:21:53.660 align:middle line:84% You know this is the thing that tells you that the particles are 01:21:53.660 --> 01:21:56.430 align:middle line:90% spiraling around field lines. 01:21:56.430 --> 01:21:58.040 align:middle line:84% And because they're spiraling, they're 01:21:58.040 --> 01:21:59.750 align:middle line:90% continuously being accelerated. 01:21:59.750 --> 01:22:02.660 align:middle line:84% And so we know straight away that these accelerated particles 01:22:02.660 --> 01:22:05.760 align:middle line:84% are going to be emitting propagating radiation. 01:22:05.760 --> 01:22:08.060 align:middle line:84% And that's the thing that we want to detect. 01:22:08.060 --> 01:22:09.560 align:middle line:84% So first of all, we'll have to solve 01:22:09.560 --> 01:22:11.040 align:middle line:90% for the equation of motion. 01:22:11.040 --> 01:22:15.110 align:middle line:84% But the second thing we'll have to do 01:22:15.110 --> 01:22:22.320 align:middle line:84% is integrate overall distribution function, 01:22:22.320 --> 01:22:28.600 align:middle line:84% f of e e3v because we will have then-- 01:22:28.600 --> 01:22:30.470 align:middle line:84% once we've solved the equation of motion, 01:22:30.470 --> 01:22:32.230 align:middle line:84% we'll have it for a single particle. 01:22:32.230 --> 01:22:34.480 align:middle line:84% But the particles have different velocities 01:22:34.480 --> 01:22:37.060 align:middle line:84% depending on where they fall in this distribution function. 01:22:37.060 --> 01:22:38.290 align:middle line:84% And they may have different velocities 01:22:38.290 --> 01:22:39.290 align:middle line:90% in different directions. 01:22:39.290 --> 01:22:41.700 align:middle line:84% They may have one velocity along the magnetic field 01:22:41.700 --> 01:22:43.660 align:middle line:84% and another velocity perpendicular 01:22:43.660 --> 01:22:44.810 align:middle line:90% to the magnetic field. 01:22:44.810 --> 01:22:47.823 align:middle line:84% And so these two steps are very non-trivial to do properly. 01:22:47.823 --> 01:22:49.240 align:middle line:84% And these are the steps that we're 01:22:49.240 --> 01:22:50.720 align:middle line:90% going to skip in this class. 01:22:50.720 --> 01:22:52.720 align:middle line:84% We're just going to give the results from this. 01:22:52.720 --> 01:22:54.742 align:middle line:84% But if you want to do this properly, 01:22:54.742 --> 01:22:56.950 align:middle line:84% if you want to work with electron cyclotron emission, 01:22:56.950 --> 01:22:59.590 align:middle line:84% you could probably go back and check that you can actually do 01:22:59.590 --> 01:23:01.270 align:middle line:90% all these intermediate steps. 01:23:01.270 --> 01:23:01.820 align:middle line:90% OK. 01:23:01.820 --> 01:23:03.070 align:middle line:90% We are well over time. 01:23:03.070 --> 01:23:04.700 align:middle line:90% I'm happy to take questions. 01:23:04.700 --> 01:23:08.040 align:middle line:84% But otherwise, I will see you on Thursday. 01:23:08.040 --> 01:23:12.000 align:middle line:90%