WEBVTT 00:00:00.000 --> 00:00:09.930 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:09.930 --> 00:00:11.700 align:middle line:84% JACK HARE: Welcome to our second lecture. 00:00:11.700 --> 00:00:14.985 align:middle line:84% Today, we are going to be talking about magnetic probes. 00:00:14.985 --> 00:00:17.670 align:middle line:90% 00:00:17.670 --> 00:00:19.500 align:middle line:84% These are surprisingly simple devices. 00:00:19.500 --> 00:00:21.660 align:middle line:84% But you can also measure a surprising amount 00:00:21.660 --> 00:00:23.580 align:middle line:90% about your plasma from these. 00:00:23.580 --> 00:00:30.900 align:middle line:90% 00:00:30.900 --> 00:00:35.390 align:middle line:84% So we'll start by considering just a simple loop of wire. 00:00:35.390 --> 00:00:40.720 align:middle line:84% So let's imagine we have a loop of wire like this. 00:00:40.720 --> 00:00:49.020 align:middle line:84% It's got some area A. And it's got some magnetic field pointing 00:00:49.020 --> 00:00:52.590 align:middle line:90% out of the page, B, like this. 00:00:52.590 --> 00:00:56.460 align:middle line:84% And we can imagine that this is a time-varying magnetic field, B 00:00:56.460 --> 00:00:58.830 align:middle line:90% of time here, OK? 00:00:58.830 --> 00:01:02.190 align:middle line:84% And then we go and grab one of Maxwell's equations. 00:01:02.190 --> 00:01:08.650 align:middle line:84% We'll take the curl of E is equal to minus 00:01:08.650 --> 00:01:10.870 align:middle line:90% the time derivative of B here. 00:01:10.870 --> 00:01:13.000 align:middle line:90% So I'm writing that as B-dot. 00:01:13.000 --> 00:01:20.190 align:middle line:84% We can take this equation and integrate over this area here, 00:01:20.190 --> 00:01:25.150 align:middle line:84% integrate up the curl of E over this surface, 00:01:25.150 --> 00:01:27.610 align:middle line:84% the dotting with the S and doing the same thing 00:01:27.610 --> 00:01:32.840 align:middle line:84% on the other side, integrating up the magnetic field 00:01:32.840 --> 00:01:35.820 align:middle line:90% dotted with S, like that. 00:01:35.820 --> 00:01:39.350 align:middle line:84% And then we can use one of our nice vector calculus identities 00:01:39.350 --> 00:01:41.330 align:middle line:84% and convert this from being a surface 00:01:41.330 --> 00:01:46.380 align:middle line:84% integral over the area into a line integral around the curve 00:01:46.380 --> 00:01:47.970 align:middle line:90% which bounds this surface here. 00:01:47.970 --> 00:01:50.690 align:middle line:84% And that will allow us to write this instead 00:01:50.690 --> 00:01:57.440 align:middle line:90% as the integral of E dot dL. 00:01:57.440 --> 00:02:00.560 align:middle line:84% And we recognize that if we're integrating an electric field 00:02:00.560 --> 00:02:03.960 align:middle line:84% around a path, this is just going to end up being a voltage. 00:02:03.960 --> 00:02:07.490 align:middle line:84% So it'll be a voltage difference between these two 00:02:07.490 --> 00:02:09.440 align:middle line:84% points-- this point here and this point here. 00:02:09.440 --> 00:02:11.148 align:middle line:84% We're not going to measure it right here. 00:02:11.148 --> 00:02:13.850 align:middle line:84% We'll attach these with cables to some digitizer 00:02:13.850 --> 00:02:15.110 align:middle line:90% or oscilloscope out here. 00:02:15.110 --> 00:02:17.900 align:middle line:84% So there'll be a potential difference around this loop. 00:02:17.900 --> 00:02:20.210 align:middle line:84% We'll call that potential difference V. 00:02:20.210 --> 00:02:24.980 align:middle line:84% And that's simply going to be equal to the integral of B-dot 00:02:24.980 --> 00:02:28.580 align:middle line:90% dotted with dS here. 00:02:28.580 --> 00:02:31.580 align:middle line:84% We've already said that the area of this surface is simply A. 00:02:31.580 --> 00:02:33.800 align:middle line:84% So we'll just write that as A. And then 00:02:33.800 --> 00:02:37.628 align:middle line:90% we'll write this as B-dot here. 00:02:37.628 --> 00:02:39.420 align:middle line:84% And if you want to increase the voltage you 00:02:39.420 --> 00:02:43.200 align:middle line:84% get for a given time-changing magnetic field, 00:02:43.200 --> 00:02:44.970 align:middle line:90% you can put more loops on this. 00:02:44.970 --> 00:02:47.640 align:middle line:84% So your probe could have one loop, 00:02:47.640 --> 00:02:49.020 align:middle line:84% or it could have two loops, or it 00:02:49.020 --> 00:02:50.790 align:middle line:90% could have n loops, like this. 00:02:50.790 --> 00:02:54.540 align:middle line:84% And we would simply multiply this by n at the end 00:02:54.540 --> 00:02:57.318 align:middle line:84% there, where n is just the number of loops we have. 00:02:57.318 --> 00:02:59.110 align:middle line:84% So that's how you make this more sensitive. 00:02:59.110 --> 00:03:03.090 align:middle line:84% And so because of this dependency on B-dot here, 00:03:03.090 --> 00:03:08.830 align:middle line:84% these are called B-dots, very inventive. 00:03:08.830 --> 00:03:09.760 align:middle line:90% OK. 00:03:09.760 --> 00:03:11.830 align:middle line:84% So this is a very simple diagnostic. 00:03:11.830 --> 00:03:13.420 align:middle line:90% It senses magnetic fields. 00:03:13.420 --> 00:03:16.030 align:middle line:84% Now, I've made a few approximations or assumptions 00:03:16.030 --> 00:03:17.680 align:middle line:90% while I've been deriving this. 00:03:17.680 --> 00:03:22.340 align:middle line:84% Can someone give me a reasonable condition on the magnetic field, 00:03:22.340 --> 00:03:25.240 align:middle line:84% particularly thinking about its spatial variation, that 00:03:25.240 --> 00:03:28.780 align:middle line:84% enables me to derive this simple equation? 00:03:28.780 --> 00:03:30.218 align:middle line:90% Yeah? 00:03:30.218 --> 00:03:31.186 align:middle line:90% STUDENT: [INAUDIBLE]? 00:03:31.186 --> 00:03:34.090 align:middle line:90% 00:03:34.090 --> 00:03:35.380 align:middle line:90% JACK HARE: Yeah, OK. 00:03:35.380 --> 00:03:38.620 align:middle line:84% And so we might have some length scale over which 00:03:38.620 --> 00:03:40.160 align:middle line:90% this magnetic field varies. 00:03:40.160 --> 00:03:42.040 align:middle line:90% So we call this LB. 00:03:42.040 --> 00:03:45.880 align:middle line:84% And we could write this as B over grad B. 00:03:45.880 --> 00:03:47.973 align:middle line:84% And you'll excuse my sloppy notation here. 00:03:47.973 --> 00:03:50.390 align:middle line:84% I've sort of dropped the fact that these are vectors here. 00:03:50.390 --> 00:03:52.600 align:middle line:84% And we have some length scale associated 00:03:52.600 --> 00:03:53.800 align:middle line:90% with the size of the probe. 00:03:53.800 --> 00:03:55.330 align:middle line:90% So we can call this LP. 00:03:55.330 --> 00:03:58.420 align:middle line:84% And that's going to go like square root of A. 00:03:58.420 --> 00:04:00.670 align:middle line:84% And so we want to make sure the magnetic field doesn't 00:04:00.670 --> 00:04:03.230 align:middle line:84% change much over this length scale here. 00:04:03.230 --> 00:04:08.910 align:middle line:84% So we might demand that LP over LP is much, much less than 1 00:04:08.910 --> 00:04:09.410 align:middle line:90% here. 00:04:09.410 --> 00:04:12.280 align:middle line:84% So effectively, your magnetic field is only changing in time. 00:04:12.280 --> 00:04:13.710 align:middle line:90% It's not changing spatially. 00:04:13.710 --> 00:04:15.370 align:middle line:84% Of course, if it is changing spatially 00:04:15.370 --> 00:04:16.745 align:middle line:84% you will still measure something. 00:04:16.745 --> 00:04:18.050 align:middle line:90% You'll still get a voltage out. 00:04:18.050 --> 00:04:20.550 align:middle line:84% It's just that your interpretation won't be quite as 00:04:20.550 --> 00:04:21.550 align:middle line:90% straightforward as this. 00:04:21.550 --> 00:04:24.010 align:middle line:84% If you know something about how the magnetic field varies 00:04:24.010 --> 00:04:26.380 align:middle line:84% spatially over your loop, then that's OK. 00:04:26.380 --> 00:04:29.690 align:middle line:84% But if you don't, it's going to make your interpretation very, 00:04:29.690 --> 00:04:30.640 align:middle line:90% very tricky. 00:04:30.640 --> 00:04:32.500 align:middle line:90% OK, any questions on this? 00:04:32.500 --> 00:04:35.260 align:middle line:90% 00:04:35.260 --> 00:04:36.804 align:middle line:90% Yeah? 00:04:36.804 --> 00:04:39.037 align:middle line:90% STUDENT: [INAUDIBLE]? 00:04:39.037 --> 00:04:39.620 align:middle line:90% JACK HARE: OK. 00:04:39.620 --> 00:04:41.600 align:middle line:84% So these are two characteristic length scales 00:04:41.600 --> 00:04:42.960 align:middle line:90% that we're comparing here. 00:04:42.960 --> 00:04:45.145 align:middle line:84% LP is a characteristic length scale of our probe. 00:04:45.145 --> 00:04:47.270 align:middle line:84% It's just going to go as a square root of the area. 00:04:47.270 --> 00:04:49.790 align:middle line:84% And LB is the characteristic length scale 00:04:49.790 --> 00:04:51.380 align:middle line:84% over which the magnetic field varies. 00:04:51.380 --> 00:04:52.880 align:middle line:84% If you look at this dimensionally, 00:04:52.880 --> 00:04:55.838 align:middle line:84% you see that it's going to have dimensions of length. 00:04:55.838 --> 00:04:58.130 align:middle line:84% And so we're just saying that our magnetic field length 00:04:58.130 --> 00:05:00.800 align:middle line:84% scale has to be much larger than our probe length scale 00:05:00.800 --> 00:05:03.710 align:middle line:84% such that the magnetic field is basically 00:05:03.710 --> 00:05:07.660 align:middle line:90% constant across the probe. 00:05:07.660 --> 00:05:10.310 align:middle line:90% Anyone else? 00:05:10.310 --> 00:05:11.185 align:middle line:90% Anyone from Columbia? 00:05:11.185 --> 00:05:14.080 align:middle line:90% 00:05:14.080 --> 00:05:16.002 align:middle line:90% OK, good stuff. 00:05:16.002 --> 00:05:17.960 align:middle line:84% So of course, now, you've got your voltage out. 00:05:17.960 --> 00:05:20.090 align:middle line:84% And you've digitized it with your oscilloscope. 00:05:20.090 --> 00:05:22.070 align:middle line:84% Or at least if you haven't digitized it, 00:05:22.070 --> 00:05:23.900 align:middle line:84% it's coming out these little cables here. 00:05:23.900 --> 00:05:26.960 align:middle line:84% And of course, you need to do something with it to get B-dot. 00:05:26.960 --> 00:05:28.700 align:middle line:90% You need to integrate it, right? 00:05:28.700 --> 00:05:32.192 align:middle line:84% So you have got an equation V equals A B-dot n. 00:05:32.192 --> 00:05:33.650 align:middle line:84% And you want an equation that looks 00:05:33.650 --> 00:05:38.930 align:middle line:90% like B-dot equals V over An. 00:05:38.930 --> 00:05:43.760 align:middle line:84% And so therefore, B is going to be the integral with respect 00:05:43.760 --> 00:05:49.520 align:middle line:84% to time of B over An like this from time 0 00:05:49.520 --> 00:05:53.240 align:middle line:84% up to whatever time T you're at at that point. 00:05:53.240 --> 00:05:53.810 align:middle line:90% OK. 00:05:53.810 --> 00:05:55.310 align:middle line:84% And there are a few ways to do this. 00:05:55.310 --> 00:05:59.810 align:middle line:84% Back in the day, you might want to try and do this automatically 00:05:59.810 --> 00:06:01.730 align:middle line:84% because doing this numerically may 00:06:01.730 --> 00:06:03.557 align:middle line:90% be computationally expensive. 00:06:03.557 --> 00:06:04.640 align:middle line:90% You can do this passively. 00:06:04.640 --> 00:06:07.790 align:middle line:84% You could have some sort of RC type filter. 00:06:07.790 --> 00:06:10.730 align:middle line:84% Those of you who know your analog electronics 00:06:10.730 --> 00:06:13.310 align:middle line:84% would recognize something like this, 00:06:13.310 --> 00:06:15.440 align:middle line:84% where we've got our B-dot loop at the end there. 00:06:15.440 --> 00:06:18.290 align:middle line:84% And we've got some resistance and some capacitance. 00:06:18.290 --> 00:06:20.242 align:middle line:84% And this forms a low pass filter which, 00:06:20.242 --> 00:06:21.950 align:middle line:84% depending on the frequency of your signal 00:06:21.950 --> 00:06:23.760 align:middle line:90% will passively integrate it up. 00:06:23.760 --> 00:06:25.220 align:middle line:90% So that's very simple to build. 00:06:25.220 --> 00:06:27.530 align:middle line:84% If you're trying to put 1,000 B-dot probes around 00:06:27.530 --> 00:06:31.010 align:middle line:84% your tokamak, you might think that this is a nice solution. 00:06:31.010 --> 00:06:33.360 align:middle line:84% If you want something a little bit more fancy, 00:06:33.360 --> 00:06:34.850 align:middle line:90% you can do this with an op amp. 00:06:34.850 --> 00:06:36.920 align:middle line:90% So we can put an op amp in here. 00:06:36.920 --> 00:06:39.710 align:middle line:84% And we can put in something where 00:06:39.710 --> 00:06:42.810 align:middle line:84% we have a resistor and a capacitor for each 00:06:42.810 --> 00:06:43.650 align:middle line:90% of the inputs. 00:06:43.650 --> 00:06:49.388 align:middle line:90% 00:06:49.388 --> 00:06:50.930 align:middle line:84% And this is nicer because, of course, 00:06:50.930 --> 00:06:52.340 align:middle line:84% it's going to buffer your signal. 00:06:52.340 --> 00:06:54.715 align:middle line:84% As well, it makes it easier to do the impedance matching. 00:06:54.715 --> 00:06:57.900 align:middle line:84% So you might want to consider an active system like this. 00:06:57.900 --> 00:07:01.220 align:middle line:84% So just to be clear, this is passive. 00:07:01.220 --> 00:07:05.440 align:middle line:84% It has no electrical components that require power. 00:07:05.440 --> 00:07:09.330 align:middle line:90% And this is an active system. 00:07:09.330 --> 00:07:12.090 align:middle line:84% And of course, the third way you can do this, especially 00:07:12.090 --> 00:07:15.420 align:middle line:84% these days, is simply take this signal and do it digitally. 00:07:15.420 --> 00:07:22.770 align:middle line:84% So we can just sum up V over An times delta T, where delta T is 00:07:22.770 --> 00:07:24.210 align:middle line:90% whatever your time base is. 00:07:24.210 --> 00:07:30.415 align:middle line:90% 00:07:30.415 --> 00:07:32.290 align:middle line:84% Now, you have to be a little bit careful here 00:07:32.290 --> 00:07:37.180 align:middle line:84% because there are other things which can induce a voltage. 00:07:37.180 --> 00:07:38.765 align:middle line:90% Oh, look, it's still there. 00:07:38.765 --> 00:07:40.390 align:middle line:84% There are other things which can induce 00:07:40.390 --> 00:07:44.707 align:middle line:84% a voltage on your probe other than just the magnetic field. 00:07:44.707 --> 00:07:46.540 align:middle line:84% But you only measure the voltage at the end. 00:07:46.540 --> 00:07:48.665 align:middle line:84% So you don't really know if something else has come 00:07:48.665 --> 00:07:49.292 align:middle line:90% and interfered. 00:07:49.292 --> 00:07:51.500 align:middle line:84% But there are some tricks to try and get around that. 00:07:51.500 --> 00:07:53.530 align:middle line:84% So let's have a chat about some other voltages 00:07:53.530 --> 00:07:56.650 align:middle line:84% that might crop up on your probe that you're not aware of. 00:07:56.650 --> 00:08:00.190 align:middle line:84% You could, for example, have some stray capacitively 00:08:00.190 --> 00:08:02.050 align:middle line:90% coupled voltage on here. 00:08:02.050 --> 00:08:06.520 align:middle line:84% So you may end up having a system where 00:08:06.520 --> 00:08:10.420 align:middle line:84% you have a loop which has a voltage on it, a B-dot, 00:08:10.420 --> 00:08:14.440 align:middle line:84% as you expect, but there is also some other capacitively 00:08:14.440 --> 00:08:16.090 align:middle line:90% coupled signal, like noise. 00:08:16.090 --> 00:08:20.900 align:middle line:84% So your signal would just look like this. 00:08:20.900 --> 00:08:23.390 align:middle line:84% But on top of it, you've got some other noisy 00:08:23.390 --> 00:08:25.900 align:middle line:90% signal that corrupts it. 00:08:25.900 --> 00:08:28.620 align:middle line:84% And as you start integrating this up, 00:08:28.620 --> 00:08:31.153 align:middle line:84% it's going to make a bit of a mess of your integration. 00:08:31.153 --> 00:08:33.570 align:middle line:84% If this is truly just noise, if it's randomly distributed, 00:08:33.570 --> 00:08:35.872 align:middle line:84% the integration is obviously going to get rid of that. 00:08:35.872 --> 00:08:37.539 align:middle line:84% It's very nice for smoothing things out. 00:08:37.539 --> 00:08:39.480 align:middle line:84% But if this has some structure-- for example, 00:08:39.480 --> 00:08:41.700 align:middle line:84% it's related to potential inside your plasma-- 00:08:41.700 --> 00:08:43.750 align:middle line:84% then this is going to screw up your measurements. 00:08:43.750 --> 00:08:46.510 align:middle line:84% So the way we get around this is we perform a differential 00:08:46.510 --> 00:08:47.010 align:middle line:90% measurement. 00:08:47.010 --> 00:08:49.260 align:middle line:90% We have two B-dots. 00:08:49.260 --> 00:08:53.550 align:middle line:84% One of the B-dots is wound in one direction, like this. 00:08:53.550 --> 00:08:56.460 align:middle line:84% And the other B-dot, which we place as close as possible, 00:08:56.460 --> 00:09:01.900 align:middle line:84% is wound in the other direction, like this. 00:09:01.900 --> 00:09:06.870 align:middle line:84% So the sense in which we evaluate this dot dL integral 00:09:06.870 --> 00:09:08.580 align:middle line:90% is in opposite directions. 00:09:08.580 --> 00:09:11.310 align:middle line:90% 00:09:11.310 --> 00:09:16.250 align:middle line:84% So for probe 1 and probe 2, we get different voltages out. 00:09:16.250 --> 00:09:18.980 align:middle line:84% For probe 1, we're going to get a voltage that 00:09:18.980 --> 00:09:26.390 align:middle line:84% looks like V1 equals A B-dot plus this capacitively coupled 00:09:26.390 --> 00:09:27.350 align:middle line:90% voltage. 00:09:27.350 --> 00:09:31.590 align:middle line:84% But for 2, the sense in which the L goes is opposite. 00:09:31.590 --> 00:09:38.390 align:middle line:84% So we get V2 equals minus A B-dot plus some capacitively 00:09:38.390 --> 00:09:39.398 align:middle line:90% coupled voltage. 00:09:39.398 --> 00:09:40.940 align:middle line:84% And so we can do a trick that we like 00:09:40.940 --> 00:09:43.910 align:middle line:84% doing in experimental physics, which is to make a differential 00:09:43.910 --> 00:09:44.823 align:middle line:90% measurement. 00:09:44.823 --> 00:09:46.490 align:middle line:84% And we can take the difference of these. 00:09:46.490 --> 00:09:51.063 align:middle line:84% We can take the difference of V1 minus V2. 00:09:51.063 --> 00:09:52.980 align:middle line:84% And you'll see that these capacitively coupled 00:09:52.980 --> 00:09:55.890 align:middle line:84% voltages, which are the same on both loops, will cancel out. 00:09:55.890 --> 00:09:58.840 align:middle line:84% But we will amplify up the signal that we actually want. 00:09:58.840 --> 00:10:04.580 align:middle line:84% And so we'll end up with 2 A B-dot, like this. 00:10:04.580 --> 00:10:08.360 align:middle line:84% So this is, as I said, a differential measurement. 00:10:08.360 --> 00:10:11.690 align:middle line:84% And whenever possible, if you're trying to get rid of noise, 00:10:11.690 --> 00:10:14.780 align:middle line:84% you want to try and do things like differential measurements. 00:10:14.780 --> 00:10:20.290 align:middle line:90% 00:10:20.290 --> 00:10:22.890 align:middle line:84% So if you're asking where does this capacity coupled signal 00:10:22.890 --> 00:10:25.080 align:middle line:84% come from, if you just imagine having 00:10:25.080 --> 00:10:30.330 align:middle line:84% your B-dot probe like this and there's some plasma floating 00:10:30.330 --> 00:10:36.250 align:middle line:84% over here, this plasma may be at some potential B plasma, 00:10:36.250 --> 00:10:37.300 align:middle line:90% like this. 00:10:37.300 --> 00:10:40.540 align:middle line:84% And your probe may be attached near ground. 00:10:40.540 --> 00:10:43.600 align:middle line:84% So it's floating at close to V equals 0. 00:10:43.600 --> 00:10:45.448 align:middle line:84% But you can see effectively there's 00:10:45.448 --> 00:10:46.990 align:middle line:84% going to be some capacitive coupling. 00:10:46.990 --> 00:10:48.580 align:middle line:84% If I drew a circuit diagram, it'll 00:10:48.580 --> 00:10:51.610 align:middle line:84% be as if there was some capacitor between these two. 00:10:51.610 --> 00:10:54.760 align:middle line:84% And that would be enough to induce some small fraction 00:10:54.760 --> 00:10:57.730 align:middle line:84% VC on here, which is going to be much, much less 00:10:57.730 --> 00:10:59.110 align:middle line:90% than the plasma potential. 00:10:59.110 --> 00:11:01.660 align:middle line:84% But the plasma potential could be a kilovolt or so 00:11:01.660 --> 00:11:04.630 align:middle line:84% if you're dealing with electron volt temperature plasmas. 00:11:04.630 --> 00:11:06.672 align:middle line:84% And so this VC can still be large enough 00:11:06.672 --> 00:11:08.380 align:middle line:84% that it will throw off your measurements. 00:11:08.380 --> 00:11:11.440 align:middle line:84% This is why it's worth doing differential measurements 00:11:11.440 --> 00:11:13.790 align:middle line:90% if you can. 00:11:13.790 --> 00:11:14.800 align:middle line:90% Any questions on that? 00:11:14.800 --> 00:11:16.013 align:middle line:90% Yes? 00:11:16.013 --> 00:11:17.985 align:middle line:84% STUDENT: What exactly [INAUDIBLE]?? 00:11:17.985 --> 00:11:22.738 align:middle line:90% 00:11:22.738 --> 00:11:24.280 align:middle line:84% JACK HARE: Yeah, that's a good point. 00:11:24.280 --> 00:11:25.050 align:middle line:90% Yes, it is. 00:11:25.050 --> 00:11:26.780 align:middle line:90% STUDENT: That ever [INAUDIBLE]. 00:11:26.780 --> 00:11:32.218 align:middle line:90% 00:11:32.218 --> 00:11:34.510 align:middle line:84% JACK HARE: Yeah, we'll talk about that more in a minute 00:11:34.510 --> 00:11:36.580 align:middle line:84% because if you've got very slowly changing magnetic fields 00:11:36.580 --> 00:11:39.013 align:middle line:84% where this is an issue, like on a superconducting tokamak, 00:11:39.013 --> 00:11:41.680 align:middle line:84% B-dots aren't really going to do any good because your signal is 00:11:41.680 --> 00:11:44.600 align:middle line:84% going to be absolutely tiny because B-dot will be very, very 00:11:44.600 --> 00:11:45.100 align:middle line:90% small. 00:11:45.100 --> 00:11:46.660 align:middle line:84% So your voltage will be very, very small. 00:11:46.660 --> 00:11:48.553 align:middle line:84% So you probably want to rely on other sensors 00:11:48.553 --> 00:11:49.345 align:middle line:90% to do that instead. 00:11:49.345 --> 00:11:52.510 align:middle line:90% 00:11:52.510 --> 00:11:55.570 align:middle line:84% Right, this really should read like delta B. 00:11:55.570 --> 00:11:57.140 align:middle line:84% This is the change in magnetic field. 00:11:57.140 --> 00:12:02.220 align:middle line:84% So you can't measure a DC magnetic field with these. 00:12:02.220 --> 00:12:04.110 align:middle line:84% Yeah, and there are lots of problems when you 00:12:04.110 --> 00:12:05.430 align:middle line:90% do this practically with drift. 00:12:05.430 --> 00:12:07.240 align:middle line:84% So if your oscilloscope isn't perfectly calibrated 00:12:07.240 --> 00:12:09.060 align:middle line:84% and it's got some slight offset to it, 00:12:09.060 --> 00:12:12.420 align:middle line:84% that's going to be integrated up as a gently ramping up 00:12:12.420 --> 00:12:14.490 align:middle line:84% magnetic field, which is another good reason 00:12:14.490 --> 00:12:16.890 align:middle line:84% to do differential measurements because you can get 00:12:16.890 --> 00:12:18.907 align:middle line:90% rid of those offsets as well. 00:12:18.907 --> 00:12:19.740 align:middle line:90% Any other questions? 00:12:19.740 --> 00:12:20.578 align:middle line:90% Yeah? 00:12:20.578 --> 00:12:22.678 align:middle line:90% STUDENT: [INAUDIBLE]? 00:12:22.678 --> 00:12:23.720 align:middle line:90% JACK HARE: Oh, well done. 00:12:23.720 --> 00:12:24.870 align:middle line:90% Yes, that's a good point. 00:12:24.870 --> 00:12:26.240 align:middle line:90% I should have mentioned it. 00:12:26.240 --> 00:12:29.360 align:middle line:84% I've assumed here that the magnetic field is perpendicular 00:12:29.360 --> 00:12:31.460 align:middle line:90% when I drew this diagram. 00:12:31.460 --> 00:12:35.130 align:middle line:84% It's parallel to the normal of this surface here. 00:12:35.130 --> 00:12:36.837 align:middle line:84% So that's where we have B-dot DS. 00:12:36.837 --> 00:12:38.420 align:middle line:84% Of course, the magnetic field could be 00:12:38.420 --> 00:12:40.100 align:middle line:90% at some arbitrary angle to it. 00:12:40.100 --> 00:12:42.830 align:middle line:84% We're only going to pick up the component which 00:12:42.830 --> 00:12:44.700 align:middle line:90% is normal to the surface. 00:12:44.700 --> 00:12:47.120 align:middle line:84% So this is insensitive to magnetic fields, for example, 00:12:47.120 --> 00:12:48.860 align:middle line:90% pointing in this direction. 00:12:48.860 --> 00:12:51.560 align:middle line:84% This is actually quite important in the tokamak where-- 00:12:51.560 --> 00:12:54.260 align:middle line:84% or in other devices, where you may not actually know what 00:12:54.260 --> 00:12:55.520 align:middle line:90% the magnetic field is. 00:12:55.520 --> 00:12:56.900 align:middle line:84% We know what the vacuum magnetic field is. 00:12:56.900 --> 00:12:58.858 align:middle line:84% But once there's current flowing in the plasma, 00:12:58.858 --> 00:13:01.140 align:middle line:84% we may not know which component we're picking up. 00:13:01.140 --> 00:13:02.660 align:middle line:84% So you may also want to have not, 00:13:02.660 --> 00:13:05.540 align:middle line:84% for example, just two loops for this measurement, but three axis 00:13:05.540 --> 00:13:06.690 align:middle line:90% B-dot probes as well. 00:13:06.690 --> 00:13:08.780 align:middle line:84% And so on laboratory astrophysics devices 00:13:08.780 --> 00:13:11.360 align:middle line:84% like LEDD and MRX, they have these arrays 00:13:11.360 --> 00:13:12.870 align:middle line:84% of probes with three different axes, 00:13:12.870 --> 00:13:15.670 align:middle line:84% so they can measure the full vector magnetic field. 00:13:15.670 --> 00:13:16.643 align:middle line:90% Yeah? 00:13:16.643 --> 00:13:19.905 align:middle line:90% STUDENT: [INAUDIBLE]? 00:13:19.905 --> 00:13:21.280 align:middle line:84% JACK HARE: Yeah, so the idea here 00:13:21.280 --> 00:13:23.090 align:middle line:84% is this could be due to anything. 00:13:23.090 --> 00:13:25.378 align:middle line:84% This is due to some high voltage being present 00:13:25.378 --> 00:13:26.420 align:middle line:90% somewhere in your system. 00:13:26.420 --> 00:13:28.780 align:middle line:84% So it could be a capacitor bank discharging. 00:13:28.780 --> 00:13:31.540 align:middle line:84% Or it could be a killer electron volt plasma. 00:13:31.540 --> 00:13:33.280 align:middle line:84% But the fact is that any conductor 00:13:33.280 --> 00:13:36.160 align:middle line:84% and any other conductor have a capacitance between them 00:13:36.160 --> 00:13:37.030 align:middle line:90% just always. 00:13:37.030 --> 00:13:39.970 align:middle line:84% There is an ability to induce some potential on one conductor 00:13:39.970 --> 00:13:40.730 align:middle line:90% from the other. 00:13:40.730 --> 00:13:43.880 align:middle line:84% And so all I'm saying is if you have some stray potential, 00:13:43.880 --> 00:13:47.350 align:middle line:84% it will couple with the same sign to these two probes. 00:13:47.350 --> 00:13:49.030 align:middle line:84% But because they're oppositely wound, 00:13:49.030 --> 00:13:52.442 align:middle line:84% we'll be able to cancel out that same sign potential there. 00:13:52.442 --> 00:13:54.400 align:middle line:84% That's the other reason why you want the probes 00:13:54.400 --> 00:13:56.890 align:middle line:84% as close together as possible because if you put them 00:13:56.890 --> 00:13:59.788 align:middle line:84% further apart, this VC may be different on the two probes. 00:13:59.788 --> 00:14:01.330 align:middle line:84% And then they won't cancel perfectly. 00:14:01.330 --> 00:14:04.000 align:middle line:84% So ideally, you'd have two loops perfectly counter 00:14:04.000 --> 00:14:07.013 align:middle line:84% wound lying directly on top of each other, yeah. 00:14:07.013 --> 00:14:08.180 align:middle line:90% Any questions from Colombia? 00:14:08.180 --> 00:14:08.847 align:middle line:90% I see two hands. 00:14:08.847 --> 00:14:11.480 align:middle line:90% 00:14:11.480 --> 00:14:12.470 align:middle line:90% Maybe Nigel first? 00:14:12.470 --> 00:14:15.130 align:middle line:90% 00:14:15.130 --> 00:14:19.390 align:middle line:84% STUDENT: Hi, could you remember to repeat questions 00:14:19.390 --> 00:14:20.920 align:middle line:90% when they're asked in person? 00:14:20.920 --> 00:14:22.045 align:middle line:90% JACK HARE: Yeah, thank you. 00:14:22.045 --> 00:14:23.520 align:middle line:90% I forgot about that. 00:14:23.520 --> 00:14:25.270 align:middle line:84% Yeah, was that Matthew's question as well? 00:14:25.270 --> 00:14:26.440 align:middle line:90% Or is it a different question? 00:14:26.440 --> 00:14:28.120 align:middle line:84% STUDENT: No, I have a separate question. 00:14:28.120 --> 00:14:31.350 align:middle line:84% Nigel, did you have anything else? 00:14:31.350 --> 00:14:32.277 align:middle line:90% STUDENT: No. 00:14:32.277 --> 00:14:33.950 align:middle line:84% STUDENT: OK, so do you have to worry 00:14:33.950 --> 00:14:36.320 align:middle line:84% about inductive coupling between two 00:14:36.320 --> 00:14:40.810 align:middle line:84% oppositely wound up probes that are very closely spaced? 00:14:40.810 --> 00:14:42.310 align:middle line:84% JACK HARE: The brief answer is you'd 00:14:42.310 --> 00:14:44.320 align:middle line:84% have to worry about that if the current flowing 00:14:44.320 --> 00:14:46.112 align:middle line:84% through the probes was sufficient to create 00:14:46.112 --> 00:14:48.790 align:middle line:84% a magnetic field that was comparable to the magnetic 00:14:48.790 --> 00:14:51.110 align:middle line:90% field, you're trying to measure. 00:14:51.110 --> 00:14:52.610 align:middle line:90% So in general, it's not. 00:14:52.610 --> 00:14:54.440 align:middle line:90% Yeah, it's a good question. 00:14:54.440 --> 00:14:58.610 align:middle line:84% But say you've got a volt or so that you're measuring. 00:14:58.610 --> 00:15:03.390 align:middle line:84% And you're measuring it over a 50 ohm terminator, 00:15:03.390 --> 00:15:06.170 align:middle line:84% which is what we normally do for the matching with 50 ohm cables. 00:15:06.170 --> 00:15:07.920 align:middle line:84% That's going to be a pretty small current. 00:15:07.920 --> 00:15:09.670 align:middle line:84% So the magnetic field you're inducing is pretty small. 00:15:09.670 --> 00:15:10.600 align:middle line:90% It's a good question. 00:15:10.600 --> 00:15:11.100 align:middle line:90% STUDENT: OK. 00:15:11.100 --> 00:15:12.660 align:middle line:90% All right, thanks. 00:15:12.660 --> 00:15:14.179 align:middle line:84% JACK HARE: Yeah, another question? 00:15:14.179 --> 00:15:15.586 align:middle line:90% STUDENT: [INAUDIBLE]? 00:15:15.586 --> 00:15:22.483 align:middle line:90% 00:15:22.483 --> 00:15:23.900 align:middle line:84% JACK HARE: Right, if you integrate 00:15:23.900 --> 00:15:25.910 align:middle line:84% a zero mean Gaussian noise, then you're right. 00:15:25.910 --> 00:15:27.890 align:middle line:90% You will get rid of VC. 00:15:27.890 --> 00:15:30.415 align:middle line:84% But it doesn't have to be a zero mean Gaussian noise. 00:15:30.415 --> 00:15:32.540 align:middle line:84% This VC, in fact, will probably represent something 00:15:32.540 --> 00:15:35.040 align:middle line:84% like-- say you're doing a discharge in a plasma. 00:15:35.040 --> 00:15:37.520 align:middle line:84% The voltage will rise and fall in a similar way 00:15:37.520 --> 00:15:39.240 align:middle line:84% to the current in a magnetic field. 00:15:39.240 --> 00:15:41.540 align:middle line:84% So you might actually have a system 00:15:41.540 --> 00:15:57.250 align:middle line:84% where, let's say, this is your signal B-dot, like this. 00:15:57.250 --> 00:16:00.940 align:middle line:84% And then your signal is doing something like-- 00:16:00.940 --> 00:16:03.740 align:middle line:90% let me draw it-- 00:16:03.740 --> 00:16:04.500 align:middle line:90% like this. 00:16:04.500 --> 00:16:06.500 align:middle line:84% So this is a capacitor bank ramping up 00:16:06.500 --> 00:16:07.617 align:middle line:90% and discharging again. 00:16:07.617 --> 00:16:09.450 align:middle line:84% And so you can see, if you average that out, 00:16:09.450 --> 00:16:10.867 align:middle line:84% you're not going to get rid of it. 00:16:10.867 --> 00:16:13.760 align:middle line:84% So this is for non-random one, yeah. 00:16:13.760 --> 00:16:17.892 align:middle line:90% STUDENT: I felt like [INAUDIBLE] 00:16:17.892 --> 00:16:19.359 align:middle line:90% 00:16:19.359 --> 00:16:22.018 align:middle line:84% JACK HARE: Oh, I guess you're right, yeah. 00:16:22.018 --> 00:16:23.810 align:middle line:84% It's going to be relatively small depending 00:16:23.810 --> 00:16:25.700 align:middle line:84% on how big your signal to noise ratio is. 00:16:25.700 --> 00:16:27.450 align:middle line:84% But you're right, it's like a random walk. 00:16:27.450 --> 00:16:31.230 align:middle line:84% So you're going to diffuse away from where you start, yeah. 00:16:31.230 --> 00:16:32.930 align:middle line:84% Yeah, I guess, as with all measurements, 00:16:32.930 --> 00:16:35.245 align:middle line:84% you want to have a good signal to noise to avoid that. 00:16:35.245 --> 00:16:36.520 align:middle line:90% STUDENT: [INAUDIBLE]? 00:16:36.520 --> 00:16:37.660 align:middle line:90% JACK HARE: It depends. 00:16:37.660 --> 00:16:39.660 align:middle line:84% You should design as a diagnostic designer 00:16:39.660 --> 00:16:41.410 align:middle line:84% a system which gives you sufficient signal 00:16:41.410 --> 00:16:43.540 align:middle line:84% to noise, which might mean increasing 00:16:43.540 --> 00:16:44.740 align:middle line:90% the area of your loop. 00:16:44.740 --> 00:16:47.020 align:middle line:84% Or it might mean increasing the number of terms 00:16:47.020 --> 00:16:50.450 align:middle line:90% in order to fulfill that. 00:16:50.450 --> 00:16:52.170 align:middle line:90% OK, I'm going to move on. 00:16:52.170 --> 00:16:55.040 align:middle line:84% So just to say, you can take these two signals here 00:16:55.040 --> 00:16:56.870 align:middle line:84% and you could feed them in automatically 00:16:56.870 --> 00:16:59.660 align:middle line:90% to an op amp similar to this. 00:16:59.660 --> 00:17:03.297 align:middle line:84% And you can make an op amp circuit that does subtraction. 00:17:03.297 --> 00:17:05.089 align:middle line:84% And then you could feed that into an op amp 00:17:05.089 --> 00:17:06.770 align:middle line:90% circuit which does integration. 00:17:06.770 --> 00:17:09.800 align:middle line:84% The trouble is, unless you inspect these two signals 00:17:09.800 --> 00:17:11.720 align:middle line:84% visually, you may find that one of them 00:17:11.720 --> 00:17:13.595 align:middle line:84% is more affected by the capacitive coupling 00:17:13.595 --> 00:17:14.220 align:middle line:90% than the other. 00:17:14.220 --> 00:17:16.178 align:middle line:84% This is a problem we have in my research, where 00:17:16.178 --> 00:17:19.880 align:middle line:84% one of your signals might look really nice, like this. 00:17:19.880 --> 00:17:22.410 align:middle line:84% And the other signal just goes like that. 00:17:22.410 --> 00:17:24.530 align:middle line:84% And if you feed these automatically into an op amp 00:17:24.530 --> 00:17:26.599 align:middle line:84% without digitizing them and just accept 00:17:26.599 --> 00:17:28.040 align:middle line:84% whatever comes out the other end, 00:17:28.040 --> 00:17:31.655 align:middle line:84% then you're not going to get a decent answer from this. 00:17:31.655 --> 00:17:32.930 align:middle line:90% This data is junk. 00:17:32.930 --> 00:17:36.890 align:middle line:84% You want your signals to look as close to opposite as possible. 00:17:36.890 --> 00:17:39.930 align:middle line:84% So probably you want to digitize them separately. 00:17:39.930 --> 00:17:42.690 align:middle line:84% Anyway, we're going to be focusing 00:17:42.690 --> 00:17:44.250 align:middle line:84% at the moment on B-dots which are 00:17:44.250 --> 00:17:45.810 align:middle line:90% placed outside of the plasma. 00:17:45.810 --> 00:17:48.150 align:middle line:84% I will talk about B-dots placed inside of the plasma 00:17:48.150 --> 00:17:50.610 align:middle line:84% because that's something that I like to do in my research. 00:17:50.610 --> 00:17:52.027 align:middle line:84% But for most of the time, if we're 00:17:52.027 --> 00:17:54.318 align:middle line:84% dealing with things like tokamaks, magnetic confinement 00:17:54.318 --> 00:17:56.880 align:middle line:84% fusion devices, you don't want to stick things inside there. 00:17:56.880 --> 00:17:59.070 align:middle line:84% They're just going to melt. So we'll 00:17:59.070 --> 00:18:01.730 align:middle line:84% be sticking with external probes. 00:18:01.730 --> 00:18:02.660 align:middle line:90% OK. 00:18:02.660 --> 00:18:09.990 align:middle line:84% There was a question earlier about measuring 00:18:09.990 --> 00:18:12.420 align:middle line:84% when B-dot is small, when we've got almost steady state 00:18:12.420 --> 00:18:13.210 align:middle line:90% magnetic fields. 00:18:13.210 --> 00:18:15.010 align:middle line:84% So let's have a quick chat about that. 00:18:15.010 --> 00:18:19.755 align:middle line:84% So when we have B-dot goes to 0, we'll call these steady states. 00:18:19.755 --> 00:18:22.317 align:middle line:90% 00:18:22.317 --> 00:18:24.900 align:middle line:84% Of course, they don't have to be absolutely not time changing. 00:18:24.900 --> 00:18:26.610 align:middle line:84% They just have to be time changing 00:18:26.610 --> 00:18:30.520 align:middle line:84% slowly enough that we can't measure the voltage easily. 00:18:30.520 --> 00:18:36.500 align:middle line:84% So this would give V on our B-dot also going to 0. 00:18:36.500 --> 00:18:38.330 align:middle line:84% So at this point, you probably want 00:18:38.330 --> 00:18:40.250 align:middle line:90% to resort to a different device. 00:18:40.250 --> 00:18:42.860 align:middle line:84% And these are often called Hall effect sensors. 00:18:42.860 --> 00:18:45.810 align:middle line:90% 00:18:45.810 --> 00:18:48.650 align:middle line:84% And another name for a Hall effect sensor is a Gauss meter. 00:18:48.650 --> 00:18:54.840 align:middle line:90% 00:18:54.840 --> 00:18:56.930 align:middle line:84% So the way these work is we have a block 00:18:56.930 --> 00:18:59.210 align:middle line:90% of a special semiconductor. 00:18:59.210 --> 00:19:05.020 align:middle line:90% 00:19:05.020 --> 00:19:07.800 align:middle line:84% So this is a terrible drawing of a cuboid. 00:19:07.800 --> 00:19:08.370 align:middle line:90% There we go. 00:19:08.370 --> 00:19:14.210 align:middle line:90% 00:19:14.210 --> 00:19:16.820 align:middle line:84% There's a magnetic field that's threading through this 00:19:16.820 --> 00:19:19.320 align:middle line:90% that we want to sense. 00:19:19.320 --> 00:19:25.410 align:middle line:84% We allow a current to flow through it in this direction, 00:19:25.410 --> 00:19:28.725 align:middle line:84% from this face to this face back here. 00:19:28.725 --> 00:19:32.730 align:middle line:90% 00:19:32.730 --> 00:19:35.420 align:middle line:84% And because we have charged particles 00:19:35.420 --> 00:19:37.250 align:middle line:84% moving in a magnetic field, there's 00:19:37.250 --> 00:19:40.080 align:middle line:84% a J cross B force which separates the charges. 00:19:40.080 --> 00:19:45.010 align:middle line:84% And so on one side, we build up a positive charge. 00:19:45.010 --> 00:19:48.680 align:middle line:84% And on the other side, we build up a negative charge. 00:19:48.680 --> 00:19:57.510 align:middle line:84% And if we attach our oscilloscope here, 00:19:57.510 --> 00:19:59.500 align:middle line:84% we can measure that potential difference. 00:19:59.500 --> 00:20:01.380 align:middle line:84% So there's going to be an electric field 00:20:01.380 --> 00:20:04.920 align:middle line:90% in this direction. 00:20:04.920 --> 00:20:07.670 align:middle line:84% And by knowing the properties of our semiconductor 00:20:07.670 --> 00:20:09.710 align:middle line:84% and measuring this electric field, 00:20:09.710 --> 00:20:13.830 align:middle line:84% we can infer what the magnetic field is here. 00:20:13.830 --> 00:20:18.950 align:middle line:84% Again, there's a J cross B force that 00:20:18.950 --> 00:20:23.270 align:middle line:84% separates the charges in these two directions. 00:20:23.270 --> 00:20:25.250 align:middle line:90% So this works pretty well. 00:20:25.250 --> 00:20:26.380 align:middle line:90% It doesn't work very fast. 00:20:26.380 --> 00:20:28.240 align:middle line:84% So you can't use these replace B-dots. 00:20:28.240 --> 00:20:30.400 align:middle line:84% If you want to have a very high frequency signal, 00:20:30.400 --> 00:20:31.480 align:middle line:90% this doesn't work. 00:20:31.480 --> 00:20:33.220 align:middle line:84% And the other problem is that the plasma 00:20:33.220 --> 00:20:35.267 align:middle line:84% is a very harsh environment, especially if you're 00:20:35.267 --> 00:20:37.600 align:middle line:84% going to a nuclear environment, like the next generation 00:20:37.600 --> 00:20:38.920 align:middle line:90% of tokamaks. 00:20:38.920 --> 00:20:42.975 align:middle line:84% And that causes degradation of this semiconductor. 00:20:42.975 --> 00:20:44.350 align:middle line:84% And then you have to work out how 00:20:44.350 --> 00:20:46.210 align:middle line:84% you're going to calibrate these things in situ. 00:20:46.210 --> 00:20:48.127 align:middle line:84% So you can't just pop them out because they're 00:20:48.127 --> 00:20:49.450 align:middle line:90% hot, radioactively hot. 00:20:49.450 --> 00:20:52.480 align:middle line:84% So these sorts of devices are very good. 00:20:52.480 --> 00:20:56.180 align:middle line:84% But we really need to have rad-hardened versions of them. 00:20:56.180 --> 00:21:02.090 align:middle line:84% And as far as I know, from at least the last time I looked, 00:21:02.090 --> 00:21:03.720 align:middle line:84% I don't think that we have a solution 00:21:03.720 --> 00:21:04.970 align:middle line:90% to this problem at the moment. 00:21:04.970 --> 00:21:07.750 align:middle line:90% But I may be wrong. 00:21:07.750 --> 00:21:09.970 align:middle line:84% Another technique you might want to use 00:21:09.970 --> 00:21:13.255 align:middle line:84% is Faraday rotation or the Faraday effect. 00:21:13.255 --> 00:21:16.972 align:middle line:90% 00:21:16.972 --> 00:21:18.930 align:middle line:84% We'll actually talk about this a bit more later 00:21:18.930 --> 00:21:22.830 align:middle line:84% on in the context of Faraday rotation inside plasmas. 00:21:22.830 --> 00:21:24.960 align:middle line:84% But in fact, the Faraday effect was first 00:21:24.960 --> 00:21:26.918 align:middle line:84% discovered by Faherty, who didn't know anything 00:21:26.918 --> 00:21:27.660 align:middle line:90% about plasmas. 00:21:27.660 --> 00:21:30.978 align:middle line:84% And he was using, well, actually, all sorts of things. 00:21:30.978 --> 00:21:32.520 align:middle line:84% If you go back to Faraday's notebook, 00:21:32.520 --> 00:21:33.640 align:middle line:90% it's kind of remarkable. 00:21:33.640 --> 00:21:36.150 align:middle line:84% He built one of the first large magnets. 00:21:36.150 --> 00:21:39.880 align:middle line:84% He also had an ability to detect the polarization of light. 00:21:39.880 --> 00:21:42.638 align:middle line:84% And he did what any reasonable experimental physicist would do. 00:21:42.638 --> 00:21:44.430 align:middle line:84% He started putting things inside the magnet 00:21:44.430 --> 00:21:46.810 align:middle line:84% and seeing if they rotated the polarization of light. 00:21:46.810 --> 00:21:48.750 align:middle line:84% And so he's got in his notebook milk. 00:21:48.750 --> 00:21:49.710 align:middle line:90% Milk does. 00:21:49.710 --> 00:21:50.700 align:middle line:90% Beef. 00:21:50.700 --> 00:21:53.220 align:middle line:84% Beef doesn't because the light didn't go through. 00:21:53.220 --> 00:21:55.290 align:middle line:90% But fair on him for trying. 00:21:55.290 --> 00:21:55.830 align:middle line:90% It's great. 00:21:55.830 --> 00:21:57.810 align:middle line:84% So yeah, there's all sorts of cool things. 00:21:57.810 --> 00:22:01.030 align:middle line:84% Now, these days, we don't use beef inside most experiments. 00:22:01.030 --> 00:22:04.350 align:middle line:84% What we use is a special type of glass which has-- 00:22:04.350 --> 00:22:07.320 align:middle line:84% it's usually called verdet glass because it's got something 00:22:07.320 --> 00:22:09.180 align:middle line:90% called a verdet constant. 00:22:09.180 --> 00:22:11.470 align:middle line:84% This is just probably some French guy's name. 00:22:11.470 --> 00:22:16.220 align:middle line:84% So if you've got some magnetic fields in this direction 00:22:16.220 --> 00:22:19.920 align:middle line:84% and you pass some linearly polarized light-- 00:22:19.920 --> 00:22:22.700 align:middle line:84% so we'll start off with a polarization, for example, 00:22:22.700 --> 00:22:24.200 align:middle line:90% in this up-down direction. 00:22:24.200 --> 00:22:27.410 align:middle line:84% When the light comes out, that polarization 00:22:27.410 --> 00:22:29.790 align:middle line:90% is going to be rotated. 00:22:29.790 --> 00:22:33.140 align:middle line:84% And the angle that it's rotated by, 00:22:33.140 --> 00:22:38.420 align:middle line:84% beta, is equal to V, which is the verdet constant-- don't get 00:22:38.420 --> 00:22:40.850 align:middle line:90% this confused with voltages. 00:22:40.850 --> 00:22:45.000 align:middle line:84% This is just a constant that is a property of this glass-- 00:22:45.000 --> 00:22:47.640 align:middle line:84% and then times by the magnetic field times 00:22:47.640 --> 00:22:52.640 align:middle line:84% by the length of this lump of glass here. 00:22:52.640 --> 00:22:53.140 align:middle line:90% OK. 00:22:53.140 --> 00:22:55.000 align:middle line:84% And so this means that if you put 00:22:55.000 --> 00:22:57.488 align:middle line:84% a little chunk of this glass inside a fiber optic, 00:22:57.488 --> 00:22:59.530 align:middle line:84% this is extremely convenient because you can just 00:22:59.530 --> 00:23:01.060 align:middle line:90% put that fiber optic somewhere. 00:23:01.060 --> 00:23:03.227 align:middle line:84% You don't have to worry about mirrors for this light 00:23:03.227 --> 00:23:05.570 align:middle line:84% because it's all trapped inside the fiber. 00:23:05.570 --> 00:23:08.110 align:middle line:84% And we have lots of really great technology and fibers 00:23:08.110 --> 00:23:10.780 align:middle line:84% for looking at polarization because the telecommunications 00:23:10.780 --> 00:23:13.030 align:middle line:84% industry uses it for multiplexing. 00:23:13.030 --> 00:23:15.760 align:middle line:84% And so you can sense this polarization. 00:23:15.760 --> 00:23:19.960 align:middle line:84% And that means that you can then infer what the magnetic field is 00:23:19.960 --> 00:23:23.740 align:middle line:84% because you know V. You know L. You measure theta. 00:23:23.740 --> 00:23:27.090 align:middle line:84% And therefore, you get B. Once again, 00:23:27.090 --> 00:23:29.590 align:middle line:84% this is B in the direction of propagation. 00:23:29.590 --> 00:23:32.070 align:middle line:84% So same with the B-dots, it's got that limitation. 00:23:32.070 --> 00:23:33.780 align:middle line:90% You don't get a vector out. 00:23:33.780 --> 00:23:39.470 align:middle line:84% And the other problem is that, again, is rad hardening. 00:23:39.470 --> 00:23:44.040 align:middle line:84% And actually not just rads, as in like neutrons and things 00:23:44.040 --> 00:23:45.907 align:middle line:84% like that, though that will definitely 00:23:45.907 --> 00:23:47.490 align:middle line:84% alter the structure here and therefore 00:23:47.490 --> 00:23:50.460 align:middle line:84% change the verdet constant, but even things like X-rays 00:23:50.460 --> 00:23:51.960 align:middle line:84% can cause blank inside this glass. 00:23:51.960 --> 00:23:54.002 align:middle line:84% So people have tried to use this on the Z-machine 00:23:54.002 --> 00:23:57.630 align:middle line:84% at Sandia, which is the world's most energetic X-ray source. 00:23:57.630 --> 00:24:00.240 align:middle line:84% And they don't work very well because the X-rays get 00:24:00.240 --> 00:24:01.260 align:middle line:90% inside the fibers. 00:24:01.260 --> 00:24:02.440 align:middle line:90% And they blank things out. 00:24:02.440 --> 00:24:05.050 align:middle line:90% So this is another possibility. 00:24:05.050 --> 00:24:08.040 align:middle line:84% But once again, it's something we need a lot more technology 00:24:08.040 --> 00:24:10.710 align:middle line:84% development on before we can use it in the harsh environments we 00:24:10.710 --> 00:24:12.970 align:middle line:90% expect in fusion devices. 00:24:12.970 --> 00:24:15.600 align:middle line:84% So any questions on the Hall effect, gas meters, 00:24:15.600 --> 00:24:17.320 align:middle line:90% or Faraday effect? 00:24:17.320 --> 00:24:18.079 align:middle line:90% Yes? 00:24:18.079 --> 00:24:20.121 align:middle line:84% And I'll try to remember to repeat your question. 00:24:20.121 --> 00:24:21.103 align:middle line:90% STUDENT: [INAUDIBLE]? 00:24:21.103 --> 00:24:30.597 align:middle line:90% 00:24:30.597 --> 00:24:31.180 align:middle line:90% JACK HARE: OK. 00:24:31.180 --> 00:24:33.097 align:middle line:84% So the question was, why can't the Hall effect 00:24:33.097 --> 00:24:35.560 align:middle line:90% be used for fast changes? 00:24:35.560 --> 00:24:38.350 align:middle line:84% Simply because there is some drift velocity associated 00:24:38.350 --> 00:24:39.520 align:middle line:90% with these carriers. 00:24:39.520 --> 00:24:41.930 align:middle line:84% And we can't make them drift fast enough. 00:24:41.930 --> 00:24:43.738 align:middle line:84% So if you have a very rapidly changing V, 00:24:43.738 --> 00:24:45.280 align:middle line:84% they'll all be sloshing to the right. 00:24:45.280 --> 00:24:46.300 align:middle line:84% And then by the time they've got there, 00:24:46.300 --> 00:24:47.842 align:middle line:84% they'll be sloshing back to the left. 00:24:47.842 --> 00:24:49.310 align:middle line:90% And they'll never make it back. 00:24:49.310 --> 00:24:52.060 align:middle line:84% So I don't know off the top of my head what the time scale is. 00:24:52.060 --> 00:24:55.755 align:middle line:84% But it may be in Hutchinson's book. 00:24:55.755 --> 00:24:57.130 align:middle line:84% Or you may be able to look it up. 00:24:57.130 --> 00:24:59.410 align:middle line:84% But my impression is that this is more 00:24:59.410 --> 00:25:02.050 align:middle line:84% for milliseconds, second kind of time scales, 00:25:02.050 --> 00:25:04.540 align:middle line:90% where B-dots can do much faster. 00:25:04.540 --> 00:25:05.710 align:middle line:90% This can be very fast. 00:25:05.710 --> 00:25:09.910 align:middle line:84% Faraday effect sensing can be as fast as your digitizer. 00:25:09.910 --> 00:25:13.630 align:middle line:84% So one can get a 50ghz digitizer these days, 00:25:13.630 --> 00:25:14.840 align:middle line:90% if you have a lot of money. 00:25:14.840 --> 00:25:18.000 align:middle line:90% So you can send very, very fast. 00:25:18.000 --> 00:25:19.117 align:middle line:90% And yeah? 00:25:19.117 --> 00:25:20.091 align:middle line:90% STUDENT: [INAUDIBLE]? 00:25:20.091 --> 00:25:23.010 align:middle line:90% 00:25:23.010 --> 00:25:25.010 align:middle line:84% JACK HARE: No, I don't know what Spark is using. 00:25:25.010 --> 00:25:26.560 align:middle line:90% So there's a good question. 00:25:26.560 --> 00:25:27.830 align:middle line:90% They presumably have a plan. 00:25:27.830 --> 00:25:30.250 align:middle line:84% I haven't seen like a diagnostics paper for Spark yet. 00:25:30.250 --> 00:25:33.130 align:middle line:84% I keep nagging Spark people to do something. 00:25:33.130 --> 00:25:34.310 align:middle line:90% But I haven't seen it yet. 00:25:34.310 --> 00:25:36.720 align:middle line:90% So it'd be cool to see it. 00:25:36.720 --> 00:25:37.970 align:middle line:90% Other questions from Columbia? 00:25:37.970 --> 00:25:41.569 align:middle line:90% 00:25:41.569 --> 00:25:42.640 align:middle line:90% STUDENT: I have one. 00:25:42.640 --> 00:25:44.380 align:middle line:84% For the Faraday effect, how do you 00:25:44.380 --> 00:25:50.110 align:middle line:84% account for you have to modulo 2 pi with the angles. 00:25:50.110 --> 00:25:52.180 align:middle line:90% You take it across multiple-- 00:25:52.180 --> 00:25:54.828 align:middle line:84% do you measure the angle at multiple stages? 00:25:54.828 --> 00:25:57.370 align:middle line:84% JACK HARE: Oh man, we're going to be talking a lot about ways 00:25:57.370 --> 00:26:01.120 align:middle line:84% to disambiguate modulo 2 pi stuff when 00:26:01.120 --> 00:26:02.360 align:middle line:90% we get into promontory. 00:26:02.360 --> 00:26:04.690 align:middle line:84% So I'll just say any technique you can use there, 00:26:04.690 --> 00:26:05.605 align:middle line:90% you can do here. 00:26:05.605 --> 00:26:07.480 align:middle line:84% So there's some really cool stuff you can do. 00:26:07.480 --> 00:26:10.990 align:middle line:84% Yeah, we can do like temporary heterodyne polarization 00:26:10.990 --> 00:26:12.580 align:middle line:90% measurements, things like that. 00:26:12.580 --> 00:26:14.980 align:middle line:90% So yeah, it's doable. 00:26:14.980 --> 00:26:18.320 align:middle line:84% This simple method here is flawed, like you pointed out. 00:26:18.320 --> 00:26:20.120 align:middle line:84% But there are other things we can do, 00:26:20.120 --> 00:26:22.640 align:middle line:84% which can get around that ambiguity. 00:26:22.640 --> 00:26:23.884 align:middle line:90% Yeah? 00:26:23.884 --> 00:26:26.443 align:middle line:84% STUDENT: Do perpendicular components of the magnetic field 00:26:26.443 --> 00:26:32.458 align:middle line:84% affect the Faraday effect sensor? 00:26:32.458 --> 00:26:34.750 align:middle line:84% JACK HARE: The question is, do perpendicular components 00:26:34.750 --> 00:26:39.050 align:middle line:84% of the magnetic field affect the Faraday effect sensor? 00:26:39.050 --> 00:26:40.460 align:middle line:90% I don't think so. 00:26:40.460 --> 00:26:41.975 align:middle line:90% There is an effect in plasmas. 00:26:41.975 --> 00:26:45.180 align:middle line:84% It's called the Cotton-Mouton effect, 00:26:45.180 --> 00:26:48.200 align:middle line:90% which can have an effect. 00:26:48.200 --> 00:26:52.073 align:middle line:84% But I don't know what happens in verdet glass. 00:26:52.073 --> 00:26:53.990 align:middle line:84% It's kind of like a solid state kind of thing, 00:26:53.990 --> 00:26:55.470 align:middle line:90% which I'm less familiar with. 00:26:55.470 --> 00:26:57.345 align:middle line:84% So I've never heard people talking about that 00:26:57.345 --> 00:26:58.400 align:middle line:90% as being a problem. 00:26:58.400 --> 00:26:59.990 align:middle line:90% Good question, though, yeah. 00:26:59.990 --> 00:27:00.966 align:middle line:90% Another question here? 00:27:00.966 --> 00:27:02.341 align:middle line:84% STUDENT: Does the verdet constant 00:27:02.341 --> 00:27:03.480 align:middle line:84% depends on the wavelength of light? 00:27:03.480 --> 00:27:04.938 align:middle line:84% JACK HARE: Yes, the verdet constant 00:27:04.938 --> 00:27:08.587 align:middle line:84% depends on the wavelength of light, maybe not very strongly. 00:27:08.587 --> 00:27:10.670 align:middle line:84% And you would know what wavelength you were using. 00:27:10.670 --> 00:27:12.003 align:middle line:90% So it wouldn't be a big problem. 00:27:12.003 --> 00:27:13.797 align:middle line:84% So for telecoms, we use 1,550 nanometers 00:27:13.797 --> 00:27:15.380 align:middle line:84% because that's what's being developed. 00:27:15.380 --> 00:27:17.350 align:middle line:84% So if you're going to do a fiber diagnostic, 00:27:17.350 --> 00:27:19.180 align:middle line:84% it's probably going to be at 1,550, 00:27:19.180 --> 00:27:24.240 align:middle line:84% unless you want to spend a lot of money on custom [INAUDIBLE].. 00:27:24.240 --> 00:27:25.755 align:middle line:90% All right, let's keep moving. 00:27:25.755 --> 00:27:29.060 align:middle line:90% 00:27:29.060 --> 00:27:29.630 align:middle line:90% OK. 00:27:29.630 --> 00:27:33.440 align:middle line:84% And now, a related diagnostic, where should we go? 00:27:33.440 --> 00:27:53.900 align:middle line:90% 00:27:53.900 --> 00:27:56.870 align:middle line:84% So a related diagnostic, another firm favorite of mine 00:27:56.870 --> 00:28:00.560 align:middle line:84% is something called a Rogowski coil, a rogowki coil, 00:28:00.560 --> 00:28:03.298 align:middle line:84% depending on what you think [INAUDIBLE].. 00:28:03.298 --> 00:28:06.930 align:middle line:90% 00:28:06.930 --> 00:28:10.095 align:middle line:84% So a Rogowski coil measures the enclosed current. 00:28:10.095 --> 00:28:14.550 align:middle line:90% 00:28:14.550 --> 00:28:15.810 align:middle line:90% Enclosed by what, you ask. 00:28:15.810 --> 00:28:17.430 align:middle line:90% Enclosed by the coil. 00:28:17.430 --> 00:28:19.860 align:middle line:90% So I will draw on in a second. 00:28:19.860 --> 00:28:21.940 align:middle line:84% And you'll see what I'm talking about. 00:28:21.940 --> 00:28:27.532 align:middle line:90% So a Rogowski coil looks like-- 00:28:27.532 --> 00:28:30.980 align:middle line:84% let's see what the best way to draw this is. 00:28:30.980 --> 00:28:34.010 align:middle line:84% We have, a little bit like a B-dot, a cable coming up. 00:28:34.010 --> 00:28:39.410 align:middle line:84% Instead of just going around in a loop like this, 00:28:39.410 --> 00:28:43.195 align:middle line:84% we have something that spirals around in a helix. 00:28:43.195 --> 00:28:47.630 align:middle line:90% 00:28:47.630 --> 00:28:50.670 align:middle line:84% And then for reasons I won't go into right now 00:28:50.670 --> 00:28:53.060 align:middle line:84% but you may feel free to ponder, we 00:28:53.060 --> 00:28:55.950 align:middle line:84% tend to not take the cable down here. 00:28:55.950 --> 00:29:01.770 align:middle line:84% But instead, we wind it back all the way around inside, 00:29:01.770 --> 00:29:02.978 align:middle line:90% like this. 00:29:02.978 --> 00:29:04.770 align:middle line:84% The answer to this is in Hutchinson's book. 00:29:04.770 --> 00:29:07.710 align:middle line:84% But I'm not going to go in right now. 00:29:07.710 --> 00:29:10.750 align:middle line:84% And the current that's enclosed-- for example, 00:29:10.750 --> 00:29:13.320 align:middle line:84% we can consider that we have some sort of current carrying 00:29:13.320 --> 00:29:18.480 align:middle line:84% rod, like this, that goes through this Rogowski coil. 00:29:18.480 --> 00:29:22.308 align:middle line:84% So we've set our Rogowski coil around some sort of conductor. 00:29:22.308 --> 00:29:23.850 align:middle line:84% Obviously, that current has to close. 00:29:23.850 --> 00:29:27.750 align:middle line:84% And it will close outside the conductor, like that. 00:29:27.750 --> 00:29:31.290 align:middle line:84% OK, so let's have a look at the geometry here, just 00:29:31.290 --> 00:29:32.220 align:middle line:90% a simpler diagram. 00:29:32.220 --> 00:29:34.050 align:middle line:90% We've got current. 00:29:34.050 --> 00:29:37.150 align:middle line:84% We've got some sort of surface like this. 00:29:37.150 --> 00:29:39.190 align:middle line:84% And we've got all these little loops. 00:29:39.190 --> 00:29:41.230 align:middle line:90% And these little loops-- 00:29:41.230 --> 00:29:45.430 align:middle line:90% hey, hey-- have an area. 00:29:45.430 --> 00:29:46.400 align:middle line:90% There's a current here. 00:29:46.400 --> 00:29:49.180 align:middle line:84% So there's going to be a magnetic field through it, 00:29:49.180 --> 00:29:51.390 align:middle line:90% like that. 00:29:51.390 --> 00:29:53.130 align:middle line:90% There's going to be two dL's. 00:29:53.130 --> 00:29:57.290 align:middle line:84% There's going to be the dL around this little loop. 00:29:57.290 --> 00:30:02.510 align:middle line:84% But there's also going to be the dL prime around this loop, 00:30:02.510 --> 00:30:03.300 align:middle line:90% like that. 00:30:03.300 --> 00:30:05.940 align:middle line:84% So we're actually going to end up with a double integral. 00:30:05.940 --> 00:30:11.050 align:middle line:84% We're going to end up with having a flux that 00:30:11.050 --> 00:30:14.020 align:middle line:90% is going through this loop. 00:30:14.020 --> 00:30:16.270 align:middle line:84% This magnetic flux is going to be 00:30:16.270 --> 00:30:19.590 align:middle line:84% equal to the number of turns per unit length 00:30:19.590 --> 00:30:21.090 align:middle line:84% because we're going to have multiple 00:30:21.090 --> 00:30:25.430 align:middle line:84% of these little surfaces all the way around like this. 00:30:25.430 --> 00:30:30.010 align:middle line:84% We're going to be integrating along L prime, which 00:30:30.010 --> 00:30:34.950 align:middle line:84% is the circumference of this Rogowski. 00:30:34.950 --> 00:30:37.110 align:middle line:84% And we're going to be integrating 00:30:37.110 --> 00:30:42.860 align:middle line:84% over A, which is the area of one of these little loops here. 00:30:42.860 --> 00:30:49.425 align:middle line:84% We're going to be integrating B dot dL-- 00:30:49.425 --> 00:30:50.800 align:middle line:84% actually, that's not quite right. 00:30:50.800 --> 00:30:54.090 align:middle line:90% 00:30:54.090 --> 00:31:00.350 align:middle line:90% B dot dS again dL, like that. 00:31:00.350 --> 00:31:03.020 align:middle line:84% Now, the key insight here is that this flux 00:31:03.020 --> 00:31:05.750 align:middle line:84% is going to be due entirely to the current that's 00:31:05.750 --> 00:31:06.860 align:middle line:90% going inside here. 00:31:06.860 --> 00:31:09.290 align:middle line:84% We know from Ampere's law that the integral 00:31:09.290 --> 00:31:12.950 align:middle line:84% of the magnetic field around some path, dL, 00:31:12.950 --> 00:31:17.920 align:middle line:84% which is this one here, using dL prime, 00:31:17.920 --> 00:31:22.240 align:middle line:84% is going to simply be mu I enclosed. 00:31:22.240 --> 00:31:27.120 align:middle line:84% So this magnetic field B here, at least on average 00:31:27.120 --> 00:31:28.890 align:middle line:84% around this whole circle, is going 00:31:28.890 --> 00:31:30.750 align:middle line:84% to be proportional just to mu, which 00:31:30.750 --> 00:31:32.917 align:middle line:84% is going to be the permeability of whatever material 00:31:32.917 --> 00:31:35.070 align:middle line:84% we're using, times the enclosed current. 00:31:35.070 --> 00:31:39.072 align:middle line:84% And the voltage that we get out, down here, 00:31:39.072 --> 00:31:40.530 align:middle line:84% which is the thing we're going for, 00:31:40.530 --> 00:31:45.090 align:middle line:84% is simply going to be the time rate of change of the total flux 00:31:45.090 --> 00:31:46.650 align:middle line:90% through this circle circuit. 00:31:46.650 --> 00:31:48.960 align:middle line:84% And so we're going to end up with something that looks 00:31:48.960 --> 00:31:57.320 align:middle line:90% like nA mu I enclosed dotted. 00:31:57.320 --> 00:31:59.223 align:middle line:84% So once again, we've got a voltage, 00:31:59.223 --> 00:32:01.640 align:middle line:84% which is proportional to the time derivative of something. 00:32:01.640 --> 00:32:04.700 align:middle line:84% It's also proportional to an area. 00:32:04.700 --> 00:32:07.070 align:middle line:84% Instead of having a total number of turns n, 00:32:07.070 --> 00:32:09.900 align:middle line:84% we now have a number of turns per unit length. 00:32:09.900 --> 00:32:16.460 align:middle line:84% So this is turns per unit length. 00:32:16.460 --> 00:32:21.440 align:middle line:90% 00:32:21.440 --> 00:32:24.170 align:middle line:84% This is not the area of the whole thing. 00:32:24.170 --> 00:32:26.760 align:middle line:84% It's the area of one of these little terms here. 00:32:26.760 --> 00:32:31.570 align:middle line:84% So this could be, for example, pi A squared 00:32:31.570 --> 00:32:36.490 align:middle line:84% if the radius of one of these little loops is A, like that. 00:32:36.490 --> 00:32:40.890 align:middle line:84% And the only reason I wrote this as mu as opposed to mu 0 00:32:40.890 --> 00:32:43.650 align:middle line:84% is because it's really mu 0 mu R. 00:32:43.650 --> 00:32:45.930 align:middle line:84% And you want to be a little bit careful here 00:32:45.930 --> 00:32:47.730 align:middle line:84% because, for some materials, if you 00:32:47.730 --> 00:32:52.860 align:middle line:84% get very strong magnetic fields, they have saturation of mu R, 00:32:52.860 --> 00:32:54.750 align:middle line:84% things like steels and stuff like that. 00:32:54.750 --> 00:32:57.075 align:middle line:84% You want to avoid those because this mu 00:32:57.075 --> 00:32:58.980 align:middle line:84% R is going to start changing depending 00:32:58.980 --> 00:33:00.340 align:middle line:90% on the strength of your signal. 00:33:00.340 --> 00:33:02.757 align:middle line:84% And so you're going to have not a nice linear relationship 00:33:02.757 --> 00:33:05.680 align:middle line:84% between voltage and current, but a non-linear relationship. 00:33:05.680 --> 00:33:07.290 align:middle line:84% So the only reason I put this in here 00:33:07.290 --> 00:33:10.380 align:middle line:84% is just to say that you shouldn't use steel 00:33:10.380 --> 00:33:15.950 align:middle line:84% for these things if you're going up to high magnetic fields. 00:33:15.950 --> 00:33:18.320 align:middle line:84% The nice thing about this is it doesn't matter where 00:33:18.320 --> 00:33:20.000 align:middle line:90% the current is inside here. 00:33:20.000 --> 00:33:22.700 align:middle line:84% Remember, Ampere's law just talks about enclosed currents. 00:33:22.700 --> 00:33:26.840 align:middle line:84% I could split this up and have two conductors, like this. 00:33:26.840 --> 00:33:31.700 align:middle line:84% I could just have one conductor over all the way on one side. 00:33:31.700 --> 00:33:33.710 align:middle line:84% I would still get the same signal out of this. 00:33:33.710 --> 00:33:35.750 align:middle line:84% That's nice because it means I don't really 00:33:35.750 --> 00:33:37.700 align:middle line:84% need to know the exact location of all 00:33:37.700 --> 00:33:39.020 align:middle line:90% the conductors inside here. 00:33:39.020 --> 00:33:41.420 align:middle line:84% That means that I don't have to position this thing quite 00:33:41.420 --> 00:33:44.220 align:middle line:84% so precisely, which is extremely useful. 00:33:44.220 --> 00:33:50.590 align:middle line:84% So this I enclosed here means that the conductor position, 00:33:50.590 --> 00:33:59.360 align:middle line:84% or should we say current channel, 00:33:59.360 --> 00:34:01.370 align:middle line:90% position is independent of this. 00:34:01.370 --> 00:34:11.199 align:middle line:90% 00:34:11.199 --> 00:34:12.010 align:middle line:90% OK. 00:34:12.010 --> 00:34:14.560 align:middle line:84% And you can go and wind one of your own of these. 00:34:14.560 --> 00:34:16.969 align:middle line:84% You can strip back a BNC cable and get some magnet wire 00:34:16.969 --> 00:34:18.469 align:middle line:84% and wrap it all the way back around, 00:34:18.469 --> 00:34:20.440 align:middle line:84% which is what Thomas Varnish has been doing 00:34:20.440 --> 00:34:21.820 align:middle line:90% for [INAUDIBLE] experiments. 00:34:21.820 --> 00:34:25.060 align:middle line:84% Or you can go online and you can get something from Pearson. 00:34:25.060 --> 00:34:26.830 align:middle line:90% So everyone buys Pearson's. 00:34:26.830 --> 00:34:28.540 align:middle line:90% These nice, green loops. 00:34:28.540 --> 00:34:30.699 align:middle line:84% And they have a passive integrator built into them, 00:34:30.699 --> 00:34:31.900 align:middle line:90% like we discussed before. 00:34:31.900 --> 00:34:33.639 align:middle line:84% And they're labeled in volts per amp. 00:34:33.639 --> 00:34:34.989 align:middle line:84% So if you just want something off the shelf 00:34:34.989 --> 00:34:37.572 align:middle line:84% and you've got lots of money, you can buy something like that. 00:34:37.572 --> 00:34:41.775 align:middle line:84% So these are absolute workhorses not just for plasma physics. 00:34:41.775 --> 00:34:43.900 align:middle line:84% These are just if you want to measure current going 00:34:43.900 --> 00:34:44.659 align:middle line:90% through something. 00:34:44.659 --> 00:34:47.650 align:middle line:84% So any questions on these before we keep moving 00:34:47.650 --> 00:34:49.638 align:middle line:84% and show how to use them in plasma systems? 00:34:49.638 --> 00:34:54.030 align:middle line:90% 00:34:54.030 --> 00:34:54.732 align:middle line:90% Mm-hmm? 00:34:54.732 --> 00:34:55.718 align:middle line:90% STUDENT: [INAUDIBLE]? 00:34:55.718 --> 00:35:01.170 align:middle line:90% 00:35:01.170 --> 00:35:02.220 align:middle line:90% JACK HARE: Yeah. 00:35:02.220 --> 00:35:03.720 align:middle line:90% Once again, yes, you're right. 00:35:03.720 --> 00:35:07.450 align:middle line:84% It only measures the current normal to this big surface here. 00:35:07.450 --> 00:35:09.560 align:middle line:90% So I guess I glossed over that. 00:35:09.560 --> 00:35:11.936 align:middle line:90% Yeah, you're quite right. 00:35:11.936 --> 00:35:12.902 align:middle line:90% STUDENT: [INAUDIBLE]? 00:35:12.902 --> 00:35:23.528 align:middle line:90% 00:35:23.528 --> 00:35:26.340 align:middle line:84% JACK HARE: It's going to happen somewhere in Ampere's law. 00:35:26.340 --> 00:35:31.560 align:middle line:84% And I think when we talk about I enclosed here, we are-- 00:35:31.560 --> 00:35:32.430 align:middle line:90% huh, Interesting. 00:35:32.430 --> 00:35:36.060 align:middle line:90% 00:35:36.060 --> 00:35:40.540 align:middle line:84% Maybe it doesn't matter because that 00:35:40.540 --> 00:35:42.850 align:middle line:84% doesn't look like we care particularly 00:35:42.850 --> 00:35:44.695 align:middle line:84% about the angle of the conductor here. 00:35:44.695 --> 00:35:46.702 align:middle line:90% 00:35:46.702 --> 00:35:48.660 align:middle line:84% I don't know the answer off the top of my head. 00:35:48.660 --> 00:35:50.560 align:middle line:84% If someone else does know the answer, shout out. 00:35:50.560 --> 00:35:52.977 align:middle line:84% Otherwise, I'll have a think about it and get back to you. 00:35:52.977 --> 00:35:54.266 align:middle line:90% STUDENT: [INAUDIBLE]. 00:35:54.266 --> 00:36:03.160 align:middle line:90% 00:36:03.160 --> 00:36:05.980 align:middle line:84% JACK HARE: OK, so I've skipped it in that line. 00:36:05.980 --> 00:36:07.400 align:middle line:90% Yeah, I think you're right. 00:36:07.400 --> 00:36:08.860 align:middle line:84% So OK, I think it does matter what 00:36:08.860 --> 00:36:10.402 align:middle line:84% the orientation of your conductor is. 00:36:10.402 --> 00:36:11.140 align:middle line:90% Yeah, cool. 00:36:11.140 --> 00:36:12.820 align:middle line:90% Thank you. 00:36:12.820 --> 00:36:13.952 align:middle line:90% Other questions? 00:36:13.952 --> 00:36:20.700 align:middle line:90% 00:36:20.700 --> 00:36:21.674 align:middle line:90% Yeah? 00:36:21.674 --> 00:36:23.560 align:middle line:90% STUDENT: [INAUDIBLE]? 00:36:23.560 --> 00:36:25.810 align:middle line:84% JACK HARE: No, you can't use this for constant current 00:36:25.810 --> 00:36:29.680 align:middle line:84% still because the signal V is proportional to I dot. 00:36:29.680 --> 00:36:32.140 align:middle line:84% So you'd have to use another current sensor 00:36:32.140 --> 00:36:34.470 align:middle line:90% if you want to do that. 00:36:34.470 --> 00:36:38.400 align:middle line:84% And these suffer from similar problems to your B-dots, 00:36:38.400 --> 00:36:40.300 align:middle line:84% where you can pick up stray voltages on them. 00:36:40.300 --> 00:36:42.390 align:middle line:84% So you might want to use an oppositely wound 00:36:42.390 --> 00:36:46.055 align:middle line:84% pair of these side by side and do differential measurements. 00:36:46.055 --> 00:36:47.430 align:middle line:84% There's also a continuous version 00:36:47.430 --> 00:36:50.520 align:middle line:84% of this, where you replace this helix with just 00:36:50.520 --> 00:36:52.145 align:middle line:90% a groove in a plate of metal. 00:36:52.145 --> 00:36:53.520 align:middle line:84% And I'll leave it to you to think 00:36:53.520 --> 00:36:55.812 align:middle line:84% about how that works because it's actually a little bit 00:36:55.812 --> 00:36:56.370 align:middle line:90% mind-bending. 00:36:56.370 --> 00:36:57.990 align:middle line:90% But it does still work. 00:36:57.990 --> 00:36:59.880 align:middle line:84% What I want to do now is talk about how 00:36:59.880 --> 00:37:02.490 align:middle line:84% to use these in the context of plasma physics. 00:37:02.490 --> 00:37:04.260 align:middle line:84% So any other questions before we move on? 00:37:04.260 --> 00:37:04.760 align:middle line:90% Columbia? 00:37:04.760 --> 00:37:08.190 align:middle line:90% 00:37:08.190 --> 00:37:09.600 align:middle line:84% All right, looks like we're good. 00:37:09.600 --> 00:37:38.640 align:middle line:90% 00:37:38.640 --> 00:37:44.970 align:middle line:84% Okey doke, let's have a think about something like a tokamak. 00:37:44.970 --> 00:37:48.430 align:middle line:84% I'm going to draw the cross-section 00:37:48.430 --> 00:37:50.920 align:middle line:90% of a tokamak like this. 00:37:50.920 --> 00:37:54.190 align:middle line:84% It's going to have a plasma inside it. 00:37:54.190 --> 00:37:57.640 align:middle line:90% 00:37:57.640 --> 00:38:00.760 align:middle line:84% And in that plasma, there's going to be some current. 00:38:00.760 --> 00:38:08.340 align:middle line:84% This is going to be the toroidal current, I phi, 00:38:08.340 --> 00:38:10.080 align:middle line:90% which loops around the torus. 00:38:10.080 --> 00:38:16.830 align:middle line:84% This toroidal current is being driven inductively 00:38:16.830 --> 00:38:18.220 align:middle line:90% by a transformer. 00:38:18.220 --> 00:38:22.200 align:middle line:90% So this is an iron core. 00:38:22.200 --> 00:38:24.000 align:middle line:90% And we have a primary winding. 00:38:24.000 --> 00:38:26.890 align:middle line:84% And the tokamak plasma forms a secondary winding. 00:38:26.890 --> 00:38:30.720 align:middle line:84% So we're driving some current using this ohmic transformer. 00:38:30.720 --> 00:38:32.850 align:middle line:84% We can drive a constant current with this. 00:38:32.850 --> 00:38:34.767 align:middle line:84% That will require us to keep ramping up 00:38:34.767 --> 00:38:36.100 align:middle line:90% the voltage on this transformer. 00:38:36.100 --> 00:38:38.250 align:middle line:90% But that's allowed. 00:38:38.250 --> 00:38:40.930 align:middle line:90% And what we'll do is we'll put-- 00:38:40.930 --> 00:38:44.090 align:middle line:84% around one part of our plasma, will put our Rogowski coil. 00:38:44.090 --> 00:38:49.880 align:middle line:84% So here comes Rogowski coil, loop de loop de loop, 00:38:49.880 --> 00:38:52.370 align:middle line:90% back wound. 00:38:52.370 --> 00:39:00.920 align:middle line:84% And we measure V goes as I phi dot. 00:39:00.920 --> 00:39:05.150 align:middle line:84% OK, so we can measure that current. 00:39:05.150 --> 00:39:07.850 align:middle line:84% I also want to draw this same system now looking 00:39:07.850 --> 00:39:09.770 align:middle line:84% from above because it makes it easier 00:39:09.770 --> 00:39:10.970 align:middle line:90% for what the next point is. 00:39:10.970 --> 00:39:15.070 align:middle line:90% Say so, this is now our torus. 00:39:15.070 --> 00:39:23.700 align:middle line:90% We've got our plasma, like this. 00:39:23.700 --> 00:39:25.230 align:middle line:84% Hopefully, it's not that unstable. 00:39:25.230 --> 00:39:29.020 align:middle line:84% And we've got some toroidal current, I phi, going around it. 00:39:29.020 --> 00:39:33.450 align:middle line:84% We've got some transformer, as already discussed, like that. 00:39:33.450 --> 00:39:39.030 align:middle line:84% We've put in our Rogowski coil, like that. 00:39:39.030 --> 00:39:41.300 align:middle line:84% And we're also going to put in another loop. 00:39:41.300 --> 00:39:44.850 align:middle line:84% And that loop is going to be our voltage loop. 00:39:44.850 --> 00:39:51.290 align:middle line:84% So this loop simply looks like this. 00:39:51.290 --> 00:39:55.480 align:middle line:84% And it measures V in the toroidal direction. 00:39:55.480 --> 00:39:59.730 align:middle line:84% The reason it does this is because this transformer 00:39:59.730 --> 00:40:03.060 align:middle line:84% is inducing some toroidal potential. 00:40:03.060 --> 00:40:03.630 align:middle line:90% It has to. 00:40:03.630 --> 00:40:06.450 align:middle line:84% That's what's driving this toroidal magnetic field. 00:40:06.450 --> 00:40:15.340 align:middle line:84% This is equal to V toroidal over the plasma resistance, whatever 00:40:15.340 --> 00:40:17.560 align:middle line:84% that means really in this context. 00:40:17.560 --> 00:40:19.960 align:middle line:84% Now, ideally, we would measure this 00:40:19.960 --> 00:40:21.722 align:middle line:84% by sticking something inside the plasma. 00:40:21.722 --> 00:40:23.680 align:middle line:84% But actually, we don't need to because it's not 00:40:23.680 --> 00:40:25.810 align:middle line:84% the plasma that's inducing V toroidal. 00:40:25.810 --> 00:40:27.920 align:middle line:84% It's this whacking great transformer. 00:40:27.920 --> 00:40:34.808 align:middle line:84% So we can put our loop, say, above the plasma 00:40:34.808 --> 00:40:35.850 align:middle line:90% and just measure it here. 00:40:35.850 --> 00:40:37.392 align:middle line:84% It doesn't matter where the break is. 00:40:37.392 --> 00:40:39.350 align:middle line:84% It just matters that when we measure difference 00:40:39.350 --> 00:40:41.870 align:middle line:84% between these two, as you traverse a circuit around here, 00:40:41.870 --> 00:40:44.705 align:middle line:84% you're going to pick up some V toroidal, which is the same V 00:40:44.705 --> 00:40:48.050 align:middle line:84% toroidal, as I said, that's driving the pattern here. 00:40:48.050 --> 00:40:50.000 align:middle line:84% It will become clear why we're trying 00:40:50.000 --> 00:40:51.590 align:middle line:90% to measure this in a moment. 00:40:51.590 --> 00:40:54.930 align:middle line:84% It's actually very much related to this plasma resistivity here. 00:40:54.930 --> 00:40:59.690 align:middle line:84% So this is an important quantity that you might want to measure. 00:40:59.690 --> 00:41:00.260 align:middle line:90% OK. 00:41:00.260 --> 00:41:02.450 align:middle line:84% So let's set this up and have a go 00:41:02.450 --> 00:41:04.460 align:middle line:84% at measuring the plasma resistivity using 00:41:04.460 --> 00:41:07.890 align:middle line:84% just two loops of wire Rogowski and V phi. 00:41:07.890 --> 00:41:09.710 align:middle line:90% OK, so we consider a plasma. 00:41:09.710 --> 00:41:13.500 align:middle line:90% 00:41:13.500 --> 00:41:19.540 align:middle line:84% It has a volume V. And it's got a surface-- 00:41:19.540 --> 00:41:24.750 align:middle line:90% 00:41:24.750 --> 00:41:26.850 align:middle line:90% what do I call it-- 00:41:26.850 --> 00:41:28.050 align:middle line:90% partial V like that. 00:41:28.050 --> 00:41:31.620 align:middle line:84% OK, and our loops are outside of here. 00:41:31.620 --> 00:41:37.515 align:middle line:84% So both the ROG and the V phi are outside this surface. 00:41:37.515 --> 00:41:41.090 align:middle line:90% 00:41:41.090 --> 00:41:42.350 align:middle line:90% OK, good. 00:41:42.350 --> 00:41:45.510 align:middle line:84% Now, let's think about energy balance in this system-- 00:41:45.510 --> 00:41:47.760 align:middle line:84% so the amount of energy in the system, 00:41:47.760 --> 00:41:50.930 align:middle line:84% how it's changing in time, and how we're injecting energy 00:41:50.930 --> 00:41:52.080 align:middle line:90% into the system. 00:41:52.080 --> 00:41:54.470 align:middle line:84% So we're going to have a few different terms here. 00:41:54.470 --> 00:41:57.260 align:middle line:84% We're going to have an ohmic heating term, 00:41:57.260 --> 00:42:00.080 align:middle line:84% E dot J. So that's going to be related 00:42:00.080 --> 00:42:03.660 align:middle line:84% to the current inside this system and the electric field, 00:42:03.660 --> 00:42:05.280 align:middle line:90% which is related to the voltage. 00:42:05.280 --> 00:42:06.980 align:middle line:84% So we can say that this is going to be-- 00:42:06.980 --> 00:42:08.130 align:middle line:84% straight away, we can see, this is going 00:42:08.130 --> 00:42:09.380 align:middle line:90% to be something with I phi. 00:42:09.380 --> 00:42:13.070 align:middle line:84% And this is going to be something to do with V phi. 00:42:13.070 --> 00:42:15.800 align:middle line:84% There's also going to be a term related 00:42:15.800 --> 00:42:19.980 align:middle line:84% to the change in total magnetic energy inside the system. 00:42:19.980 --> 00:42:25.580 align:middle line:84% So the magnetic energy is B squared over 2 mu 0. 00:42:25.580 --> 00:42:28.130 align:middle line:84% And we want to have the time rate of change 00:42:28.130 --> 00:42:30.467 align:middle line:90% of that to make it a power. 00:42:30.467 --> 00:42:32.300 align:middle line:84% I'm going to write that slightly differently 00:42:32.300 --> 00:42:33.967 align:middle line:84% to make it clearer what's going on here. 00:42:33.967 --> 00:42:37.500 align:middle line:90% 00:42:37.500 --> 00:42:43.360 align:middle line:84% We want the time rate of change of the magnetic energy density, 00:42:43.360 --> 00:42:43.930 align:middle line:90% like this. 00:42:43.930 --> 00:42:48.210 align:middle line:84% And all of this is integrated over this volume, dV. 00:42:48.210 --> 00:42:50.900 align:middle line:90% 00:42:50.900 --> 00:42:53.090 align:middle line:84% And this is going to be equal to-- 00:42:53.090 --> 00:42:55.120 align:middle line:84% this change in internal energy of the system 00:42:55.120 --> 00:42:56.870 align:middle line:84% is going to be equal the energy that we're 00:42:56.870 --> 00:42:57.953 align:middle line:90% injecting into the system. 00:42:57.953 --> 00:43:00.050 align:middle line:84% We're injecting the energy into the system 00:43:00.050 --> 00:43:01.550 align:middle line:90% through this transformer. 00:43:01.550 --> 00:43:06.140 align:middle line:84% And this is going to be equal to 1 upon mu 0 00:43:06.140 --> 00:43:11.930 align:middle line:84% the integral through this surface partial V of E cross 00:43:11.930 --> 00:43:17.180 align:middle line:90% B integrated over the surface. 00:43:17.180 --> 00:43:21.060 align:middle line:84% So this is the Poynting flux from electromagnetism. 00:43:21.060 --> 00:43:26.630 align:middle line:90% 00:43:26.630 --> 00:43:28.250 align:middle line:90% This is ohmic heating. 00:43:28.250 --> 00:43:30.770 align:middle line:90% 00:43:30.770 --> 00:43:34.490 align:middle line:84% And this is effectively the inductive power, 00:43:34.490 --> 00:43:37.130 align:middle line:84% the energy we have to spend to change the magnetic field. 00:43:37.130 --> 00:43:41.820 align:middle line:90% 00:43:41.820 --> 00:43:43.320 align:middle line:84% And these are balanced because we're 00:43:43.320 --> 00:43:45.880 align:middle line:84% going to be discussing a system in steady state. 00:43:45.880 --> 00:43:47.520 align:middle line:84% So we're not going to allow the energy 00:43:47.520 --> 00:43:51.810 align:middle line:84% to change without us putting some power into it. 00:43:51.810 --> 00:43:55.800 align:middle line:84% So I'm not sure I need that assumption, but there we go. 00:43:55.800 --> 00:43:58.350 align:middle line:84% OK, questions on this setup before we 00:43:58.350 --> 00:44:01.060 align:middle line:84% try and work out some of these terms? 00:44:01.060 --> 00:44:02.522 align:middle line:90% Yeah? 00:44:02.522 --> 00:44:07.453 align:middle line:84% STUDENT: You said it's [INAUDIBLE] 00:44:07.453 --> 00:44:08.120 align:middle line:90% JACK HARE: Yeah. 00:44:08.120 --> 00:44:09.425 align:middle line:90% STUDENT: [INAUDIBLE]? 00:44:09.425 --> 00:44:13.020 align:middle line:90% 00:44:13.020 --> 00:44:14.640 align:middle line:84% JACK HARE: Yeah, so imagine we're 00:44:14.640 --> 00:44:16.050 align:middle line:90% microwaving a lump of chicken. 00:44:16.050 --> 00:44:19.830 align:middle line:84% This left-hand side here is the temperature, the thermal energy 00:44:19.830 --> 00:44:20.800 align:middle line:90% density of the chicken. 00:44:20.800 --> 00:44:22.620 align:middle line:84% So we're integrating over the chicken, OK? 00:44:22.620 --> 00:44:24.430 align:middle line:84% The right-hand side is the Poynting flux, 00:44:24.430 --> 00:44:26.280 align:middle line:84% which is the microwaves penetrating 00:44:26.280 --> 00:44:28.290 align:middle line:90% the surface of the chicken, OK? 00:44:28.290 --> 00:44:30.580 align:middle line:84% And so we're setting those two equal to each other. 00:44:30.580 --> 00:44:33.760 align:middle line:84% So this is the internal energy of the plasma. 00:44:33.760 --> 00:44:37.072 align:middle line:84% And this is the power that we're putting into the plasma. 00:44:37.072 --> 00:44:38.530 align:middle line:84% So this is not the internal energy. 00:44:38.530 --> 00:44:42.982 align:middle line:84% This is the change in internal energy as a function of time. 00:44:42.982 --> 00:44:44.190 align:middle line:90% That's an ohmic heating rate. 00:44:44.190 --> 00:44:45.000 align:middle line:90% That's a power. 00:44:45.000 --> 00:44:47.035 align:middle line:84% This is the time derivative of energy. 00:44:47.035 --> 00:44:47.910 align:middle line:90% So it's also a power. 00:44:47.910 --> 00:44:51.820 align:middle line:84% And this is a power integrated with [INAUDIBLE].. 00:44:51.820 --> 00:44:52.820 align:middle line:90% Yeah? 00:44:52.820 --> 00:44:53.820 align:middle line:90% STUDENT: [INAUDIBLE]? 00:44:53.820 --> 00:45:02.300 align:middle line:90% 00:45:02.300 --> 00:45:04.220 align:middle line:84% JACK HARE: If we counterwound the B phi, 00:45:04.220 --> 00:45:05.962 align:middle line:90% we would measure nothing. 00:45:05.962 --> 00:45:06.926 align:middle line:90% STUDENT: [INAUDIBLE]? 00:45:06.926 --> 00:45:12.230 align:middle line:90% 00:45:12.230 --> 00:45:14.960 align:middle line:84% JACK HARE: Yes, you could have two oppositely wound 00:45:14.960 --> 00:45:16.175 align:middle line:90% loops in B phi. 00:45:16.175 --> 00:45:17.970 align:middle line:90% Yeah, sure. 00:45:17.970 --> 00:45:19.710 align:middle line:84% And yeah, any questions from Colombia? 00:45:19.710 --> 00:45:24.010 align:middle line:90% 00:45:24.010 --> 00:45:24.610 align:middle line:90% Yeah, Grant? 00:45:24.610 --> 00:45:25.604 align:middle line:90% STUDENT: [INAUDIBLE]? 00:45:25.604 --> 00:45:30.562 align:middle line:90% 00:45:30.562 --> 00:45:32.270 align:middle line:84% JACK HARE: We are not including radiation 00:45:32.270 --> 00:45:33.978 align:middle line:84% because we don't want to be here all day. 00:45:33.978 --> 00:45:35.630 align:middle line:84% Yes, OK, cool, but good question. 00:45:35.630 --> 00:45:37.640 align:middle line:90% Yes, there should be radiation. 00:45:37.640 --> 00:45:39.260 align:middle line:90% Where would radiation be? 00:45:39.260 --> 00:45:41.030 align:middle line:90% Left or right-hand side? 00:45:41.030 --> 00:45:42.140 align:middle line:90% STUDENT: [INAUDIBLE]? 00:45:42.140 --> 00:45:45.395 align:middle line:90% 00:45:45.395 --> 00:45:47.270 align:middle line:84% JACK HARE: Yeah, so if we included radiation, 00:45:47.270 --> 00:45:48.440 align:middle line:84% it would also be a Poynting flux. 00:45:48.440 --> 00:45:50.300 align:middle line:84% It'd be a Poynting flux going outwards instead. 00:45:50.300 --> 00:45:52.675 align:middle line:84% And we'd see that because we carefully examine E cross B. 00:45:52.675 --> 00:45:55.050 align:middle line:84% And we would see what was going out that way. 00:45:55.050 --> 00:45:55.770 align:middle line:90% It's in there. 00:45:55.770 --> 00:45:56.550 align:middle line:90% It's in there. 00:45:56.550 --> 00:45:58.050 align:middle line:84% I'm really not going to consider it, 00:45:58.050 --> 00:45:59.410 align:middle line:90% as you'll see in the next slide. 00:45:59.410 --> 00:46:00.420 align:middle line:90% Other questions, yes? 00:46:00.420 --> 00:46:01.386 align:middle line:90% STUDENT: [INAUDIBLE]? 00:46:01.386 --> 00:46:04.513 align:middle line:90% 00:46:04.513 --> 00:46:05.430 align:middle line:90% JACK HARE: Absolutely. 00:46:05.430 --> 00:46:07.650 align:middle line:84% So you can get current here from other effects. 00:46:07.650 --> 00:46:10.087 align:middle line:84% We're considering a very simple model of it. 00:46:10.087 --> 00:46:11.670 align:middle line:84% I mean, the bootstrap current you only 00:46:11.670 --> 00:46:13.980 align:middle line:84% get if there's some other current driven in the system. 00:46:13.980 --> 00:46:15.420 align:middle line:84% You have to have some other current, 00:46:15.420 --> 00:46:17.550 align:middle line:84% gradients of that current drive the bootstrap current. 00:46:17.550 --> 00:46:19.008 align:middle line:84% So you need to have something else. 00:46:19.008 --> 00:46:20.938 align:middle line:84% We could also have non-inductive current drive 00:46:20.938 --> 00:46:22.230 align:middle line:90% and all sorts of clever things. 00:46:22.230 --> 00:46:23.940 align:middle line:84% But none of those have really been demonstrated. 00:46:23.940 --> 00:46:25.395 align:middle line:90% So we're sticking with this. 00:46:25.395 --> 00:46:27.400 align:middle line:84% This also is a very simple model, 00:46:27.400 --> 00:46:29.220 align:middle line:84% which allows us to make some progress. 00:46:29.220 --> 00:46:29.748 align:middle line:90% Yeah? 00:46:29.748 --> 00:46:31.290 align:middle line:84% There was another hand up over there? 00:46:31.290 --> 00:46:33.260 align:middle line:90% No? 00:46:33.260 --> 00:46:34.250 align:middle line:90% Any other questions? 00:46:34.250 --> 00:46:38.250 align:middle line:84% We're going to try and make this slightly more comprehensible 00:46:38.250 --> 00:46:38.750 align:middle line:90% in a moment. 00:46:38.750 --> 00:46:41.292 align:middle line:84% I've already given you a hint about where some of these terms 00:46:41.292 --> 00:46:42.412 align:middle line:90% are going to come from. 00:46:42.412 --> 00:46:43.620 align:middle line:90% STUDENT: Hey, quick question. 00:46:43.620 --> 00:46:48.120 align:middle line:84% How did we measure the loop voltage? 00:46:48.120 --> 00:46:49.538 align:middle line:90% The toroidal voltage? 00:46:49.538 --> 00:46:51.330 align:middle line:84% JACK HARE: We measured the toroidal voltage 00:46:51.330 --> 00:46:53.040 align:middle line:84% in a really dumb way, which is we simply 00:46:53.040 --> 00:46:57.990 align:middle line:84% just put a loop of wire around the tokamak above the plasma. 00:46:57.990 --> 00:47:02.790 align:middle line:84% And the same loop voltage, which is induced inside the plasma, 00:47:02.790 --> 00:47:05.740 align:middle line:84% will also be induced inside this loop of wire. 00:47:05.740 --> 00:47:06.240 align:middle line:90% STUDENT: OK. 00:47:06.240 --> 00:47:07.740 align:middle line:84% JACK HARE: And so we are just using the wire 00:47:07.740 --> 00:47:09.943 align:middle line:84% as a proxy bit of plasma because we can conveniently 00:47:09.943 --> 00:47:11.610 align:middle line:84% stick electrodes into it, which we can't 00:47:11.610 --> 00:47:14.350 align:middle line:90% do with our plasma itself. 00:47:14.350 --> 00:47:16.110 align:middle line:84% And so we just have this loop here. 00:47:16.110 --> 00:47:19.875 align:middle line:84% That's what measures loop voltage for us. 00:47:19.875 --> 00:47:20.375 align:middle line:90% Cool. 00:47:20.375 --> 00:47:23.690 align:middle line:90% 00:47:23.690 --> 00:47:24.230 align:middle line:90% OK. 00:47:24.230 --> 00:47:30.980 align:middle line:84% So we now have a look at taking apart some of these terms here. 00:47:30.980 --> 00:47:38.030 align:middle line:84% The Poynting term-- in reality, this 00:47:38.030 --> 00:47:40.730 align:middle line:84% looks like electromagnetic radiation going in. 00:47:40.730 --> 00:47:43.640 align:middle line:84% So we could try and work it all out with E and B properly. 00:47:43.640 --> 00:47:45.830 align:middle line:84% But we can also just use some circuit theory 00:47:45.830 --> 00:47:47.190 align:middle line:90% and see what's going on there. 00:47:47.190 --> 00:47:49.400 align:middle line:84% So a circuit theory would tell us 00:47:49.400 --> 00:47:53.810 align:middle line:84% that the power is equal to I times V. 00:47:53.810 --> 00:47:55.790 align:middle line:84% And so we're going to decompose that. 00:47:55.790 --> 00:48:01.070 align:middle line:84% It's going to be V phi I phi-- so these are 00:48:01.070 --> 00:48:02.600 align:middle line:90% both in the toroidal direction. 00:48:02.600 --> 00:48:05.240 align:middle line:84% And there could be a poloidal component of this, 00:48:05.240 --> 00:48:07.840 align:middle line:90% V theta, I theta. 00:48:07.840 --> 00:48:12.220 align:middle line:84% There's unlikely to be a radial component of this. 00:48:12.220 --> 00:48:13.780 align:middle line:84% And that's because it's pretty hard 00:48:13.780 --> 00:48:15.778 align:middle line:84% to arrange for currents to go radially outwards 00:48:15.778 --> 00:48:17.570 align:middle line:84% because you violate the divergence theorem. 00:48:17.570 --> 00:48:20.470 align:middle line:90% So let's get rid of that. 00:48:20.470 --> 00:48:23.200 align:middle line:84% Now, it also turns out that, in something 00:48:23.200 --> 00:48:26.640 align:middle line:90% that's in steady state-- 00:48:26.640 --> 00:48:31.192 align:middle line:84% and so this could be a tokamak or a stellarator, 00:48:31.192 --> 00:48:32.525 align:middle line:90% or some other device like that-- 00:48:32.525 --> 00:48:38.230 align:middle line:90% 00:48:38.230 --> 00:48:42.050 align:middle line:84% this V theta would only come about 00:48:42.050 --> 00:48:48.100 align:middle line:84% if we had a change in the toroidal magnetic field in time. 00:48:48.100 --> 00:48:51.220 align:middle line:84% Because we're in steady state, our toroidal magnetic field 00:48:51.220 --> 00:48:51.760 align:middle line:90% is constant. 00:48:51.760 --> 00:48:53.620 align:middle line:84% We've set it with, say, our big superconducting 00:48:53.620 --> 00:48:54.912 align:middle line:90% magnets or something like that. 00:48:54.912 --> 00:48:56.890 align:middle line:90% That doesn't happen. 00:48:56.890 --> 00:48:58.890 align:middle line:90% So that's 0. 00:48:58.890 --> 00:49:00.270 align:middle line:90% And this whole term is 0. 00:49:00.270 --> 00:49:02.563 align:middle line:84% So as you might have predicted from the start, 00:49:02.563 --> 00:49:04.980 align:middle line:84% the energy that we're pushing in just from a circuit model 00:49:04.980 --> 00:49:08.040 align:middle line:84% is related to the voltage that we're 00:49:08.040 --> 00:49:11.140 align:middle line:84% dropping around the loop and the current that we're driving here. 00:49:11.140 --> 00:49:14.490 align:middle line:84% So this is just treating this as some ohmic resistor, which 00:49:14.490 --> 00:49:15.960 align:middle line:90% is dissipating some energy. 00:49:15.960 --> 00:49:19.260 align:middle line:84% There's nothing particularly plasma physics inside here. 00:49:19.260 --> 00:49:22.430 align:middle line:90% This all makes sense so far? 00:49:22.430 --> 00:49:23.660 align:middle line:90% OK. 00:49:23.660 --> 00:49:27.730 align:middle line:84% So then we can write power is going 00:49:27.730 --> 00:49:35.530 align:middle line:84% to be equal to still E dot J, which is the ohmic power still. 00:49:35.530 --> 00:49:41.890 align:middle line:84% And then we're going to set that equal to B phi I phi. 00:49:41.890 --> 00:49:43.465 align:middle line:90% So this is the Poynting flux. 00:49:43.465 --> 00:49:48.550 align:middle line:90% 00:49:48.550 --> 00:49:52.730 align:middle line:84% I'll just label that ohmic as well. 00:49:52.730 --> 00:49:55.250 align:middle line:84% And then I'm going to take this inductive term 00:49:55.250 --> 00:49:56.250 align:middle line:90% to the other side. 00:49:56.250 --> 00:49:58.340 align:middle line:84% So I'm going to put it as a minus sign here. 00:49:58.340 --> 00:50:01.730 align:middle line:84% And I'm going to write it as a partial derivative 00:50:01.730 --> 00:50:03.740 align:middle line:84% not of the integral of the magnetic energy 00:50:03.740 --> 00:50:09.020 align:middle line:84% density over the volume, but in terms, again, of circuit 00:50:09.020 --> 00:50:09.600 align:middle line:90% quantity. 00:50:09.600 --> 00:50:16.810 align:middle line:84% So this is a 1/2 L I phi squared, like that. 00:50:16.810 --> 00:50:18.520 align:middle line:90% So I'm running out of space. 00:50:18.520 --> 00:50:19.950 align:middle line:90% So where shall I write this? 00:50:19.950 --> 00:50:24.030 align:middle line:84% So this L here is the inductance. 00:50:24.030 --> 00:50:29.912 align:middle line:90% 00:50:29.912 --> 00:50:32.120 align:middle line:84% And it's a circuit property that probably many of you 00:50:32.120 --> 00:50:33.180 align:middle line:90% are familiar with. 00:50:33.180 --> 00:50:35.630 align:middle line:84% We're going to define our inductance here 00:50:35.630 --> 00:50:43.430 align:middle line:84% as 1 over mu 0 I phi squared integral of B theta 00:50:43.430 --> 00:50:49.343 align:middle line:84% squared dV, so integration of the volume. 00:50:49.343 --> 00:50:51.760 align:middle line:84% And if you look at that closely, you realize all I've done 00:50:51.760 --> 00:50:53.470 align:middle line:84% is something incredibly tautological. 00:50:53.470 --> 00:50:56.118 align:middle line:84% So I've just defined L to include all of this stuff 00:50:56.118 --> 00:50:57.160 align:middle line:90% that we still don't know. 00:50:57.160 --> 00:50:58.510 align:middle line:84% But this now looks a lot more familiar 00:50:58.510 --> 00:51:00.018 align:middle line:84% because we've got L. That contains 00:51:00.018 --> 00:51:01.060 align:middle line:90% everything we don't know. 00:51:01.060 --> 00:51:03.400 align:middle line:84% But I phi is stuff that we do know because we're measuring it 00:51:03.400 --> 00:51:04.210 align:middle line:90% with a recalcitrant. 00:51:04.210 --> 00:51:05.230 align:middle line:90% So don't worry about it. 00:51:05.230 --> 00:51:06.110 align:middle line:90% I haven't done anything. 00:51:06.110 --> 00:51:07.902 align:middle line:84% This is just sleight of hand at the moment. 00:51:07.902 --> 00:51:09.182 align:middle line:90% But we will resolve it. 00:51:09.182 --> 00:51:11.140 align:middle line:84% What we're going to resolve is try and work out 00:51:11.140 --> 00:51:12.640 align:middle line:90% what's going on with L here. 00:51:12.640 --> 00:51:14.140 align:middle line:84% Perhaps we can just get rid of it. 00:51:14.140 --> 00:51:15.550 align:middle line:90% OK. 00:51:15.550 --> 00:51:17.650 align:middle line:84% I just want to point out the exact value of L 00:51:17.650 --> 00:51:19.600 align:middle line:84% doesn't just depend on the current that's 00:51:19.600 --> 00:51:20.970 align:middle line:90% flowing inside here. 00:51:20.970 --> 00:51:22.720 align:middle line:84% It depends on the distribution of current. 00:51:22.720 --> 00:51:26.590 align:middle line:84% And so if I have, for example, a tokamak where 00:51:26.590 --> 00:51:29.230 align:middle line:84% the current density as a function of radius 00:51:29.230 --> 00:51:33.490 align:middle line:84% looks like this, that has a very different inductance from one 00:51:33.490 --> 00:51:36.110 align:middle line:84% where the current density looks like that. 00:51:36.110 --> 00:51:38.740 align:middle line:84% And so that L is a measure, in some sense, 00:51:38.740 --> 00:51:40.870 align:middle line:90% of the geometry of your system. 00:51:40.870 --> 00:51:43.353 align:middle line:84% It actually doesn't care about the strength of the current. 00:51:43.353 --> 00:51:45.520 align:middle line:84% It just cares about the distribution of the current. 00:51:45.520 --> 00:51:48.470 align:middle line:90% 00:51:48.470 --> 00:51:48.980 align:middle line:90% OK. 00:51:48.980 --> 00:51:50.563 align:middle line:84% And the other thing I want to say here 00:51:50.563 --> 00:51:53.960 align:middle line:84% is that the definition of L we're using here 00:51:53.960 --> 00:51:57.880 align:middle line:84% is sometimes referred to as the energy inductance. 00:51:57.880 --> 00:52:03.760 align:middle line:84% There's another version, which is called the flux inductance. 00:52:03.760 --> 00:52:09.400 align:middle line:84% And that one is defined as L is equal to the magnetic flux 00:52:09.400 --> 00:52:11.820 align:middle line:90% times the current. 00:52:11.820 --> 00:52:14.250 align:middle line:84% These definitions are often similar. 00:52:14.250 --> 00:52:16.290 align:middle line:90% They often give similar results. 00:52:16.290 --> 00:52:19.360 align:middle line:84% But actually, there are some really subtle differences. 00:52:19.360 --> 00:52:21.510 align:middle line:84% And if you want to go plunge into Jackson 00:52:21.510 --> 00:52:23.700 align:middle line:84% or some other equivalent electromagnetism textbook, 00:52:23.700 --> 00:52:24.810 align:middle line:90% you can have a look at it. 00:52:24.810 --> 00:52:26.550 align:middle line:84% But the purposes of this, if you've ever 00:52:26.550 --> 00:52:28.050 align:middle line:84% seen this definition of inductance, 00:52:28.050 --> 00:52:29.717 align:middle line:84% it's not the one we're using here today. 00:52:29.717 --> 00:52:31.390 align:middle line:84% We're using this definition inductance, 00:52:31.390 --> 00:52:35.400 align:middle line:84% which is the total magnetic energy squared [INAUDIBLE] 00:52:35.400 --> 00:52:38.850 align:middle line:84% total magnetic energy over the current squared instead. 00:52:38.850 --> 00:52:39.780 align:middle line:90% OK. 00:52:39.780 --> 00:52:41.760 align:middle line:84% Questions on this before we move on? 00:52:41.760 --> 00:52:50.956 align:middle line:90% 00:52:50.956 --> 00:52:54.305 align:middle line:84% Oh, we're running out of time, I see. 00:52:54.305 --> 00:53:04.190 align:middle line:90% 00:53:04.190 --> 00:53:04.745 align:middle line:90% Okey doke. 00:53:04.745 --> 00:53:40.170 align:middle line:90% 00:53:40.170 --> 00:53:41.870 align:middle line:90% Working with this equation here. 00:53:41.870 --> 00:53:42.630 align:middle line:90% OK. 00:53:42.630 --> 00:53:47.840 align:middle line:84% So we want to ask ourselves, what is the time derivative 00:53:47.840 --> 00:53:55.270 align:middle line:84% of a 1/2 L I phi squared, like that. 00:53:55.270 --> 00:53:56.830 align:middle line:90% Well, we can chain rule this. 00:53:56.830 --> 00:54:03.460 align:middle line:90% We'll get L I phi I phi dot. 00:54:03.460 --> 00:54:13.170 align:middle line:84% We'll get plus 1/2 I phi squared L dot. 00:54:13.170 --> 00:54:14.250 align:middle line:90% OK. 00:54:14.250 --> 00:54:15.930 align:middle line:84% So there's actually two terms here. 00:54:15.930 --> 00:54:17.910 align:middle line:84% We can have a change in the magnetic energy 00:54:17.910 --> 00:54:19.725 align:middle line:84% because this, again, just still represents 00:54:19.725 --> 00:54:20.850 align:middle line:90% the stored magnetic energy. 00:54:20.850 --> 00:54:23.145 align:middle line:84% It can change because the amount of current flowing 00:54:23.145 --> 00:54:24.270 align:middle line:90% through the system changes. 00:54:24.270 --> 00:54:25.450 align:middle line:90% That kind of makes sense. 00:54:25.450 --> 00:54:26.340 align:middle line:84% We know if we change the current, 00:54:26.340 --> 00:54:28.048 align:middle line:84% we're going to change the magnetic field. 00:54:28.048 --> 00:54:30.110 align:middle line:84% So we'll change the magnetic energy density. 00:54:30.110 --> 00:54:32.870 align:middle line:84% But we'll also get a change if we change L. 00:54:32.870 --> 00:54:35.690 align:middle line:84% So I gave you this example earlier, 00:54:35.690 --> 00:54:37.800 align:middle line:84% that L is related to the current distribution. 00:54:37.800 --> 00:54:40.970 align:middle line:84% So say, for example, you have a sawtooth crash or something 00:54:40.970 --> 00:54:43.520 align:middle line:84% like that inside your tokamak that redistributes current. 00:54:43.520 --> 00:54:46.400 align:middle line:84% That current redistribution is going to drive a change in L. 00:54:46.400 --> 00:54:49.250 align:middle line:84% And so that will change the magnetic energy. 00:54:49.250 --> 00:54:51.830 align:middle line:84% And so that magnetic energy will have 00:54:51.830 --> 00:54:54.810 align:middle line:84% to be paid for by, for example, the Poynting flux. 00:54:54.810 --> 00:54:56.790 align:middle line:84% So they'll have to be some balance here. 00:54:56.790 --> 00:54:58.460 align:middle line:84% But for our purposes, we're going 00:54:58.460 --> 00:55:00.413 align:middle line:90% to set this all equal to 0. 00:55:00.413 --> 00:55:02.330 align:middle line:84% We're going to set it equal to 0 because we're 00:55:02.330 --> 00:55:06.290 align:middle line:84% saying that this is 0 because we're in steady state. 00:55:06.290 --> 00:55:09.530 align:middle line:84% And slightly more weaker, we're going to just sort of say, 00:55:09.530 --> 00:55:12.383 align:middle line:84% yeah, this is roughly 0 compared to the other things. 00:55:12.383 --> 00:55:15.050 align:middle line:84% And the reason we're doing that is we can't set it to exactly 0. 00:55:15.050 --> 00:55:16.940 align:middle line:84% Otherwise, we won't be able to sense it with our Rogowski. 00:55:16.940 --> 00:55:19.220 align:middle line:84% And we will have completely failed in this task. 00:55:19.220 --> 00:55:21.110 align:middle line:84% But we're going to say that it's small enough 00:55:21.110 --> 00:55:24.410 align:middle line:84% that our Rogowski, which is super sensitive, can get it. 00:55:24.410 --> 00:55:26.660 align:middle line:84% But it's not big enough to actually cause a change 00:55:26.660 --> 00:55:28.680 align:middle line:90% to our overall picture here. 00:55:28.680 --> 00:55:32.400 align:middle line:84% So we're going to be able to get rid of this term entirely. 00:55:32.400 --> 00:55:36.340 align:middle line:90% 00:55:36.340 --> 00:55:38.470 align:middle line:84% So this means that the power balance for our plasma 00:55:38.470 --> 00:55:41.170 align:middle line:84% is just going to be external heating 00:55:41.170 --> 00:55:45.280 align:middle line:90% is equal to the external-- 00:55:45.280 --> 00:55:48.280 align:middle line:84% the externally injected power is going to go purely into ohmic 00:55:48.280 --> 00:55:50.590 align:middle line:90% heating our plasma. 00:55:50.590 --> 00:55:53.830 align:middle line:84% And so that means that the power coming in 00:55:53.830 --> 00:55:57.190 align:middle line:84% is going to be equal to the integral over the volume 00:55:57.190 --> 00:56:02.930 align:middle line:84% of the current density over the local conductivity 00:56:02.930 --> 00:56:07.940 align:middle line:84% of our plasma, sigma, integrated over the volume here. 00:56:07.940 --> 00:56:10.900 align:middle line:84% This is just a rearrangement where 00:56:10.900 --> 00:56:15.350 align:middle line:84% we say that J is equal to sigma E, like that. 00:56:15.350 --> 00:56:23.100 align:middle line:84% So we replace J dot E with J squared over sigma. 00:56:23.100 --> 00:56:27.460 align:middle line:84% And you will remember that, of course, in general, sigma 00:56:27.460 --> 00:56:30.490 align:middle line:84% is a tensor, especially in plasmas. 00:56:30.490 --> 00:56:34.870 align:middle line:84% It can be a tensor, which is very much not a multiple 00:56:34.870 --> 00:56:36.080 align:middle line:90% of the identity matrix. 00:56:36.080 --> 00:56:39.980 align:middle line:84% But here, we're just going to treat it as a scalar. 00:56:39.980 --> 00:56:42.500 align:middle line:84% And there can be some subtleties involved in this technique 00:56:42.500 --> 00:56:45.170 align:middle line:84% if you think about the tensor nature of this instead. 00:56:45.170 --> 00:56:46.470 align:middle line:90% OK. 00:56:46.470 --> 00:56:48.300 align:middle line:84% So now, we've got this squared here. 00:56:48.300 --> 00:56:53.630 align:middle line:84% And again, J is just going to be equal to I 00:56:53.630 --> 00:56:56.727 align:middle line:90% phi over pi A squared. 00:56:56.727 --> 00:56:58.310 align:middle line:84% I probably forgot to mention somewhere 00:56:58.310 --> 00:57:00.560 align:middle line:84% that our tokamak has a minor radius of A. 00:57:00.560 --> 00:57:02.100 align:middle line:90% But we use this a lot. 00:57:02.100 --> 00:57:11.600 align:middle line:84% So this is-- actually, sorry, I'm being slightly vague here. 00:57:11.600 --> 00:57:13.940 align:middle line:84% So this is like the average J here 00:57:13.940 --> 00:57:15.190 align:middle line:90% that we're going to work with. 00:57:15.190 --> 00:57:17.110 align:middle line:84% And there's going to be some subtlety between going 00:57:17.110 --> 00:57:18.940 align:middle line:84% through J squared averaged over the volume 00:57:18.940 --> 00:57:20.957 align:middle line:84% to just J average of the volume squared. 00:57:20.957 --> 00:57:22.540 align:middle line:84% I'm going to sweep that under the rug. 00:57:22.540 --> 00:57:25.540 align:middle line:84% We're just going to do the second one, yeah. 00:57:25.540 --> 00:57:30.270 align:middle line:84% And so that means we end up with an-- 00:57:30.270 --> 00:57:32.710 align:middle line:90% let's see here. 00:57:32.710 --> 00:57:38.920 align:middle line:84% We also have a volume of our system, which is pi A squared 00:57:38.920 --> 00:57:43.120 align:middle line:84% dotted with 2 pi capital R. That's the volume of a torus 00:57:43.120 --> 00:58:03.990 align:middle line:84% with minor radius and major radius R. 00:58:03.990 --> 00:58:05.580 align:middle line:90% OK, good. 00:58:05.580 --> 00:58:08.690 align:middle line:84% So that means we end up by being able to write 00:58:08.690 --> 00:58:17.210 align:middle line:84% our conductivity volume averaged conductivity as equal to I 00:58:17.210 --> 00:58:23.780 align:middle line:84% phi squared over the power times by-- 00:58:23.780 --> 00:58:31.680 align:middle line:84% this is going to be 2 pi R over pi A squared here. 00:58:31.680 --> 00:58:34.170 align:middle line:84% And this I phi squared over the power 00:58:34.170 --> 00:58:41.480 align:middle line:84% is just going to be equal to I phi over D phi, like this, 00:58:41.480 --> 00:58:46.940 align:middle line:84% still times by 2 R over A squared. 00:58:46.940 --> 00:58:51.050 align:middle line:84% So we can now measure the conductivity of our plasma 00:58:51.050 --> 00:58:53.525 align:middle line:84% because we have measured this using our Rogowski. 00:58:53.525 --> 00:58:57.060 align:middle line:90% 00:58:57.060 --> 00:59:02.080 align:middle line:84% We've measured this using our flux loop. 00:59:02.080 --> 00:59:05.380 align:middle line:84% And I sincerely hope we know the size of our tokamak. 00:59:05.380 --> 00:59:07.960 align:middle line:84% Otherwise, we've kind of lost the stuff. 00:59:07.960 --> 00:59:11.130 align:middle line:84% So this is a pretty neat result. And the reason it's neat 00:59:11.130 --> 00:59:13.530 align:middle line:84% is because of what theory tells us 00:59:13.530 --> 00:59:15.570 align:middle line:90% about this scalar conductivity. 00:59:15.570 --> 00:59:19.290 align:middle line:84% So theory tells us that the conductivity sigma is something 00:59:19.290 --> 00:59:29.140 align:middle line:84% like 2 times 10 to the 4 TE the 3/2 over z, 00:59:29.140 --> 00:59:31.390 align:middle line:84% which is we'll talk about it in a moment, 00:59:31.390 --> 00:59:32.820 align:middle line:84% but a bit like the atomic charge, 00:59:32.820 --> 00:59:34.980 align:middle line:84% but not quite here because I'm going to give it a little z 00:59:34.980 --> 00:59:37.590 align:middle line:84% sigma to show it's not quite the z you've been expecting-- 00:59:37.590 --> 00:59:43.640 align:middle line:84% and then times by the Coulomb logarithm, that way up. 00:59:43.640 --> 00:59:48.880 align:middle line:84% And this is in units of 1 over ohm meters. 00:59:48.880 --> 00:59:51.640 align:middle line:84% And I think, to get this coefficient out front, 00:59:51.640 --> 00:59:53.800 align:middle line:84% this is in units of electron volts. 00:59:53.800 --> 00:59:56.330 align:middle line:84% So if you look inside the NRL Plasma Formulary, 00:59:56.330 --> 00:59:57.897 align:middle line:90% you'll get this nice equation. 00:59:57.897 --> 00:59:59.980 align:middle line:84% So this is to do with the Spitzer resistive, which 00:59:59.980 --> 01:00:01.750 align:middle line:90% you saw in part 1. 01:00:01.750 --> 01:00:05.080 align:middle line:84% And so there's some kind of nice things in here. 01:00:05.080 --> 01:00:07.780 align:middle line:90% We've now measured sigma. 01:00:07.780 --> 01:00:10.900 align:middle line:84% The Coulomb logarithm is a very slowly changing function 01:00:10.900 --> 01:00:13.390 align:middle line:84% of the plasma parameters because it's in a logarithm. 01:00:13.390 --> 01:00:16.060 align:middle line:84% And for something like a tokamak, it's about 10. 01:00:16.060 --> 01:00:17.290 align:middle line:90% So you can use 10. 01:00:17.290 --> 01:00:18.800 align:middle line:90% You don't really care that much. 01:00:18.800 --> 01:00:20.930 align:middle line:90% This is interesting. 01:00:20.930 --> 01:00:24.250 align:middle line:84% So z, or z sigma, is always going to be greater 01:00:24.250 --> 01:00:28.540 align:middle line:90% than the true z of your plasma. 01:00:28.540 --> 01:00:30.642 align:middle line:84% It's a little bit like z effective, 01:00:30.642 --> 01:00:32.350 align:middle line:84% which you may have come across in-- yeah, 01:00:32.350 --> 01:00:33.980 align:middle line:90% we did it in the fusion class. 01:00:33.980 --> 01:00:34.480 align:middle line:90% It's great. 01:00:34.480 --> 01:00:35.950 align:middle line:84% It's a little bit like z effective, 01:00:35.950 --> 01:00:37.630 align:middle line:84% in that it also takes into account 01:00:37.630 --> 01:00:39.760 align:middle line:84% the fact that you have some impurities inside your plasma. 01:00:39.760 --> 01:00:41.927 align:middle line:84% And they increase the number of electrons available. 01:00:41.927 --> 01:00:46.102 align:middle line:90% But it's also not z effective. 01:00:46.102 --> 01:00:47.810 align:middle line:84% And if you want to understand what it is, 01:00:47.810 --> 01:00:51.920 align:middle line:84% you have to go do some detailed calculations where 01:00:51.920 --> 01:00:54.620 align:middle line:84% you take into account what those extra electrons from impurities 01:00:54.620 --> 01:00:55.560 align:middle line:90% are doing. 01:00:55.560 --> 01:00:58.280 align:middle line:84% However, if you have a perfectly pure plasma, 01:00:58.280 --> 01:01:02.480 align:middle line:84% like a pure hydrogen, or deuterium, or tritium, 01:01:02.480 --> 01:01:07.220 align:middle line:84% then you do end up with z sigma equals z equals 1. 01:01:07.220 --> 01:01:09.380 align:middle line:84% So let's say you've done a really good job 01:01:09.380 --> 01:01:11.540 align:middle line:84% and you have got the very nice and pure plasma. 01:01:11.540 --> 01:01:15.980 align:middle line:84% You could probably set that equal to 1 or at least 1-ish. 01:01:15.980 --> 01:01:17.640 align:middle line:84% Let's not worry about it too much. 01:01:17.640 --> 01:01:19.535 align:middle line:84% So that means, by measuring sigma-- 01:01:19.535 --> 01:01:39.260 align:middle line:90% 01:01:39.260 --> 01:01:46.160 align:middle line:84% by measuring sigma from your Rogowski and your flux loop 01:01:46.160 --> 01:01:50.300 align:middle line:84% and the geometry of your tokamak, 01:01:50.300 --> 01:01:54.702 align:middle line:84% you can get out the plasma temperature, 01:01:54.702 --> 01:01:56.910 align:middle line:84% which is not a bad measurement for two loops of wire. 01:01:56.910 --> 01:02:01.280 align:middle line:90% 01:02:01.280 --> 01:02:04.390 align:middle line:84% The volume average plasma temperature admittedly, 01:02:04.390 --> 01:02:06.390 align:middle line:84% which temperatures are going to be dominated by. 01:02:06.390 --> 01:02:09.700 align:middle line:90% 01:02:09.700 --> 01:02:12.190 align:middle line:84% It's going to be dominated by the cold plasma at the edge 01:02:12.190 --> 01:02:14.866 align:middle line:90% or the hot plasma at the core. 01:02:14.866 --> 01:02:15.366 align:middle line:90% [INAUDIBLE]? 01:02:15.366 --> 01:02:20.292 align:middle line:90% STUDENT: 01:02:20.292 --> 01:02:21.000 align:middle line:90% JACK HARE: Maybe. 01:02:21.000 --> 01:02:23.790 align:middle line:90% 01:02:23.790 --> 01:02:25.890 align:middle line:90% Any advances on that? 01:02:25.890 --> 01:02:28.370 align:middle line:90% STUDENT: [INAUDIBLE]. 01:02:28.370 --> 01:02:32.180 align:middle line:84% JACK HARE: Yeah, so the hot plasma is much more conductive. 01:02:32.180 --> 01:02:34.370 align:middle line:84% We aren't including inductance here. 01:02:34.370 --> 01:02:37.370 align:middle line:84% So inductance would force the plasma to the outside 01:02:37.370 --> 01:02:38.870 align:middle line:84% because it's going to want to reside 01:02:38.870 --> 01:02:41.520 align:middle line:84% on the skin of the conductor, if we've got a non-steady state 01:02:41.520 --> 01:02:42.020 align:middle line:90% system. 01:02:42.020 --> 01:02:44.270 align:middle line:84% But in steady state, the inductance doesn't matter. 01:02:44.270 --> 01:02:48.027 align:middle line:84% The current has time to soak through the entire conductor. 01:02:48.027 --> 01:02:49.610 align:middle line:84% So this would be a pretty good measure 01:02:49.610 --> 01:02:51.277 align:middle line:84% of where most of the current is flowing, 01:02:51.277 --> 01:02:53.070 align:middle line:84% which is in the core of the plasma. 01:02:53.070 --> 01:02:55.250 align:middle line:84% So that's actually pretty cool because this is 01:02:55.250 --> 01:02:56.660 align:middle line:90% a really hard thing to measure. 01:02:56.660 --> 01:02:58.160 align:middle line:84% I mean, the temperature is something 01:02:58.160 --> 01:03:00.350 align:middle line:84% that people invest a huge amount of money in doing. 01:03:00.350 --> 01:03:02.660 align:middle line:84% Now, this isn't a great measurement. 01:03:02.660 --> 01:03:06.320 align:middle line:84% These loops are not exactly the most accurate things. 01:03:06.320 --> 01:03:10.280 align:middle line:84% And of course, you've got a T to the 3/2, right? 01:03:10.280 --> 01:03:15.200 align:middle line:84% And T to the 3/2 means that any small error in sigma is going 01:03:15.200 --> 01:03:17.450 align:middle line:84% to be amplified a bit when we measure the temperature. 01:03:17.450 --> 01:03:21.170 align:middle line:84% But it is still remarkably easy to do and MC of plasma. 01:03:21.170 --> 01:03:24.980 align:middle line:84% So this, again, applies for any plasma. 01:03:24.980 --> 01:03:25.610 align:middle line:90% Let's see. 01:03:25.610 --> 01:03:27.180 align:middle line:90% It needs to be steady state. 01:03:27.180 --> 01:03:29.520 align:middle line:84% We've derived it with toroidal symmetry. 01:03:29.520 --> 01:03:32.160 align:middle line:84% But I don't see any reason why you couldn't try and redo this 01:03:32.160 --> 01:03:35.400 align:middle line:90% for like a z-pinch plasma. 01:03:35.400 --> 01:03:39.090 align:middle line:84% I think we probably have made some assumptions. 01:03:39.090 --> 01:03:41.730 align:middle line:84% I don't think we've made that many assumptions in terms 01:03:41.730 --> 01:03:45.120 align:middle line:84% of the axisymmetry of the plasma because the current has 01:03:45.120 --> 01:03:46.240 align:middle line:90% to be constant around it. 01:03:46.240 --> 01:03:47.660 align:middle line:84% So even if it's a stellarator, you 01:03:47.660 --> 01:03:49.410 align:middle line:84% still need one measurement with a Rogowski 01:03:49.410 --> 01:03:50.993 align:middle line:84% to get the current through the plasma. 01:03:50.993 --> 01:03:54.140 align:middle line:90% So yeah, questions on this? 01:03:54.140 --> 01:03:55.563 align:middle line:90% Yeah? 01:03:55.563 --> 01:03:56.716 align:middle line:90% STUDENT: [INAUDIBLE]? 01:03:56.716 --> 01:04:00.180 align:middle line:90% 01:04:00.180 --> 01:04:02.515 align:middle line:84% JACK HARE: Yeah, so always when we 01:04:02.515 --> 01:04:04.140 align:middle line:84% say things like steady state and things 01:04:04.140 --> 01:04:05.700 align:middle line:90% like that, what we should say-- 01:04:05.700 --> 01:04:07.710 align:middle line:84% it's like saying large or something. 01:04:07.710 --> 01:04:10.200 align:middle line:84% We should come up with a dimensionless parameter 01:04:10.200 --> 01:04:11.610 align:middle line:90% which characterizes that. 01:04:11.610 --> 01:04:16.550 align:middle line:84% So what dimensionless parameter should we 01:04:16.550 --> 01:04:19.790 align:middle line:84% have to enable us to use the steady state assumption, which 01:04:19.790 --> 01:04:22.970 align:middle line:84% was dropping this term and actually also this term as well? 01:04:22.970 --> 01:04:25.615 align:middle line:90% 01:04:25.615 --> 01:04:26.990 align:middle line:84% We're talking about steady state. 01:04:26.990 --> 01:04:28.290 align:middle line:90% We're talking about times. 01:04:28.290 --> 01:04:32.720 align:middle line:84% So there should be some sort of timescale that we're measuring. 01:04:32.720 --> 01:04:35.230 align:middle line:84% So there should be like an inductive timescale 01:04:35.230 --> 01:04:37.420 align:middle line:84% here where we could form a timescale out of this. 01:04:37.420 --> 01:04:38.890 align:middle line:90% STUDENT: [INAUDIBLE]. 01:04:38.890 --> 01:04:41.140 align:middle line:90% 01:04:41.140 --> 01:04:42.890 align:middle line:84% JACK HARE: Yeah, rate of change of current 01:04:42.890 --> 01:04:44.570 align:middle line:84% or the current over the rate of change of the current 01:04:44.570 --> 01:04:45.780 align:middle line:90% will give you some timescale. 01:04:45.780 --> 01:04:46.940 align:middle line:84% So that would be a good timescale. 01:04:46.940 --> 01:04:47.960 align:middle line:90% But that needs to be small. 01:04:47.960 --> 01:04:49.400 align:middle line:84% That needs to be much, much less-- 01:04:49.400 --> 01:04:53.210 align:middle line:84% not much less than 1 because that has units of time. 01:04:53.210 --> 01:04:55.320 align:middle line:84% It needs to be smaller than the pulse length. 01:04:55.320 --> 01:04:57.900 align:middle line:84% So the current must change on the times-- 01:04:57.900 --> 01:04:59.150 align:middle line:90% I got it the wrong way around. 01:04:59.150 --> 01:05:01.550 align:middle line:84% The current must change on a time scale which is long 01:05:01.550 --> 01:05:03.560 align:middle line:90% compared to the current pulse. 01:05:03.560 --> 01:05:05.330 align:middle line:90% So it changes slowly, right? 01:05:05.330 --> 01:05:08.000 align:middle line:84% So if the current changes sufficiently slowly, 01:05:08.000 --> 01:05:09.900 align:middle line:84% you'll be able to use these equations. 01:05:09.900 --> 01:05:13.640 align:middle line:84% And if the change in current is, say, on the order of 10 01:05:13.640 --> 01:05:15.710 align:middle line:84% to the minus 2, then you'll have errors 01:05:15.710 --> 01:05:18.200 align:middle line:84% on the order of 10 to the minus 2 or something in here 01:05:18.200 --> 01:05:20.147 align:middle line:84% because I think this is mostly linear. 01:05:20.147 --> 01:05:21.980 align:middle line:84% And that will go slightly wrong when you try 01:05:21.980 --> 01:05:25.420 align:middle line:84% and work out the temperature from sigma 01:05:25.420 --> 01:05:27.170 align:middle line:84% because it's been linear up to this point. 01:05:27.170 --> 01:05:28.580 align:middle line:84% And then we'll have some slight non-linearity. 01:05:28.580 --> 01:05:29.670 align:middle line:90% But you get the idea. 01:05:29.670 --> 01:05:32.120 align:middle line:84% So you with all of these things, whenever 01:05:32.120 --> 01:05:34.880 align:middle line:84% we make these assumptions, these assumptions can be justified. 01:05:34.880 --> 01:05:37.580 align:middle line:84% There'll be some dimensionless parameter that allows 01:05:37.580 --> 01:05:39.110 align:middle line:90% us to justify that assumption. 01:05:39.110 --> 01:05:41.030 align:middle line:90% Yeah, other questions? 01:05:41.030 --> 01:05:44.907 align:middle line:90% 01:05:44.907 --> 01:05:45.865 align:middle line:90% Anything from Colombia? 01:05:45.865 --> 01:05:51.810 align:middle line:90% 01:05:51.810 --> 01:05:52.784 align:middle line:90% STUDENT: [INAUDIBLE]? 01:05:52.784 --> 01:06:20.160 align:middle line:90% 01:06:20.160 --> 01:06:21.750 align:middle line:84% JACK HARE: Well, we're averaging-- 01:06:21.750 --> 01:06:24.960 align:middle line:84% really, what we're averaging is this quantity here, J dot E. 01:06:24.960 --> 01:06:28.710 align:middle line:84% So E is going to be practically uniform throughout the entire 01:06:28.710 --> 01:06:31.318 align:middle line:84% thing because we're driving this flux of the transformer. 01:06:31.318 --> 01:06:33.360 align:middle line:84% Or we're driving this potential as a transformer. 01:06:33.360 --> 01:06:34.950 align:middle line:84% That's not going to change very much. 01:06:34.950 --> 01:06:36.520 align:middle line:90% I'm not worried about that. 01:06:36.520 --> 01:06:41.940 align:middle line:90% J I agree is peaked in general. 01:06:41.940 --> 01:06:45.550 align:middle line:84% For your typical tokamak plasma, it's going to be peaked. 01:06:45.550 --> 01:06:49.252 align:middle line:84% So I think it's possible that we are biased towards the hotter 01:06:49.252 --> 01:06:50.460 align:middle line:90% temperatures because of that. 01:06:50.460 --> 01:06:51.570 align:middle line:84% And that's probably where it comes in, 01:06:51.570 --> 01:06:53.700 align:middle line:84% as I was saying, that they are the ones carrying the current. 01:06:53.700 --> 01:06:56.033 align:middle line:84% And so we're biased towards measuring their temperature. 01:06:56.033 --> 01:06:58.740 align:middle line:84% So yeah, it's also interesting, like what does it 01:06:58.740 --> 01:07:00.900 align:middle line:84% mean to measure an average temperature? 01:07:00.900 --> 01:07:03.300 align:middle line:84% Because if you have a small bit of plasma 01:07:03.300 --> 01:07:05.100 align:middle line:84% that's very hot but very low density, 01:07:05.100 --> 01:07:06.850 align:middle line:84% it has very little thermal energy density. 01:07:06.850 --> 01:07:08.517 align:middle line:84% And if you have a large amount of plasma 01:07:08.517 --> 01:07:10.410 align:middle line:84% that's cold, if you mix those together, 01:07:10.410 --> 01:07:12.115 align:middle line:84% like mixing hot water with cold water, 01:07:12.115 --> 01:07:13.740 align:middle line:84% you don't just average the temperature. 01:07:13.740 --> 01:07:15.782 align:middle line:84% You don't say, this is boiling, this is freezing, 01:07:15.782 --> 01:07:17.100 align:middle line:90% the average is 50. 01:07:17.100 --> 01:07:19.750 align:middle line:84% You would weight it by the number of particles you have. 01:07:19.750 --> 01:07:26.710 align:middle line:84% And so yeah, I think all of this is kind of slightly hand-wavy. 01:07:26.710 --> 01:07:29.530 align:middle line:84% And again, I think Hutchinson goes into a bit more detail 01:07:29.530 --> 01:07:31.690 align:middle line:84% and says that, given the accuracy that you can make 01:07:31.690 --> 01:07:34.510 align:middle line:84% these measurements with, some of the subtleties of the theory 01:07:34.510 --> 01:07:36.700 align:middle line:84% are not that important because your noise is 01:07:36.700 --> 01:07:40.458 align:middle line:84% larger than the inadequacy of your models. 01:07:40.458 --> 01:07:41.500 align:middle line:90% But it's a good question. 01:07:41.500 --> 01:07:43.750 align:middle line:84% If you really wanted to do this to measure temperature 01:07:43.750 --> 01:07:45.640 align:middle line:84% on like ITER, you might want to spend a little bit 01:07:45.640 --> 01:07:46.682 align:middle line:90% more time thinking about. 01:07:46.682 --> 01:07:47.980 align:middle line:90% Yeah, cool. 01:07:47.980 --> 01:07:48.954 align:middle line:90% STUDENT: [INAUDIBLE]? 01:07:48.954 --> 01:07:52.850 align:middle line:90% 01:07:52.850 --> 01:07:54.020 align:middle line:90% JACK HARE: Yes, absolutely. 01:07:54.020 --> 01:07:58.140 align:middle line:84% Yeah, so this is really just a steady state section, yeah. 01:07:58.140 --> 01:07:58.640 align:middle line:90% Cool. 01:07:58.640 --> 01:08:01.210 align:middle line:90% 01:08:01.210 --> 01:08:02.740 align:middle line:90% Yeah. 01:08:02.740 --> 01:08:03.830 align:middle line:90% OK. 01:08:03.830 --> 01:08:05.080 align:middle line:90% We've still got a little time. 01:08:05.080 --> 01:08:06.288 align:middle line:90% So we're going to keep going. 01:08:06.288 --> 01:08:10.254 align:middle line:90% 01:08:10.254 --> 01:08:12.346 align:middle line:90% Let me clear some space. 01:08:12.346 --> 01:08:36.689 align:middle line:90% 01:08:36.689 --> 01:08:38.919 align:middle line:90% So another use of this in MTF-- 01:08:38.919 --> 01:08:44.559 align:middle line:90% 01:08:44.559 --> 01:08:46.700 align:middle line:84% here, we have a system in which we 01:08:46.700 --> 01:08:49.819 align:middle line:84% create a well inside our magnetic pressure, which 01:08:49.819 --> 01:08:51.930 align:middle line:90% our plasma sits inside. 01:08:51.930 --> 01:08:55.680 align:middle line:84% So if I have a diagram like this, 01:08:55.680 --> 01:09:00.229 align:middle line:84% where this is the minor radius of our machine-- 01:09:00.229 --> 01:09:02.359 align:middle line:84% actually, just some spatial coordinates. 01:09:02.359 --> 01:09:04.399 align:middle line:84% I don't think I've drawn this correctly here. 01:09:04.399 --> 01:09:09.560 align:middle line:84% But we have some magnetic field pressure profile, like this. 01:09:09.560 --> 01:09:13.814 align:middle line:90% 01:09:13.814 --> 01:09:15.580 align:middle line:84% And we've got a little dip inside it. 01:09:15.580 --> 01:09:17.529 align:middle line:84% And that little dip is designed such 01:09:17.529 --> 01:09:23.080 align:middle line:84% that the thermal pressure of our plasma, E equals NT, 01:09:23.080 --> 01:09:24.319 align:middle line:90% sits inside it. 01:09:24.319 --> 01:09:27.342 align:middle line:84% And so the total pressure is constant here. 01:09:27.342 --> 01:09:29.550 align:middle line:84% So this means that the system is in pressure balance. 01:09:29.550 --> 01:09:30.810 align:middle line:90% It's in equilibrium. 01:09:30.810 --> 01:09:32.300 align:middle line:90% OK. 01:09:32.300 --> 01:09:35.870 align:middle line:84% The neat thing about that is, because the magnetic field is 01:09:35.870 --> 01:09:37.970 align:middle line:90% the thing that's allowing-- 01:09:37.970 --> 01:09:39.680 align:middle line:84% the magnetic field is the thing that's 01:09:39.680 --> 01:09:42.620 align:middle line:84% confining the pressure, by measuring the magnetic field 01:09:42.620 --> 01:09:46.399 align:middle line:84% in an MTF device, we can indirectly measure the pressure. 01:09:46.399 --> 01:09:47.930 align:middle line:84% And you'll remember that a pressure 01:09:47.930 --> 01:09:50.569 align:middle line:84% is a pretty key quantity for our fusion 01:09:50.569 --> 01:09:53.120 align:middle line:84% because we want to have something that goes like-- 01:09:53.120 --> 01:09:54.680 align:middle line:84% what's that nice propaganda formula? 01:09:54.680 --> 01:10:00.140 align:middle line:90% We've got like p squared-- 01:10:00.140 --> 01:10:01.780 align:middle line:90% anyway, whatever. 01:10:01.780 --> 01:10:05.900 align:middle line:84% I can't remember how they write it on the board in NW-17. 01:10:05.900 --> 01:10:08.692 align:middle line:84% But the fusion power is going to go as the pressure squared. 01:10:08.692 --> 01:10:10.400 align:middle line:84% So we'd like to know what the pressure is 01:10:10.400 --> 01:10:12.600 align:middle line:84% because we want to know what our fusion power is. 01:10:12.600 --> 01:10:14.930 align:middle line:84% And we can do that by making very careful measurements 01:10:14.930 --> 01:10:16.040 align:middle line:90% of the magnetic field. 01:10:16.040 --> 01:10:18.820 align:middle line:84% And we're going to talk about that in detail now. 01:10:18.820 --> 01:10:21.070 align:middle line:84% But we're going to get a little bit lost in the weeds. 01:10:21.070 --> 01:10:23.445 align:middle line:84% I'm not going to have a chance to finish it this lecture. 01:10:23.445 --> 01:10:25.340 align:middle line:84% So I just want you to focus on the fact 01:10:25.340 --> 01:10:33.600 align:middle line:84% that what we're trying to do is measure P, the pressure, 01:10:33.600 --> 01:10:35.470 align:middle line:90% from the magnetic field. 01:10:35.470 --> 01:10:38.910 align:middle line:84% So again, we're trying to get at some interesting quantities 01:10:38.910 --> 01:10:42.780 align:middle line:84% inside the plasma just using a few B-dot probes, which, 01:10:42.780 --> 01:10:45.000 align:middle line:90% again, is kind of cool. 01:10:45.000 --> 01:10:45.540 align:middle line:90% OK. 01:10:45.540 --> 01:10:47.040 align:middle line:84% So we'll start by looking at this 01:10:47.040 --> 01:10:49.080 align:middle line:90% in terms of a force balance. 01:10:49.080 --> 01:10:52.890 align:middle line:84% So our steady state MHD equation is 01:10:52.890 --> 01:10:59.340 align:middle line:84% J cross B minus the gradient of pressure equals 0, like that. 01:10:59.340 --> 01:11:02.490 align:middle line:84% If we insert Maxwell's equation-- 01:11:02.490 --> 01:11:04.950 align:middle line:84% one of Maxwell's equations, ignoring the displacement 01:11:04.950 --> 01:11:09.270 align:middle line:84% current-- so curl of B equals mu 0 J-- 01:11:09.270 --> 01:11:11.580 align:middle line:84% into this equation, we end up with something 01:11:11.580 --> 01:11:14.490 align:middle line:84% that we can eventually rewrite in the form 01:11:14.490 --> 01:11:20.790 align:middle line:84% minus the gradient of pressure plus B squared 01:11:20.790 --> 01:11:34.600 align:middle line:84% over 2 mu 0 plus 1 over mu 0 B dot the gradient of B equals 0. 01:11:34.600 --> 01:11:37.883 align:middle line:84% So this balance equation here-- again, 01:11:37.883 --> 01:11:39.050 align:middle line:90% it's equivalent to this one. 01:11:39.050 --> 01:11:40.610 align:middle line:90% We've just eliminated J-- 01:11:40.610 --> 01:11:42.890 align:middle line:90% has two sets of terms in it. 01:11:42.890 --> 01:11:46.970 align:middle line:84% These terms look like pressure gradients. 01:11:46.970 --> 01:11:49.910 align:middle line:84% The thermal pressure is definitely a pressure gradient. 01:11:49.910 --> 01:11:53.630 align:middle line:84% But the magnetic energy density also 01:11:53.630 --> 01:11:56.182 align:middle line:84% looks like a sort of magnetic pressure gradient. 01:11:56.182 --> 01:11:57.515 align:middle line:90% So these two terms look similar. 01:11:57.515 --> 01:12:00.120 align:middle line:90% 01:12:00.120 --> 01:12:04.070 align:middle line:84% And this term here is the tension force. 01:12:04.070 --> 01:12:08.090 align:middle line:84% Or you might want to call it the curvature force 01:12:08.090 --> 01:12:10.342 align:middle line:84% because you can see from B dot grad B, 01:12:10.342 --> 01:12:12.800 align:middle line:84% there's going to be some term associated with the curvature 01:12:12.800 --> 01:12:13.790 align:middle line:90% of the magnetic field. 01:12:13.790 --> 01:12:18.610 align:middle line:90% 01:12:18.610 --> 01:12:22.630 align:middle line:84% Now, this is a very hard equation to solve in general. 01:12:22.630 --> 01:12:24.092 align:middle line:84% So this is fully three dimensional. 01:12:24.092 --> 01:12:25.800 align:middle line:84% If you're trying to solve this in a fully 01:12:25.800 --> 01:12:28.380 align:middle line:84% three dimensional object, like a stellarator 01:12:28.380 --> 01:12:31.200 align:middle line:84% or even inside a tokamak where you 01:12:31.200 --> 01:12:34.450 align:middle line:84% haven't made any assumptions, this is very, very, very hard. 01:12:34.450 --> 01:12:36.480 align:middle line:84% So what we're going to do is solve 01:12:36.480 --> 01:12:38.800 align:middle line:90% this instead in a cylinder. 01:12:38.800 --> 01:12:42.300 align:middle line:84% So we're going to take our tokamak. 01:12:42.300 --> 01:12:45.190 align:middle line:90% We're going to cut. 01:12:45.190 --> 01:12:48.130 align:middle line:90% And we're going to unfold it. 01:12:48.130 --> 01:12:52.000 align:middle line:84% And we're going to turn it into a long, thin cylinder, 01:12:52.000 --> 01:12:52.810 align:middle line:90% like this. 01:12:52.810 --> 01:12:55.630 align:middle line:84% You can equivalently attempt to do this 01:12:55.630 --> 01:12:58.330 align:middle line:84% by making the aspect ratio of this very, very large. 01:12:58.330 --> 01:13:00.430 align:middle line:84% But rather than that, we're just going 01:13:00.430 --> 01:13:02.240 align:middle line:90% to do it this way around here. 01:13:02.240 --> 01:13:05.980 align:middle line:84% So in our standard tokamak toroidal geometry, 01:13:05.980 --> 01:13:09.930 align:middle line:84% we have the phi coordinates and the theta 01:13:09.930 --> 01:13:12.720 align:middle line:84% coordinate, like that, the toroidal and poloidal 01:13:12.720 --> 01:13:13.720 align:middle line:90% coordinates. 01:13:13.720 --> 01:13:15.960 align:middle line:84% And now, we're going to have the phi 01:13:15.960 --> 01:13:17.980 align:middle line:90% coordinates in this direction. 01:13:17.980 --> 01:13:19.890 align:middle line:84% So I may occasionally refer it to it 01:13:19.890 --> 01:13:23.280 align:middle line:84% as z instead because it's now cylindrical coordinates. 01:13:23.280 --> 01:13:25.980 align:middle line:84% And our theta coordinate is still 01:13:25.980 --> 01:13:29.470 align:middle line:84% going to wrap around our plasma, like that. 01:13:29.470 --> 01:13:30.960 align:middle line:84% So we're going to look at solutions 01:13:30.960 --> 01:13:33.210 align:middle line:84% to this equation in this geometry 01:13:33.210 --> 01:13:34.320 align:middle line:90% because it's much simpler. 01:13:34.320 --> 01:13:36.030 align:middle line:84% We'll be trying to find ways where 01:13:36.030 --> 01:13:38.820 align:middle line:84% we can use our B-dot probes to measure pressure 01:13:38.820 --> 01:13:40.080 align:middle line:90% in this geometry. 01:13:40.080 --> 01:13:42.090 align:middle line:84% It is possible to use all of these techniques 01:13:42.090 --> 01:13:43.290 align:middle line:90% in a full toroidal geometry. 01:13:43.290 --> 01:13:45.990 align:middle line:84% You just have to go back and solve this equation instead, 01:13:45.990 --> 01:13:47.130 align:middle line:90% much more complicated. 01:13:47.130 --> 01:13:49.050 align:middle line:84% Any questions on this before we keep going? 01:13:49.050 --> 01:13:52.382 align:middle line:90% 01:13:52.382 --> 01:13:53.810 align:middle line:90% STUDENT: [INAUDIBLE]? 01:13:53.810 --> 01:13:57.663 align:middle line:90% 01:13:57.663 --> 01:13:59.080 align:middle line:84% JACK HARE: Yeah, so you would have 01:13:59.080 --> 01:14:01.450 align:middle line:84% to solve this to get the [INAUDIBLE].. 01:14:01.450 --> 01:14:04.435 align:middle line:84% Or I think in [INAUDIBLE] you still assume axisymmetry. 01:14:04.435 --> 01:14:06.310 align:middle line:84% So you can still have a reduced form of this. 01:14:06.310 --> 01:14:07.810 align:middle line:90% But it's still very complicated. 01:14:07.810 --> 01:14:09.230 align:middle line:90% Yeah, exactly. 01:14:09.230 --> 01:14:12.730 align:middle line:84% So again, just for demonstrating how this works intuitively, 01:14:12.730 --> 01:14:14.710 align:middle line:84% we're going to use this very simple geometry. 01:14:14.710 --> 01:14:18.030 align:middle line:84% But these techniques still apply for real tokamak geometries 01:14:18.030 --> 01:14:18.530 align:middle line:90% as well. 01:14:18.530 --> 01:14:20.860 align:middle line:84% So don't think that this is only relevant to z-pinches. 01:14:20.860 --> 01:14:22.965 align:middle line:84% This is very much relevant to other stuff. 01:14:22.965 --> 01:14:26.990 align:middle line:84% I'm just going to make my z-pinch a little bit shorter so 01:14:26.990 --> 01:14:30.538 align:middle line:84% that I can fit in an equation underneath here 01:14:30.538 --> 01:14:32.080 align:middle line:84% because one of the things we're going 01:14:32.080 --> 01:14:38.810 align:middle line:84% to be interested in measuring is the magnetic field B theta here. 01:14:38.810 --> 01:14:43.710 align:middle line:84% And this magnetic field will take some arbitrary values. 01:14:43.710 --> 01:14:47.300 align:middle line:84% But as with any other function, we can decompose this. 01:14:47.300 --> 01:14:48.840 align:middle line:84% We can do a Fourier decomposition. 01:14:48.840 --> 01:14:50.840 align:middle line:84% And so we're going to do a Fourier decomposition 01:14:50.840 --> 01:14:51.817 align:middle line:90% on this magnetic field. 01:14:51.817 --> 01:14:53.900 align:middle line:84% And then we're going to talk about different terms 01:14:53.900 --> 01:14:57.110 align:middle line:84% in that Fourier decomposition as if they were separate things. 01:14:57.110 --> 01:14:58.130 align:middle line:90% They're not separate. 01:14:58.130 --> 01:14:59.630 align:middle line:84% They're just our way of representing 01:14:59.630 --> 01:15:00.620 align:middle line:90% this magnetic field. 01:15:00.620 --> 01:15:04.220 align:middle line:84% But it does allow us intuitively to get at some of the physics 01:15:04.220 --> 01:15:05.280 align:middle line:90% term by term. 01:15:05.280 --> 01:15:08.300 align:middle line:84% So if we write this down as a Fourier decomposition, 01:15:08.300 --> 01:15:12.800 align:middle line:90% we'll have a DC term, C0 over 2. 01:15:12.800 --> 01:15:14.840 align:middle line:84% The 2 here is just our convention when 01:15:14.840 --> 01:15:16.460 align:middle line:90% we're doing Fourier transforms. 01:15:16.460 --> 01:15:17.550 align:middle line:90% We've seen this before. 01:15:17.550 --> 01:15:21.290 align:middle line:84% And then we're going to have a sum over an index M 01:15:21.290 --> 01:15:25.440 align:middle line:90% from 1 to infinity. 01:15:25.440 --> 01:15:35.330 align:middle line:84% And we're going to have a term CM cosine of M theta plus SM 01:15:35.330 --> 01:15:40.320 align:middle line:90% sine of M theta, like this. 01:15:40.320 --> 01:15:43.370 align:middle line:84% So these are different coefficients, 01:15:43.370 --> 01:15:44.490 align:middle line:90% which we don't know. 01:15:44.490 --> 01:15:47.600 align:middle line:84% But of course, we can find them by doing 01:15:47.600 --> 01:15:49.550 align:middle line:84% the Fourier transform of this or the Fourier 01:15:49.550 --> 01:15:52.640 align:middle line:90% decomposition of this. 01:15:52.640 --> 01:15:54.560 align:middle line:84% And we do that by doing things like-- 01:15:54.560 --> 01:15:57.350 align:middle line:84% I'm sure you've seen this before, but just to remind you, 01:15:57.350 --> 01:15:59.960 align:middle line:84% this is going to be 1 over pi the integral 01:15:59.960 --> 01:16:09.520 align:middle line:84% of 0 to 2 pi of B theta cosine of M theta B theta 01:16:09.520 --> 01:16:11.620 align:middle line:84% and similarly for the sine component. 01:16:11.620 --> 01:16:12.120 align:middle line:90% OK. 01:16:12.120 --> 01:16:13.300 align:middle line:90% So we have our signal. 01:16:13.300 --> 01:16:15.010 align:middle line:84% We can decompose it into these modes. 01:16:15.010 --> 01:16:17.460 align:middle line:84% And then we can give these modes names, right? 01:16:17.460 --> 01:16:20.730 align:middle line:84% And so we're going to particularly consider the modes 01:16:20.730 --> 01:16:29.280 align:middle line:84% M equals 0, which is this term, and the M equals 1 mode here. 01:16:29.280 --> 01:16:31.590 align:middle line:84% These are the lowest order modes in our system. 01:16:31.590 --> 01:16:33.883 align:middle line:84% There are modes all the way up to infinity. 01:16:33.883 --> 01:16:35.550 align:middle line:84% But these are the ones which are usually 01:16:35.550 --> 01:16:39.120 align:middle line:84% the most serious in terms of plasma instabilities. 01:16:39.120 --> 01:16:41.800 align:middle line:84% And they're also the ones that provide the most information. 01:16:41.800 --> 01:16:44.620 align:middle line:84% So once we've measured our poloidal magnetic field, 01:16:44.620 --> 01:16:47.250 align:middle line:84% we've done this decomposition, we can learn an awful lot 01:16:47.250 --> 01:16:51.240 align:middle line:84% by studying first the M equals 0 component of the magnetic field, 01:16:51.240 --> 01:16:53.260 align:middle line:84% and then the M equals 1 component. 01:16:53.260 --> 01:16:55.590 align:middle line:84% If you want to, you can keep extending this analysis 01:16:55.590 --> 01:16:56.800 align:middle line:90% all the way up to infinity. 01:16:56.800 --> 01:16:59.383 align:middle line:84% But generally, people tend to get bored before they get there. 01:16:59.383 --> 01:17:01.500 align:middle line:90% So cool. 01:17:01.500 --> 01:17:04.530 align:middle line:84% In the homework, which has now uploaded, 01:17:04.530 --> 01:17:08.880 align:middle line:84% you are asked to sketch these modes. 01:17:08.880 --> 01:17:11.850 align:middle line:84% Does anyone want to tell me what shapes 01:17:11.850 --> 01:17:15.500 align:middle line:90% these two first modes give us? 01:17:15.500 --> 01:17:16.040 align:middle line:90% Yeah? 01:17:16.040 --> 01:17:18.170 align:middle line:90% STUDENT: [INAUDIBLE] 01:17:18.170 --> 01:17:23.357 align:middle line:84% JACK HARE: OK, M equals 0 is a nice circle, like that. 01:17:23.357 --> 01:17:24.940 align:middle line:84% So this is a symmetric magnetic field. 01:17:24.940 --> 01:17:28.600 align:middle line:84% It doesn't depend on the azimuthal or deployed line here, 01:17:28.600 --> 01:17:29.320 align:middle line:90% yeah. 01:17:29.320 --> 01:17:31.630 align:middle line:90% What about the M equals 1? 01:17:31.630 --> 01:17:32.130 align:middle line:90% Hm? 01:17:32.130 --> 01:17:33.450 align:middle line:90% STUDENT: [INAUDIBLE]. 01:17:33.450 --> 01:17:34.170 align:middle line:90% JACK HARE: Oval. 01:17:34.170 --> 01:17:34.740 align:middle line:90% OK. 01:17:34.740 --> 01:17:36.150 align:middle line:90% Is it an oval like this? 01:17:36.150 --> 01:17:38.910 align:middle line:90% Or an oval like this? 01:17:38.910 --> 01:17:42.012 align:middle line:90% STUDENT: [INAUDIBLE]. 01:17:42.012 --> 01:17:42.720 align:middle line:90% JACK HARE: Sorry? 01:17:42.720 --> 01:17:44.050 align:middle line:90% STUDENT: [INAUDIBLE]. 01:17:44.050 --> 01:17:45.670 align:middle line:90% JACK HARE: Right, OK. 01:17:45.670 --> 01:17:48.720 align:middle line:90% It's here. 01:17:48.720 --> 01:17:50.574 align:middle line:90% So which oval is it? 01:17:50.574 --> 01:17:52.715 align:middle line:90% STUDENT: [INAUDIBLE]. 01:17:52.715 --> 01:17:54.090 align:middle line:84% JACK HARE: It's a trick question. 01:17:54.090 --> 01:17:54.540 align:middle line:90% Come on. 01:17:54.540 --> 01:17:55.415 align:middle line:90% STUDENT: [INAUDIBLE]. 01:17:55.415 --> 01:17:56.510 align:middle line:90% JACK HARE: It's neither. 01:17:56.510 --> 01:17:58.230 align:middle line:90% STUDENT: [INAUDIBLE]. 01:17:58.230 --> 01:17:59.560 align:middle line:90% JACK HARE: Yeah, exactly. 01:17:59.560 --> 01:18:02.850 align:middle line:90% So that's the M equals 2. 01:18:02.850 --> 01:18:03.450 align:middle line:90% Cool. 01:18:03.450 --> 01:18:07.230 align:middle line:84% OK, so you're telling me the M equals 1 mode is shifted. 01:18:07.230 --> 01:18:10.770 align:middle line:84% So if I had my initial position here, it's now up like this. 01:18:10.770 --> 01:18:12.540 align:middle line:90% Or is it like this? 01:18:12.540 --> 01:18:13.930 align:middle line:90% Or is it like this? 01:18:13.930 --> 01:18:16.068 align:middle line:90% Which of these is it? 01:18:16.068 --> 01:18:17.355 align:middle line:90% STUDENT: [INAUDIBLE]. 01:18:17.355 --> 01:18:17.980 align:middle line:90% JACK HARE: Yes. 01:18:17.980 --> 01:18:21.370 align:middle line:90% OK, so it depends on CM and SM. 01:18:21.370 --> 01:18:23.590 align:middle line:84% These are effectively telling you which direction 01:18:23.590 --> 01:18:24.757 align:middle line:90% it's going to be shifted in. 01:18:24.757 --> 01:18:27.460 align:middle line:84% For example, CM is probably in this direction. 01:18:27.460 --> 01:18:29.060 align:middle line:90% And SM is in this direction. 01:18:29.060 --> 01:18:31.480 align:middle line:84% So your relative contributions of these two 01:18:31.480 --> 01:18:34.000 align:middle line:84% is going to tell you how much your plasma has shifted. 01:18:34.000 --> 01:18:37.095 align:middle line:84% So straight away, that is a useful measurement. 01:18:37.095 --> 01:18:38.470 align:middle line:84% This measurement straight away is 01:18:38.470 --> 01:18:40.263 align:middle line:84% going to tell you if your plasma is moving. 01:18:40.263 --> 01:18:41.680 align:middle line:84% Perhaps, if your plasma is moving, 01:18:41.680 --> 01:18:42.640 align:middle line:90% it's going to hit the wall. 01:18:42.640 --> 01:18:44.098 align:middle line:84% And you can do some active feedback 01:18:44.098 --> 01:18:45.800 align:middle line:90% and stop it from doing that. 01:18:45.800 --> 01:18:48.250 align:middle line:84% But it turns out we'll learn even more by looking at the M 01:18:48.250 --> 01:18:52.910 align:middle line:84% equals 1, mode even for plasmas which are relatively stable. 01:18:52.910 --> 01:18:53.480 align:middle line:90% OK. 01:18:53.480 --> 01:18:55.313 align:middle line:84% We are coming up on the end of this session. 01:18:55.313 --> 01:18:58.070 align:middle line:84% So I don't want to start on the next section, which 01:18:58.070 --> 01:19:02.810 align:middle line:84% is going to be using M equals 0 to measure diamagnetism. 01:19:02.810 --> 01:19:06.560 align:middle line:90% 01:19:06.560 --> 01:19:08.690 align:middle line:84% But I am happy to take any questions 01:19:08.690 --> 01:19:11.340 align:middle line:84% for the next few minutes before we pause today. 01:19:11.340 --> 01:19:12.200 align:middle line:90% Any questions? 01:19:12.200 --> 01:19:13.494 align:middle line:90% Yes? 01:19:13.494 --> 01:19:14.462 align:middle line:90% STUDENT: [INAUDIBLE]? 01:19:14.462 --> 01:19:26.272 align:middle line:90% 01:19:26.272 --> 01:19:27.980 align:middle line:84% JACK HARE: We're totally neglecting-- are 01:19:27.980 --> 01:19:29.480 align:middle line:84% we keeping the tension term in there 01:19:29.480 --> 01:19:31.250 align:middle line:90% if we're [INAUDIBLE] tension? 01:19:31.250 --> 01:19:33.110 align:middle line:90% We neglect the toroidal tension. 01:19:33.110 --> 01:19:35.648 align:middle line:84% The only thing that remains is the poloidal tension. 01:19:35.648 --> 01:19:37.190 align:middle line:84% These magnetic field lines themselves 01:19:37.190 --> 01:19:38.317 align:middle line:90% have curvature, right? 01:19:38.317 --> 01:19:39.650 align:middle line:90% So we keep the poloidal tension. 01:19:39.650 --> 01:19:41.108 align:middle line:84% We get rid of the toroidal tension. 01:19:41.108 --> 01:19:41.950 align:middle line:90% So this is the same. 01:19:41.950 --> 01:19:44.450 align:middle line:84% This is why I said, if you make this very, very large aspect 01:19:44.450 --> 01:19:46.160 align:middle line:84% ratio, so the curvature is very small, 01:19:46.160 --> 01:19:48.620 align:middle line:84% you also get a similar kind of picture. 01:19:48.620 --> 01:19:50.120 align:middle line:84% So that is something that is missing 01:19:50.120 --> 01:19:51.770 align:middle line:84% from this model that may or may not 01:19:51.770 --> 01:19:55.130 align:middle line:90% be important in a real system. 01:19:55.130 --> 01:19:58.000 align:middle line:90% OK, yeah, it's a good question. 01:19:58.000 --> 01:20:02.000 align:middle line:90%