WEBVTT 00:00:00.085 --> 00:00:02.800 This is College Physics Answers with Shaun Dychko. 00:00:03.742 --> 00:00:08.617 Two glass slides are separated at one end by a piece of hair 00:00:08.617 --> 00:00:11.075 that is 0.1 millimeters in diameter. 00:00:11.702 --> 00:00:13.748 And so the distance here is 0.1 millimeters. 00:00:13.994 --> 00:00:16.280 Then the pieces of glass are touching at this end. 00:00:16.577 --> 00:00:20.897 So this forms a triangular wedge of air between the pieces of glass. 00:00:22.457 --> 00:00:26.502 Now light incident here will interfere. 00:00:26.691 --> 00:00:31.617 It'll one Ray will bounce off of this interface between glass and air 00:00:32.050 --> 00:00:33.468 and then go upwards. 00:00:33.811 --> 00:00:37.445 And then the another Ray will bounce off the bottom piece of glass 00:00:37.550 --> 00:00:38.908 and then go back up. 00:00:39.217 --> 00:00:42.011 And this drawing is not quite accurate 00:00:42.011 --> 00:00:45.841 because it says that the light is incident perpendicularly. 00:00:45.965 --> 00:00:51.948 So all these rays are in fact overlapping but it would be too difficult to draw that 00:00:52.217 --> 00:00:54.920 and explain where the different rays are coming from. 00:00:55.222 --> 00:00:56.940 And so that's why it's showing an angle. 00:00:56.940 --> 00:00:59.885 But keep in mind that it's meant to be overlapping 00:00:59.885 --> 00:01:02.210 so rays one and two will interfere. 00:01:03.074 --> 00:01:05.514 And in this case because we're looking for dark bands 00:01:05.760 --> 00:01:07.891 the interference is going to be destructive. 00:01:09.771 --> 00:01:16.251 And then at some position later further down here some distance delta x 00:01:16.360 --> 00:01:19.542 you'll have the next dark band occurring when 00:01:19.794 --> 00:01:21.588 there is destructive interference again 00:01:21.760 --> 00:01:25.411 as a result of this increased path length difference. 00:01:27.125 --> 00:01:32.411 So let's figure out expressions for the phase shifts of Rays one and two 00:01:32.710 --> 00:01:37.925 and then and then we'll see how that changes with horizontal position. 00:01:39.771 --> 00:01:42.434 Now the phase shift of Ray one is zero 00:01:42.790 --> 00:01:46.480 because this reflection here occurs at an interface 00:01:46.480 --> 00:01:50.811 where you're beginning in a medium with high index refraction and some kind of glass 00:01:51.220 --> 00:01:55.234 and then going to a medium of low index refraction which is air. 00:01:55.617 --> 00:01:57.188 And so there is no phase shift there. 00:01:58.474 --> 00:01:59.902 And then for Ray two however 00:02:00.222 --> 00:02:03.377 there's a reflection off of this interface between the air in the glass 00:02:03.377 --> 00:02:06.989 and this does have a phase shift automatically of half of a wavelength 00:02:07.670 --> 00:02:11.360 when going from a low index refraction to a high index refraction 00:02:12.400 --> 00:02:14.700 and then additionally there's a phase shift 00:02:14.700 --> 00:02:18.794 due to this additional path length travelled by ray two. 00:02:19.622 --> 00:02:24.622 And so we calculate the number of wavelengths that path length corresponds to 00:02:24.622 --> 00:02:26.562 and then multiply by the wavelength. 00:02:27.011 --> 00:02:31.131 And this works well these are actually going to be the same 00:02:31.131 --> 00:02:33.212 because normally I have a subscript n here 00:02:33.212 --> 00:02:40.556 to say you know the additional path divided by the wavelength in that medium. 00:02:40.980 --> 00:02:43.108 But in this case the medium is air. 00:02:45.194 --> 00:02:47.097 So these cancel and we have a two there 00:02:47.097 --> 00:02:50.841 because the ray is going down once over that thickness 00:02:50.950 --> 00:02:54.029 and then up again over the thickness it's doing a round trip 00:02:54.091 --> 00:02:57.257 traveling that thickness twice the thickness of that air gap there. 00:02:58.114 --> 00:03:01.988 And this works out to Pi over two plus two times the thickness of the air gap. 00:03:04.280 --> 00:03:06.742 So now we have phase shifts for each of the two rays. 00:03:07.022 --> 00:03:10.028 Now the total phase shift when you add them together 00:03:10.159 --> 00:03:14.859 has to be some integer plus a half times the wavelength 00:03:14.859 --> 00:03:18.117 in order to have destructive interference 00:03:20.862 --> 00:03:25.468 and this total phase shift is going to be Delta Phi 1 plus Delta Phi 2 00:03:25.840 --> 00:03:27.594 equals m plus half times wavelength. 00:03:28.474 --> 00:03:31.902 And now we can substitute these expressions for Delta Phi 00:03:31.902 --> 00:03:36.619 that we found before in blue zero four every one 00:03:36.930 --> 00:03:43.131 and then lambda over two plus 2t for ray two and we're going to solve this for t . 00:03:44.805 --> 00:03:47.897 And so I multiply both sides by two here 00:03:48.070 --> 00:03:50.291 and then distribute the lambda into the brackets 00:03:50.520 --> 00:03:53.194 and end up with Lambda plus 40 equals 00:03:56.702 --> 00:03:59.257 I guess I just multiplied the two into the brackets there. 00:03:59.257 --> 00:04:01.291 So that's 2m plus one times lambda 00:04:01.291 --> 00:04:03.458 and you end up with 2m lambda plus lambda 00:04:04.019 --> 00:04:09.030 and then subtract a lambda from both sides 00:04:11.982 --> 00:04:16.651 and you have 40 equals 2m lambda and then t is m lambda over two. 00:04:19.051 --> 00:04:22.805 So this is an expression for the thickness of the air gap 00:04:23.020 --> 00:04:25.262 needed to have destructive interference. 00:04:25.714 --> 00:04:28.840 And there will be many such thicknesses that cause destructive interference 00:04:28.840 --> 00:04:33.668 because we can choose any value any integer value for m. 00:04:37.811 --> 00:04:42.754 So we're going to have here's the use with the glass pieces are touching 00:04:43.620 --> 00:04:47.320 and that's some thickness t1 will have destructive interference. 00:04:47.748 --> 00:04:52.680 And that's at a horizontal position x1 from this point where they're touching 00:04:53.240 --> 00:04:57.462 and then again will have destructive interference that happening again at t2 . 00:04:57.657 --> 00:04:59.690 So this will be corresponding to m equals one 00:04:59.690 --> 00:05:02.577 and this corresponds to m equals two these two thicknesses. 00:05:03.380 --> 00:05:05.234 We would plug in the number one or the number two 00:05:05.234 --> 00:05:06.577 into this expression for thickness 00:05:06.960 --> 00:05:09.022 and this will happen at a position x2 . 00:05:09.262 --> 00:05:11.085 And the question is asking us 00:05:11.690 --> 00:05:14.742 what is the difference between these horizontal positions. 00:05:15.514 --> 00:05:18.217 How far apart are the dark bands horizontally. 00:05:18.348 --> 00:05:19.834 What is delta x in other words. 00:05:20.777 --> 00:05:26.931 So these are similar triangles and because the angles are all the same. 00:05:27.131 --> 00:05:31.022 I mean they share this common angle here and this is a 90 degree angles. 00:05:31.022 --> 00:05:34.847 That means this angles the same two and because they're similar 00:05:34.847 --> 00:05:37.441 that means that the ratio of corresponding sides is the same. 00:05:38.131 --> 00:05:42.617 So when you take x2 divided by t2 that's gonna be the same as x1 divided by t1 . 00:05:42.874 --> 00:05:45.531 So let's just call it x over d . 00:05:51.531 --> 00:05:54.222 And d being, this is, 00:05:55.280 --> 00:05:59.680 this here because we know what the ratio for this triangle is 00:06:00.177 --> 00:06:02.320 and the ratios where all the triangles inside here 00:06:02.320 --> 00:06:04.222 are all going to be the same because they're all similar. 00:06:04.222 --> 00:06:07.230 And I mean similar in a mathematical technical sense of the word 00:06:07.630 --> 00:06:10.217 meaning that all the time all the angles in the Triangle are the same 00:06:10.580 --> 00:06:13.371 and therefore ratios of corresponding sides are the same. 00:06:15.182 --> 00:06:20.045 And we know the opposite and the adjacent for this triangle here 00:06:20.210 --> 00:06:23.742 it's 0.1 millimetres high and 7.5 centimeters long. 00:06:25.497 --> 00:06:31.251 So we can say that x2 is t2 times x over d and x1 is t1 times x over d 00:06:33.650 --> 00:06:36.617 and we'll substitute for each of those in this expression for delta x. 00:06:37.280 --> 00:06:40.485 And that's what I've done here and the x over d can be factored out 00:06:40.780 --> 00:06:44.085 and it's gonna be t2 minus t1 times x over d 00:06:44.588 --> 00:06:49.640 and Delta x then substituting for t2 and t1 using this expression here 00:06:50.030 --> 00:06:52.611 substituting the numbers two and one for n . 00:06:54.148 --> 00:06:58.108 We have two times lambda over two minus one times lambda over two times x over d 00:06:58.417 --> 00:07:04.297 and then this works out to Lambda over two and then we can substitute in numbers. 00:07:04.554 --> 00:07:07.310 So that's 589 nanometres is the wavelength we told 00:07:07.310 --> 00:07:10.165 divided by two times seven and a half centimeters 00:07:10.165 --> 00:07:11.828 written as times ten to the minus two meters 00:07:12.190 --> 00:07:15.662 divided by 0.1 millimetres written as times ten to the minus three meters 00:07:15.970 --> 00:07:20.097 gives us this many nanometres which is 0.221 mm. 00:07:20.508 --> 00:07:23.388 So this is the horizontal separation between dark bands. 00:07:24.462 --> 00:07:27.828 Now what you would expect in this picture is 00:07:27.828 --> 00:07:35.345 to see lines like this spaced apart 0.221 millimeters 00:07:36.085 --> 00:07:39.137 and in reality though you don't see that 00:07:39.440 --> 00:07:43.880 because that's showing you that there are imperfections in these pieces of glass 00:07:43.880 --> 00:07:47.988 it means that they're not perfectly straight as shown in the drawing 00:07:48.411 --> 00:07:52.302 and so that's kind of interesting and this is one of the ways 00:07:52.302 --> 00:07:58.842 that they actually engineer really nearly perfect mirrors for telescopes and so on 00:07:59.074 --> 00:08:01.411 is by looking at interference patterns like this.