 Hello, this is example 5.3.1 on page 207 of the Duffy and Beckman text So this problem is about a single glass cover system asking to calculate the transmissivity reflectivity and absorptivity of that glass cover, so Single this cover, what that means is that we can look up in Table 0.1.1 That the index of refraction Is 1.526 That's the information that we gather from the fact that it's a single glass cover Thickness L length is a 2.3 millimeters That's given and that equals 0.0023 meters Extinction coefficient K is equal to 32 units per meter and the incidence angle theta is Given as 60 degrees and we are asked to find transmissivity tau reflectivity rho and absorptivity alpha And I've also added that I want to check that the sum of all of these tau plus rho plus alpha equals 1 because it should be because The incidental radiation is either transmitted through the material reflected off the material or absorbed in the material And that's it. That's a It's a one-sum game something's got to happen to each photon and those are the three things that can happen to them so To kick this off First we have to calculate the refraction angle so We do that using equation 5.1.4 at theta 2 Equal to the sine inverse the incidence angle Divided by the index of refraction that results in theta 2 equal to 34.58 degrees And so this allows us to calculate the extinction coefficient optical path product is KL over cosine theta 2 So K is 32 was zero zero two point zero zero two three divided by the cosine of 34.58 So what we get is 0.08 94 the next piece of the puzzle is To calculate tau sub a so here we're going to use equation five point two point two and we get Tau sub a equals Exponential of negative zero point zero eight nine four the number we just calculated above What we get is zero point nine one five. So note that this is not the transmissivity. This is tau sub a We still have to average for the parallel and perpendicular components of Polarization in the glass so This is an intermediate step on our way to the solution So the next piece of the puzzle We need to use Equation five point one point two Two five point one point two five point one point two and Five point one point one. We're gonna use five point one point one first Calculate the perpendicular component of reflectance That's equal to Sign squared of the angle we calculated above minus the incidence angle divided by Sign squared of thirty four point five eight Plus the incidence angle. So we get zero point one eight four divided by zero point nine nine four which is equal to zero point one eight five And then the parallel component Is it is almost the same equation instead of signs? We use tangents thirty four point five eight minus sixty divided by the tan squared of thirty four point five eight plus Sixty that gives us zero point two two six Divided by actually a very large number 55 So what we end up with because we're dividing by such a large number is we end up with a very small number for the Parallel component there and so what we end up with here is a way to calculate Tau transmissivity, so therefore tau is equal to zero point nine one five which is the Tau sub a number divided by two times one minus zero point one eight five divided by one plus zero point one eight five which is the perpendicular component times one minus zero point one eight five squared divided by one minus zero point one eight five times zero point nine one five quantity squared Close that parentheses Plus now we have to do the parallel component one minus zero point zero zero one Over one plus zero zero zero one multiplied by one minus zero point zero zero one squared divided by one minus zero zero zero one times zero point nine one five quantity Squared and close all those parentheses and brackets This gives us zero point five times zero point six two five plus zero point nine one two Which is equal to zero point seven six eight And I'll answer for tau So it's definitely a long drawn-out process and now we have tau good part is that that's the hard part Calculating reflectivity and absorptivity is relatively straightforward at this point so In light of the lack of room in this little area here we're going to move up into the empty space up here for the next part of the problem so That was for calculating tau so next We're gonna calculate reflectivity next so to do that we use equation five point three point two For reflectance That gives us just row equals zero point five I'm zero point one eight five again. We reuse a lot of these same numbers That we calculated before so we've already done the bulk of the work getting this far One plus zero point nine one five I'm zero point six two five Plus zero point zero zero one times one plus zero point nine one five Times zero point nine one two So when you crunch all those numbers You end up with reflectivity equals to zero point one four seven Next we're going to calculate advertivity We do that using equation five point three point three Alpha equals one minus zero point nine one five divided by two multiplied by one minus zero point one eight five which is again that perpendicular component of of our over here divided by One minus zero point one eight five time is zero point nine one five Zero point nine one five is that Tau sub a again over here plus one minus zero point zero zero one which is the parallel component over here divided by one minus zero point zero zero one times zero point nine one five Tau sub a again close the parentheses Crunch all those numbers And you end up with alpha equals zero point zero eight five So we've calculated each of these three pieces and now it's good to just check it. So we're gonna Check that Tau plus row plus alpha equals one if you punch those numbers in your calculator Point seven six eight plus Point one four seven Plus point zero eight five you end up indeed that they do equal one check so we've accounted for all the photons essentially and That's example five point three point one. So thank you so much for your time and for listening