Figure P35.56 shows an interferometer known as Fresnel's...

Figure P35.56 shows an interferometer known as $Fresnel's biprism$. The magnitude of the prism angle $A$ is extremely small. (a) If $S_0$ is a very narrow source slit, show that the separation of the two virtual coherent sources $S_1$ and $S_2$ is given by $d = 2aA(n - 1)$, where $n$ is the index of refraction of the material of the prism. (b) Calculate the spacing of the fringes of green light with wavelength 500 nm on a screen 2.00 m from the biprism. Take $a$ = 0.200 m, A = 3.50 mrad, and $n$ = 1.50.

Transcript

00:01 This is chapter 35 problem number 56.

00:04 We have fresnals by prism.

00:07 So if it's not as a narrow split, show that the separation of two virtual coherent sources, that's one of the issue.

00:15 It's given by d equals to a, towards the distance from the sources to the screen times a and minus one.

00:28 Okay, this is the first.

00:30 Portion that we need to prove.

00:32 Let me draw the diagram real quick, s1, s not, and s2.

00:38 So then this entire distance is d, and then we have the prism in front of the screen, a, away from it at an angle capital a.

00:49 So what we need to do is basically we want to look at the distance between the slits.

00:57 So then we're going to have to look at the angle at which two beams come from each source, the difference in the angle, right? and then we can go to the distance between the slits.

01:12 So for that we're going to have to apply snl's laws several times at this interface, and then, well, pick one portion, doesn't matter at this interface.

01:22 So then we're going to have an incoming beam here, let's call it data, and then the way that it gets refracted by the prism is we're going to have a different angle, as you know, but closer to normal.

01:36 It's kind of hard for me to draw here.

01:38 But let's call that as theta i.

01:41 And then it's going to hit the prism again.

01:44 And there's another.

01:44 We're going to have to write another snell's law here as the angle goes back to initial.

01:51 Then finally, well, we're going to call this angle theta k.

01:54 And then finally we're going to establish the relationship between, well, the outgoing angle and the initial angle.

02:01 So in order to do that, nelslow remember n -air times sine theta equals n -prism times sign of theta initial.

02:12 Right? so n -prism, sorry, the n -air is one sine theta...