Suppose that, on some surreal world, raindrops had a square...

Suppose that, on some surreal world, raindrops had a square cross section and always fell with one face horizontal. Figure $33-56$ shows such a falling drop, with a white beam of sunlight incident at $\theta=70.0^{\circ}$ at point $P$. The part of the light that enters the drop then travels to point $A,$ where some of it refracts out into the air and the rest reflects. That reflected light then travels to point $B$. where again some of the light refracts out into the air and the rest reflects. What is the difference in the angles of the red light $(n=1.331)$ and the blue light $(n=1.343)$ that emerge at (a) point $A$ and (b) point $B ?$ (This angular difference in the light emerging at, say, point $A$ would be the rainbow's angular width.)

Key Concepts

Snell's Law

Snell's Law describes how light bends when it passes from one medium into another with a different refractive index. This law is fundamental in calculating the paths of rays as they refract at interfaces, determining the angles of incidence and refraction which are crucial for understanding how light is directed within and out of optical elements.

Internal Reflection

Internal reflection occurs when light hits an interface from within a medium and reflects rather than refracts, based on the angle of incidence and refractive indices. In systems involving multiple interactions between light and surfaces, internal reflection plays a key role in determining the overall path of the light ray and contributes to phenomena like concentrated beams or repeated deviations within the medium.

Dispersion

Dispersion refers to the dependence of a material’s refractive index on the wavelength of light, meaning that different colors (wavelengths) refract by slightly different amounts. This concept is critical in explaining why light separates into a spectrum of colors, as seen in phenomena such as rainbows, and leads to differences in the deviation angles between various wavelengths.

Geometric Optics and Ray Tracing

Geometric optics and ray tracing involve approximating light as rays that travel in straight lines when in uniform media and bend at interfaces according to laws like Snell's Law. This approach is essential for analyzing optical systems, predicting the paths of light as they refract and reflect within an object, and determining overall angular deviations for light emerging from the system.

Angular Deviation

Angular deviation measures the difference between the initial direction of an incident ray and the final direction of the emerging ray after undergoing refraction and reflection. Calculating angular deviation is central to understanding and predicting the formation and width of phenomena like rainbows, where variations in angular displacement between different wavelengths result in a spread of colors.

Transcript

00:01 In this question, we are looking at dispersion by a square raindrop, okay? so yeah, so this is like an interesting world that we have a square ring drop.

00:17 Then light comes in at 70 degrees anger incidence.

00:25 Okay, so this is data 1 and then it gets reflected.

00:29 Then some of it gets reflected here and then some of it gets reflected here.

00:37 So here is point a, here is point b.

00:42 In this question, we want to find the difference in the angles of the red light and the blue light, the images at a and point a and at point b.

00:54 Okay.

00:57 So we are given the reflective index for the red light, okay, is equal to 1 .331.

01:08 Then we are also given the refractive index of dual light to be 1 .343.

01:18 All right, so we can calculate on 2 -2 -2 part a.

01:23 First we need to calculate the data 2 here.

01:29 And data 3 here so data 2 data 2 rate for that using snail's law is now says that n1 side data 1 equals to n2 side data 2 so n1 is 1 and then n2 is the refractive indices of the of the light in the class or in the water drop it okay and then data 1 is 70 degrees okay we want to find data 2 for red and blue light okay so sine data 2 r is psi 70 degrees debug by 1 .331 okay so we get 0 .706 so we can calculate data 2 r is 44 .9 degrees, 0 .91 degrees.

02:55 Okay, and then we do repeat the same step for blue light.

03:07 So sine 70 degrees divide by 1 .34 3.

03:12 Okay, we get 0 .6996.

03:17 And then this gives us data 2b to be, 4.

03:23 0 .40 degrees, okay? and then the incident angle at point a.

03:40 So the data ir is, maybe data 3r, okay, is 90 degrees minus data 2r and then data 3b is equal to 90 degrees minus data 3b.

04:00 Okay, i'm going to look at data 4 which is the light that images from the point a okay so using cell so again we have n 1 side data 3 and n sine data 3 goes to psi data 4 okay because n 4 is 1 so for red light, sine data 4r is equal to nr, side data 3r, we have 1 .331 times sine 90 degrees minus 44 .91 degrees...

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