The prism in the figure is made of crown glass. Its index of...

The prism in the figure is made of crown glass. Its index of refraction ranges from $1.517$ for the longest visible wavelengths to $1.538$ for the shortest. Find
the range of refraction angles for the light transmitted into air through the right side of the prism.

Key Concepts

Snell's Law

Snell's Law describes how light bends when it passes from one medium into another by relating the angles of incidence and refraction to the indices of refraction of the two media. This principle is foundational in understanding how light changes direction at boundaries and is critical for solving problems involving multiple media, such as a light beam exiting a prism.

Index of Refraction

The index of refraction quantifies how much light slows down in a material compared to its speed in a vacuum. It is a fundamental property of materials and plays a crucial role in determining how light bends and disperses when it encounters interfaces between different media.

Dispersion

Dispersion is the phenomenon where the index of refraction of a material varies with the wavelength of light. This variation causes different colors (wavelengths) of light to refract by different amounts, leading to a spread or range of refraction angles. Understanding dispersion is essential when analyzing optical devices like prisms that separate light into its component colors.

Optical Prism

An optical prism uses multiple refractions at its surfaces to alter the direction of light beams, often splitting white light into its constituent spectral colors. The analysis of light passing through a prism involves applying Snell's Law at each interface, accounting for the material's index of refraction and its wavelength dependency due to dispersion.

Transcript

00:01 For this problem, you're asked to find the range of refraction angles as light passes through this particular prism.

00:08 It's a partial diagram from what's given.

00:12 The first thing you have to do is find the first angle of a fraction.

00:16 I call that theta r1 here at the first boundary.

00:22 And so we can use snell's law, n1, sine theta 1, equals n2, n2, n2.

00:33 To sign theta 2 and n1 for error is 1 so leave that alone.

00:40 Sign theta 1 is 55 degrees equals n2 for crown glass.

00:46 You're given that with the first refraction of index for the longer wavelengths is 1 .517 sine theta 2 and you solve that for theta 2 doing the algebra and you get an answer for that angle of 32 .68 degrees.

01:05 Now from that you have to find the next angle of instance.

01:09 And so there's a little bit of geometry you have to do to find that angle.

01:13 First, you have to find this angle here.

01:16 And you can do that using the fact that the quadrilateral, i'll draw it in blue, this quadrilateral right here.

01:24 The sums of the angles in that quadrilateral is 360 degrees.

01:28 And you know that this angle is 90 and that this angle is 90 because that is the normal line.

01:35 So let me get that stuff off the picture.

01:36 So what i'm going to say is, let's call that angle beta, just to give it a variable letter.

01:43 So this angle here i'm going to call beta.

01:47 So we know the angle at the top is 60 degrees.

01:50 So 60 degrees plus 90 degrees plus 90 degrees plus 90 degrees plus beta has to equal 360 degrees.

02:01 It sums the angles of quadrottal is 360.

02:04 So you solve that for beta and you get beta is equal to 120 degrees.

02:09 Well, then you have to find the angle of incidence on the opposite side.

02:13 So that is going to be this angle here.

02:18 And so for that angle, you have to use the sums of the angles of the triangle is 180.

02:24 So i'll call that theta incident two, because it's incident on the opposite side of the prism.

02:37 And so that angle plus the 32 .68, that we're already calculated plus the 120, has to equal 180 degrees.

02:47 So theta incident 2 is equal to 27 .32 if you do the algebra degrees.

02:57 So now you can find the angle of a fraction on the far side.

03:01 That's going to be the first angle we need.

03:02 That's going to be this angle here.

03:04 Again, using snell's law...

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