(III) Suppose a ray strikes the left face of the prism in Fig. 52 at 45.0^∘ as shown, but is tot

(III) Suppose a ray strikes the left face of the prism in...

Key Concepts

Snell's Law

Snell’s Law is the fundamental relationship between the angles of incidence and refraction when a light ray passes from one medium into another. It relates the sine of the incident angle to the sine of the refracted angle with the indices of refraction of the two media. In problems involving prisms, Snell’s Law is used to determine how a ray bends as it enters or exits the prism, which is essential to understand the overall light path inside the prism.

Total Internal Reflection

Total internal reflection occurs when light attempts to pass from a medium with a higher index of refraction to one with a lower index at an angle greater than the medium’s critical angle, causing the light to be completely reflected back into the higher-index medium. This concept is crucial in the given problem because the ray is totally internally reflected inside the prism, placing constraints on the index of refraction relative to the incident angles.

Critical Angle

The critical angle is defined as the angle of incidence above which total internal reflection occurs for light trying to move from a denser medium to a less dense medium. It is calculated using the ratio of the indices of refraction of the two media. In the context of prism problems, knowing the critical angle allows one to relate the geometry of the internal ray path to the minimum required index of refraction for total internal reflection to take place.

Prism Geometry

Prism geometry involves understanding the shape and angles within the prism, particularly the apex angle, which governs the relationships between the paths of light inside the prism. The angles at which light enters, travels through, and reflects off the surfaces of the prism are all interconnected through the prism’s geometry. In this problem, the 60° apex and the incidence angles dictate the conditions for which the light remains confined within the prism via total internal reflection, thereby influencing the necessary index of refraction.

Transcript

00:01 Okay, in this problem we have a prism where light comes in from the left refrax into as it enters the prism and then totally internally reflects on the right side of the prism so i show some arrows just indicating which way this ray is going we have all these various angles.

00:19 It's first angle of contact with the prism which is 45 degrees we have this theta 2 for the refraction of the light ray and of course the at the 8 3 and theta 4 for the second interaction at the right surface of the prism and and it's problem we're asked to find a relationship, some kind of say something about the index of refraction, n of the prism.

00:39 So it's a pretty general problem.

00:40 It's a pretty difficult problem, but we're going to try to get through it.

00:43 We're going to be making use of snell's law, which simply says that n1 sine theta 1 for the light ray in one medium is the same as n2 sine theta 2 for that same light ray on the other side of the medium.

00:58 That's snells law.

00:59 That's valid for refraction.

01:03 We're going to be making heavy use of it.

01:04 It in addition to a lot of trigonometry, as you might expect.

01:09 So to start us off, let's consider that first interaction of the light ray with the left side of the prism, left interface.

01:16 The light ray is initially an error before it enters the prism.

01:19 So we have nair, sine theta 1 equals n sine theta 2, where n is just unknown.

01:25 It's the end of the prism.

01:28 And of air is just 1.

01:30 So we're left with sine theta 1 equal n sine theta 2.

01:34 And then finally, we can get this intermediate step sine theta 2 has to be equal to 1 over n sine theta 1.

01:49 So we're going to leave that for now.

01:52 Now at the second surface, we should write this at second surface just so we're clear what we're talking about.

02:02 The angle of incidence is going to be 90 degrees minus theta 2.

02:09 Just need to justify that to yourself, why that's the case.

02:15 And then we also add plus 90 degrees minus theta 3 plus third angle, the angle of the prison, a.

02:26 These all have to add up to 180 degrees.

02:28 So we're considering the big triangle.

02:32 Once we have this, we can break it down a little bit.

02:34 We have theta 3 minus, or rather, theta 3 equals a minus theta 2, which is also equal to which a is simply 6 degrees.

02:46 We can plug it in now.

02:47 So we have 6 degrees minus theta 2, and that's what theta 3 is.

02:51 We're now going to apply snell's law again and move to a second page, or rather third page.

02:59 So the index, to model the refraction at the surface of the second surface, we have n -sign theta -3, which we were able to get a relation for, equalling n -air, sine data -4.

03:17 Okay, so doing a bit of simplification, n -air is one, as i mentioned before.

03:26 So we have sine theta -4, and the maximum value of things.

03:37 Of theta 4 for the internal reflection is 90 degrees...

Please give Ace some feedback