A thin layer of ice (n = 1.309) floats on the surface of water (n = 1.333) in a bucket. A ray of

A thin layer of ice (n = 1.309) floats on the surface of...

Key Concepts

Total Internal Reflection

Total internal reflection occurs when light strikes the boundary of a medium at an angle greater than the critical angle, resulting in the light being completely reflected back into the original medium rather than refracted into the second medium. This phenomenon is crucial in applications like optical fibers and lens design, and it also governs the conditions under which light can exit a layered medium such as floating ice on water.

Critical Angle

The critical angle is the specific angle of incidence in the denser medium—the medium with a higher index of refraction—at which the refracted ray emerges parallel to the boundary. Beyond this angle, total internal reflection occurs, meaning no light passes through the interface. This concept is key to understanding the limits of refraction at the interface between water, ice, and air.

Index of Refraction

The index of refraction quantifies how much light slows down when it travels through a medium compared to its speed in a vacuum. It is essential in determining the bending of light at interfaces between substances, and differences in refractive indices are the reason light refracts or reflects differently when passing between media like water, ice, and air.

Snell's Law

Snell's Law relates the angles of incidence and refraction for a light ray passing through an interface between two media with different indices of refraction. It is a fundamental principle in optics that allows us to calculate how light rays bend when entering a new medium, making it crucial for solving problems involving layered materials such as water and ice.

Transcript

00:01 To be looking at refraction, and in particular, snell's law, with a fairly complicated set of boundaries, not too bad.

00:10 But what is happening is there is one of these underwater lights inside of a bucket.

00:15 And the light is propagating up through both the water and the layer of ice before it gets into the air.

00:23 Now, at some point, the angle of the rays will not be able to make it out.

00:28 Because of total internal reflection.

00:32 So let's take a look at snells law.

00:35 It says that between two sides of an interface, the index on one side times the sign of the angle with respect to the normal is the same product of index and sign of angle on the other side.

00:53 Now, what can happen if one of the indices is smaller than the other? so if you are starting in a denser material going to a less dense, what will happen is there will be a critical angle.

01:12 So let's say n1 is denser and n2 is less dense.

01:18 What will happen is the angle can come out at a 90 degree.

01:24 So n1 sine of theta 1 is equal to n2.

01:31 Times sign of 90 degrees.

01:35 And this can happen because the, let's see, the angle on the right -hand side can be less than the angle on, sorry, the angle on the left -hand side can be smaller than the angle on the right -hand side because of the denser material.

02:00 And this is called total internal reflection.

02:04 That angle inside the denser material at which this happens is called the critical angle.

02:11 So let's take a look at the ice air interface.

02:16 Ice has an index of refraction much higher than air, and so the ray can get trapped inside of the ice, such that n1, well, let's just go ahead and put in for the n of ice 1 .309 times the sign of the critical angle is equal to the n for air, which 2 -3 -sigphids a decimal place, 1 .000 times, of course, sign of 90 is 1.

02:54 And then we can see how this critical angle can occur.

02:59 Sign of angle must be less than one.

03:03 And this works out to be an angle of, let's try for sigphix, just to be sure, 49 .81, we'll call it.

03:24 So pointing out in my figure, the theta critical for the ray of light inside the ice is 49 .81 degrees, in which case it comes out into the air at a 90 degree angle.

03:40 Okay, let's now compute the angle that the ray must be coming into the water.

03:49 And for that, we're just going to use simple snellslaw...

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