Two specks of dirt are trapped in a crystal ball, one at the center and the other halfway to the

Two specks of dirt are trapped in a crystal ball, one at the...

Key Concepts

Refraction

Refraction is the bending of light as it passes from one medium to another with a different optical density. This change occurs because light travels at different speeds in different media, and the amount of bending is governed by Snell's law. In many applications, understanding refraction is key to explaining why objects viewed through a transparent medium appear displaced from their true positions.

Apparent Depth

Apparent depth refers to the phenomenon where an object immersed in a medium appears to be closer to the surface than it really is. This illusion occurs because of refraction; when light exits a denser medium into a less dense one, the object’s image is shifted. The apparent depth is typically modeled as the real depth divided by the refractive index, which is essential in determining how far an object seems to be from the viewer compared to its actual position.

Spherical Surface Refraction

Spherical surface refraction deals with the way light rays are bent upon emerging from or entering a spherical interface, such as a crystal ball. When viewing objects inside a sphere along a line that passes through its center, many of the complexities of curved surfaces simplify — particularly when the rays strike normally. This allows the use of formulas analogous to those for flat surfaces (like the apparent depth relationship) to analyze the displacement of images within the sphere.

Transcript

00:01 So in 57 says two specks of dirt are trapped in a crystal ball, one at the center, and one halfway to the surface.

00:09 If you peer into the ball on a line joining the two specks, the outer one appears to be only one third the way to the other.

00:17 Find the refractive index of the ball.

00:19 So what we're given is that the one of the specs, which we'll call this spec 1, and this is spec 2.

00:28 The location of spec 1, the object itself, is the radius.

00:37 The location of spec 2 is half the radius away from the curve surface, because the refraction is happening here at this curved surface.

00:48 We have to use the version of the curve surface equation that is a little bit more complicated because we have two indices of refraction.

00:59 We have n1 over s plus n2 over s prime, and that's equal to the difference of the indices divided by the radius of that curve surface.

01:13 So for spec 1, let me start way over here, so we have space.

01:16 For spec 1, n1 over, spec 1 is at r distance away from the curve surface.

01:24 So s becomes r plus 1 over s prime, but 1 is the second index of a fraction where this is the index of the ball and index number two is error.

01:36 So that's 1.

01:36 So 1 over s prime, 1 prime, so i'll keep it clear that this is spec 1 we're talking about, is equal to the difference in indices, 1 minus n1, divided by the radius.

01:51 So i'm going to take that equation and solve it for s prime.

01:55 And then i'm going to do something with the ratios of s prime for spec 1 and s prime for spec 2.

02:01 So just rearranging this, we have 1 over s prime equals 1 minus n1 over r minus n1 over r.

02:15 That's conveniently has the same denominator.

02:19 So 1 minus n1 minus n1 is 2 n1s divided by r.

02:27 And that's 1 over s prime.

02:29 So s prime is equal to r over 1 minus 2n.

02:34 So i'm just going to put that in a little box for now.

02:38 We'll come back to it momentarily.

02:41 We're going to do the same thing for spec 2.

02:44 Spec 2.

02:46 Oops.

02:47 Spec 2 has an initial object distance of half the radius.

02:54 So my spec 2 is, the equation goes n1 over, r over, 2 plus 1 over the image distance for spec 2 equals the same numerator over here, 1 minus n1 over r...

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