A shallow glass dish is 4.00 cm wide at the bottom, as shown in Figure P 35.63 . When an observe

step 1 Here we were given a glass, and it says it is four centimeters high. Oh, sorry, four centimeters

A shallow glass dish is 4.00 cm wide at the bottom, as shown...

Key Concepts

Refraction

Refraction is the phenomenon where light changes direction as it passes from one medium to another due to a change in its speed. In this problem, light is bending when it passes from water to air, which causes a shift in the apparent location of the bottom of the dish observed by the eye.

Snell's Law

Snell's Law quantitatively describes the relationship between the angles of incidence and refraction when light passes between two media with different refractive indices. This principle is essential for determining how the direction of light changes upon exiting the water, thus altering the perceived position of the bottom of the dish.

Apparent Depth

The concept of apparent depth explains why objects submerged in a medium such as water appear to be at a shallower depth than they actually are. The refraction at the water-air interface shifts the perceived position of the bottom, which is key to understanding how the observer sees different points on the dish under different conditions.

Geometric Optics

Geometric optics is the framework used to analyze the paths of light rays and their interactions with surfaces. By constructing ray diagrams and applying principles like Snell's Law, one can relate the physical dimensions of optical systems—such as the dish’s height—to the observed paths of the rays, thus solving for unknown distances.

Transcript

00:01 Here we were given a glass, and it says it is four centimeters high.

00:09 Oh, sorry, four centimeters wide at the bottom.

00:17 Four centimeters.

00:18 The observer's eye is placed.

00:21 The observer sees the edge of the bottom of the empty dish.

00:25 So i'm going to say his eyeball is over here, and he sees the edge of the dish.

00:35 When it's empty.

00:36 And then when the dish is filled with water, i'm going to use green water.

00:42 He no longer sees the edge because the light refracts and he sees the face.

00:49 And so they're asking when it's filled all the way with water, so i guess let's fill it all the way up.

00:55 What is the height of the glass? okay, we've drawn the picture a little bit just to make it a little clearer.

01:07 So i'm still trying to find that that height.

01:10 And so let's use our angles.

01:12 We know it's filled with water, which will help us because the end of the water is given at 1 .33 and of the a zero.

01:23 And so we're going to try to figure out what that height is.

01:27 Some lengths that are given to me, i do know that this is two centimeters and this is also two centimeters, which might be a useful tool for us.

01:37 I'm going to call this h.

01:40 And the angles that i'm going to be using are, this is my initial angle of the air, and then here's our angle of the water.

01:54 And so let's break out some snell's law and go n of the air times sign of the angle in the air is equal to n of the water times the sign of angle of the water.

02:08 Real quick, just so i don't have to actually plug anything in.

02:10 The sign of the angle of the water is the opposite divided by the hypotenuse.

02:16 There's just so many keep in mind for that.

02:24 Now i'm going to go back and define some more angles real quick.

02:27 So i know that this is our angle theta a, which means that's also this angle here.

02:34 And so i want to define another angle right here, right? which is because of this triangle here i'm gonna i'm gonna call this theta one just to give it a different name i know that theta one plus the angle of the air is equal to 90 degrees so theta 1 is equal to angle of the air sorry 90 minus the angle of the air angle the air because that this triangle is actually gonna be pretty helpful so i know that the tangent of theta 1 is equal to h over 4 centimeters i also want to use this now this red triangle right here right and go tangent of theta water right is going to be equal to two centimeters divided by h now if i can just reconcile, and i've got three equations and three unknowns, which is helpful.

03:58 So i've got my snells law equation.

04:00 I've got the equation from the red right triangle and one from the black right triangle.

04:05 If you notice, h is not something i know.

04:08 And then the angles, the respective angles, are not something that i know.

04:13 They're two of the angles.

04:14 And so we can solve this with some nasty algebra.

04:20 So let's simplify these down.

04:22 Because i define theta 1 as 90 minus theta a, this is actually also equal to the co -tangent of theta a is going to be the same as h over 4 centimeters.

04:37 And so then i can take this and turn this into the tangent of theta a is equal to 4 centimeters over h.

04:49 So that equation will be a little bit easier to use.

04:52 Have the unknowns as h, theta, a, you know, h, and theta w.

05:00 I'm just going to rewrite some of these to clean it up a little bit.

05:03 I'm going to solve for h.

05:04 So from the bottom right equation, h is going to equal 4 divided by the tangent of theta a.

05:11 I also know that h is going to be equal to 2 divided by the tangent of theta w.

05:22 And i know that we solve for one of these.

05:29 Let's go sine of theta a is going to be equal to 1 .33, which is the index for water times the sign of theta w.

05:52 Sine of theta w, right over 1, so i can just leave that there.

06:05 Then if i plug in here, i'm just going to do 4...

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