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# solutions3 - Homework Set 3 Due Thursday 07/14 Problem 1 A...

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Homework Set 3 Due Thursday, 07/14 Problem 1 A room contains two parallel wall mirrors, on opposite walls 5 meters apart. The mirrors are 8 meters long. Suppose that one person stands in a doorway, in line with mirror 2 and one meter away from it, and another person (the observer) stands one meter away from the other edge of the mirror. If the observer looks at mirror 1, how many images of the other person does he see? Solution: Each possible path of light between the person and the observer produces an image. There is a distance of 1 meters between the person and the corner of the mirror, so the path with the most possible reflections goes from the person to the edge of mirror 1, a shift of 1 meter. The reflection angle is the same as the incidence angle, so the light gets back to mirror 2 at a point 1 meter from the edge. It goes back to mirror 1 at 2 meters from the edge, then to mirror 2 at 3 meters, mirror 1 at 4 meters, mirror 2 at 5 meters, 1 at 6 meters, 2 at 7 meters, and finally hits mirror 1 at 8 meters and is reflected back to the observer, making a total of 9 reflections: It is clearly possible to have a path with just 1 reflection (light goes to center of mirror 1, then to the observer): The next possible path has 3 reflections (twice off mirror 1, once off mirror 2): Clearly any odd number of reflections up to 9 is possible. Thus, there are 5 possible paths (with 1, 3, 5, 7 and 9 reflections), and so the observer will see 5 images.

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Problem 2 A pencil going through the surface of the water appears to bend slightly where it crosses the surface. The pencil enters the water with an angle of 45 degrees to the vertical. If an observer is looking at the pencil approximately from above, at what angle does the pencil appear to bend (by how many degrees does its image diverge from a straight line)? Assume that the index of refraction of water is 1.33. Solution: Suppose a length L of pencil is submerged underwater. Then, the depth of the pencil's tip is L cos 45 ο . Due to refraction of light through the water, the depth of the tip appears to be less than it actually is. Using the equation for refraction of light by a spherical surface in the limit where R goes to infinity, we obtain The apparent depth of the pencil tip is thus less than the actual depth: The angle α made by the image of the pencil with the vertical is cos = 0.75 cos 45 ο , or = 58 o . The difference between this angle and the actual angle of the pencil is 8 o , so the pencil appears to be bent by 8 o .
Problem 3 A transparent sphere forms an image of the Sun precisely on its surface, on the side opposite to the direction of the Sun. What is the index of refraction of the material of the sphere? Solution:

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solutions3 - Homework Set 3 Due Thursday 07/14 Problem 1 A...

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