Question

Your eye is designed to work in air. Surrounding it with water impairs its ability to form images. Consequently, scuba divers wear masks to allow them to form images properly underwater. However, this does affect the perception of distance, as you will calculate. Consider a flat piece of plastic (index of refraction npnpn_p) with water (index of refraction nwnwn_w) on one side and air (index of refraction nanan_a) on the other. If light is to move from the water into the air, it will be refracted twice: once at the water/plastic interface and once at the plastic/air interface.

Part A

If the light strikes the plastic (from the water) at an angle θw , at what angle θa does it emerge from the plastic (into the air)?

Express your answer in terms of nw , np , na , and θw . Remember that the inverse sine of a number x should be entered as asin(x) in the answer box.

Part B

What is the distance to the object in terms of θw and l ?

Express your answer in terms of θw and l .

Part C

If the distance to the object is more than about 0.4 m , then you can use the small-angle approximation tan(θ)≈θ . What is the formula for the distance D to the object, if you make use of this approximation?

Express your answer in terms of θw and l .

Part D

Now use the expression found in Part C for the distance between your eyes and the object at point O, and find the ratio of the apparent distance to the real distance, d/D . Remember that the apparent distance is the distance calculated by your eyes using the angle θa instead of the angle θw . Since we are dealing with small angles, you may use the approximations sin(x)≈x and asin(x)≈x .

Express your answer in terms of nw and na .

Answer 1

Check the explanation

Step-by-step explanation:

A) There are two important angles within the plastic: the angle immediately after the first refraction (the water/plastic interface) and the angle immediately before the second refraction (the plastic/air interface).

To find out how they relate, draw a picture with the path the light follows in the plastic and the normal to both surfaces.

Once you have labeled both angles, keep in mind that the surfaces are parallel, and thus their normal are parallel lines. An important theorem from geometry will give you the relationship between the angles.

Using Snell's Law, θa = asin[(nw/na)*sin(θw)]

B) D = l/tan(θw)

C) D = l/θw

D) d/D = na/nw