Question

- You find a smooth, spherical crystal ball of radius R = 4 cm, and you wish to determine its index of refraction n. (Note, the crystal ball is transparent, made of a uniform material, and has no bubbles or defects inside.) When the sun is directly overhead, an image of the sun forms on the ground when the ball is a distance ℎ = 2.35 cm above the ground. For your work below, treat the sun as a point source of light that is infinitely far away. (f) Measuring from the center of the sphere, what is the focal length f of the crystal ball for paraxial rays? In order to derive an expression for the focal length of the crystal ball, you should consider how light is refracted upon entering the upper surface of the sphere (northern hemisphere) and how it is refracted when it exits the lower surface of the sphere (the southern hemisphere). Use the following questions below to motivate your symbolic/algebraic derivation of the focal length. For all of your work below, express your final answers in terms of the symbols n and R. (g) Measuring from the “north pole” of the sphere, what is the image distance s'1 for the intermediate image of the sun formed by the upper surface of the sphere? (h) Measuring from the “south pole” of the sphere, what is the object distance ? for the intermediate image? (i) Measuring from the “south pole” of the sphere, what is the image distance for the final image of the sun formed by the lower surface of the sphere? (j) Measuring from the “equatorial plane”, what is the focal length f of the crystal ball? Now that you have a symbolic expression for the focal length of the sphere, you can use the numbers given in the problem (R = 4 cm and ℎ = 2.35 cm) to find the sphere’s index of refraction. (k) What is the index of refraction of the material that was used to make the crystal ball? What material might this be?

Answer 1