Fig. 15.3 Angles, distances, and geometry for image formation behind a spherical surface, which is a simple model for how our eyes focus. As sketched, the refractive index inside the sphere is larger than the refractive index outside the sphere, corresponding to the situation for our eyes Problem 3: The Thin Lens Formula The goal of this problem is to derive the \"thin lens formula,\" which relates the location of an image, formed by a lens, to the location of the object and the focal length of the lens. (a) Reproduce the left-hand portion Fig. 15.3, including rays and angles just to the right of the spherical surface. Now, add to the right-hand side of your sketch a near mirror image of the left-hand portion that you already sketched. which shows the path of light rays through a second nearby spherical surface. Indicate the new angles &', B, ", 6/ (inside the lens), and , (outside the lens), analogous to the angles already introduced. (b) In the chapter, we found o niw +ni = (ny nj)y (15.44) for the first spherical surface. Following steps similar to those in the chapter, find a similar equation that relates the new angles ', B', and ', and the refractive indices, n;. and n;. (c) How is A related to B? (d) Combine Eq. 15.44, and your answers to (b) and (c) to find an equation that relates &, o', y. ', ni. and n;. Using a small angle approximation. as in the chapter, relate . @', y. and ' to the object-to-lens distance, p. the lens-to-image distance, g. the radius of curvature, R, and /. (It is a thin lens, so i' = h.) e (e