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 vour sketch a near mirror image of the left-hand portion that vou already sketched, which shows the path of light rays through a second nearby spherical surface. Indicate the new angles o, . 7', Elt' {inside the lens), and B:_" (outside the lens), analogous to the angles already introduced. (b) In the chapter, we found no + an = (nr - nv (15.44) for the first spherical surface. Following steps similar to those in the chapter, find a similar equation that relates the new angles o, [, and v . and the refractive indices, n. and n__ 6.=v - B t el. I:Ir iR _'rrr L ny an =na' +ny na' tnB' = ny' -ny' na + n B' = (n - n)y' Since B =- B' i na - nB = (n- n )y' (c) How is B related to B'? B =- B (d) Combine Eq. 15.44, and your answers to (b) and (c) to find an equation that relates a, a : Y : Y , n; and n. natnB = (n-n)y - eql na' tnB' = (n - n)y' Since B =- B' na' - nB = (n - n )y' ---eq2 Eq1+ eq2 natna = (n - n ) (y + y') (e) Using a small angle approximation, as in the chapter, relate a, a', y, and y' to the object-to-lens distance, p, the lens-to-image distance, q, the radius of curvature, R, and h. (It is a thin lens, so h = h.) Using a small angle approximation, which is valid when the angles are small enough that sine ~ 0 (where 0 is radians ), we can relate the angles to the geometric parameters of the system. Since it is a thin lens, h ~ h' and R' ~ R natna' = (n - n) (y +y) ( ) + n ( # ) = ( n - n ) ( #+# ) n n 2 (n -n) + = p 9 R