Chapter 44: Problem 15

Question: Show that the focal length for a thin lens whose index of refraction is n and which is immersed in a fluid whose index of refraction is $n^\prime$ is given by

$${1\over f}~=~{n-n^\prime\over n^\prime} \left( {1\over R_1}-{1\over R_2}\right)$$

Solution: Starting with the relation between object and image for one spherical surface, we have that

$${n_1\over O}+{n_2\over I_1}~=~(n_2-n_1){1\over R_1}$$

where I₁ is the image from the curved surface, which serves as the object for the second curved surface. Taking into account the -ve sign due to the location of the image, we have that the object for the second surface is O₂=-I₁. For the second surface, we have that

$${n_2\over O_2}+{n_1\over I}~=~(n_1-n_2){1\over R_2}$$

The answer follows.