Question

how do you derive the lens maker's equation?

Answer 1

For a spherical lens:

1/s1+n/s1' = (n-1)/(R1)

where:
1 = index of refraction of air
n = index of refraction of lens
s1 = distance to object
s1' = distance to image
R1 = radius of curvature of the lens

and applying this to the other side of the lens:

-n/s1'+1/s2' = (1-n)/R2 = -(n-1)/R2

with

s2' = distance to final image
R2 = radius of curvature of other side of lens

adding these two equations:

1/s1+n/s1' + -n/s1'+1/s2' = 1/s1+1/s2'


1/s1+1/s2' = (n-1)/R1 - (n-1)/R2
= (n-1)[1/R1-1/R2]

and the thins lens equation gives 1/s+1/s'=1/f

where f is the focal length

and substituting this gives the lens maker equation

1/f = (n-1)[1/R1-1/R2]