Finding the radii of curvature for an achromatic lens

[youmei0426]

Homework Statement

An achromatic lens is to be made by combining plano-convex and plano-concave lenses made from two different types of glass, A and B. These have the following refractive indexes:

red light: A - 1.51 B - 1.64
blue light: A - 1.53 B - 1.68

What radii of curvature should be used to produce a combination equivalent to a converging lens with a focal length of 500mm?

Relevant Equations

1/f = 1/f_1 + 1/f_2
1/f = (n-1)(1/R_1+1/R_2)
R - curvature
n - refractive index
f - focal length

Since the two lenses are plano-concave and plano-convex, the lens maker equation can be simplified into containing only 1 R for each of the lens. Substituting the values for the red light, I got: 1/f₁=0.51/R₁ and 1/f₂ = 0.64/R₂. Adding these two equations and equating them to 1/(500*10^-3).

However, this equation clearly has two variables. I'm sure we need to use the data for the blue light, but I am not sure how I can incorporate them into the equations? The curvatures of the lenses should not change as the wavelength changes...

Thanks in advance for the help!

 

[BvU]

this equation clearly has two variables

Yes, so you need another equation -- the one for blue light!