Radius of Curvature to Correct Myopia

[TeslaCoil137]

Homework Statement

A person's eye has a near point of 7 cm. The cornea at the outer surface of the eye has a refractive index of n_c = 1.376 and forms a convex shape with a radius of curvature of R_2 = 8 mm against air. The figure below shows the same eye with a contact lens (refractive index of n_L = 1.5) mounted against the cornea such that second (right) surface matches the curvature of the cornea (i.e. R_2= 8 mm). Determine the radius R_1 of the first surface of the contact lens that will correct the near point to the normal 25 cm distance from the eye. Assume paraxial and thin lens conditions.

Homework Equations

(1) 1/f =1/f_1 + 1/f_2, (2) 1/f_1 = (n/c -1) (1/R_1 -1/R_2), (3) 1/f_2 = (n_L -1)(1/R_2), (4) 1/u + 1/v = 1/f

The Attempt at a Solution

The focal length of the combined lenses required to correct the present myopia is found from the Gaussian thin lens equation as 1/-7 + 1/-25 = 1/f ⇒ f= -5.47. Using paraxial optics and Fermat's principle of least time relevant equations (1)-(3) are easily found by requiring that all paths through the lens take equal time to reach the focus. Substituting (2) and (3) into (1) with the given numbers we find R_1 =-17.98 cm.

 

[andrevdh]

The object distance is real +25 cm, but I do not think this is the correct approach.

The applicable formula that I could find is

n/a + n''/b' = (n'-n)/r + (n''-n')/r'

where the object distance for the first surface of the contact lens (curvature radius r) being a, and the final image being formed at b' by the second surface (curvature radius r') and the refractive indices n (air), n' (contact lens), and n'' (fluid)

This was derived from the formula for refraction at a single spherical surface

n/a + n'/b = (n'-n)/r

being applied at both surfaces and assuming that the distance between the two surfaces is negligible

Maybe you are suppose to derive the 1st formula?

 

[TeslaCoil137]

Ok, thank you. So what's the conceptual issue I have that lead to my approach?

 

[andrevdh]

Well, for one (4), Gauss's equation for thin lenses, assume that the lens is in air.

What you can also do it apply the second equation in my previous post at both surfaces if you do not have the first one in you handbook and then assume that the distance between the two surfaces are negligible.