Most Generalized Lensmaker's Equation

[TaylorRiles]

Homework Statement

Problem Statement: A glass sphere with radius R = 10 cm and index n = 1.52 is coated with a reflecting layer over one hemisphere. An object with a height of h = 1 cm is placed within 15 cm in front of the clear surface of the sphere. Determine the position, the size, and the character of the final image.

My issue is that there are too many generalizations to account for, and I can't find the right equations. We cannot use the thin lens equation because the lens is a sphere; I don't think we can use the ordinary mirror formula because the reflective surface is inside a medium with non-unity index of refraction. And how does object height fit into all this?

Homework Equations

Lensmaker's formula that accounts for thick lenses:
1/f = (n - 1)(1/R1 - 1/R2 + ((n-1)d)/(n*R1*R2)
where n is index of refraction of lens, d is thickness of lens (in this case diameter of sphere, I believe), R1 is radius of lens facing object, R2 is radius facing away (will be negative).

Lensmaker's formula that accounts for objects in different media:
1/f = [ (n of lens/n of outside medium) -1][ (R1-R2)/R1*R2]

Mirror formula (whose assumptions may or may not include immersion in not-air medium)
1/p + 1/q = -2/r
where r is radius of curvature, p is object distance, q is image distance.

The Attempt at a Solution

First considered lens. Used lensmaker's formula for thick lenses and used the formula
1/f = (n lens) - 1)(1/R1 - 1/R2 + ((n-1)d)/(n*R1*R2), with R1 = 10,R2 = -10, n = 1.52, d = 20. Got f of lens equal to -7.16. The solutions manual says that's incorrect, and they use this formula:
1/(object distance) + (n lens)/(image distance) = ((n lens)-1)/R.
I can see where the right side came from, but not the n lens on top of image distance. Help?
And then once the mirror comes in, I'm lost. The image should form way to the right of the mirror but never does. How do we deal with that?
Thanks so much!

 

[Simon Bridge]

Welcome to PF;
The most general formulation of par-axial ray optics is probably the ray-transfer matrix formulation.
It will even handle mirrors.

First considered lens. Used lensmaker's formula for thick lenses and used the formula
1/f = (n lens) - 1)(1/R1 - 1/R2 + ((n-1)d)/(n*R1*R2), with R1 = 10,R2 = -10, n = 1.52, d = 20. Got f of lens equal to -7.16. The solutions manual says that's incorrect, and they use this formula:
1/(object distance) + (n lens)/(image distance) = ((n lens)-1)/R.
I can see where the right side came from, but not the n lens on top of image distance. Help?
And then once the mirror comes in, I'm lost. The image should form way to the right of the mirror but never does. How do we deal with that?

I know it is convenient to just plug numbers into equations and believe whatever comes out but that is not how physics is done.
You are best letting the physics be your guide and running through the calculation in several steps.

The fat-lense lensmakers equation is not appropriate fr this situation because you don't have a fat lens.

In the par-axial approximation, what you have is a 3-element system ...
1. spherical interface
2. length of glass
3. spherical mirror