Image of an Object Inside Hollow Glass Sphere

[Kaan99]

Hello,
This question is from 1990 Turkey National Physics Olympics. I tried my best to translate it clearly.

1. Homework Statement
https://s23.postimg.org/cotn29afv/Hollow+Spherical+Glass.jpg

The sphere of radius 2R has an empty sphere inside with radius R. In order for the image of an object on the inner surface of the sphere to be R/5 far away from the object (which is shown with an asterisk*) for an observer at the indicated position (göz), what should be the index of refraction of the glass? Where is the focal point of this optical system?

Homework Equations

Thin Lens Equation
Lensmaker's Equation

The Attempt at a Solution

The lens is made up of to circles of radii R and 2R, using lensmaker and thin lens eq.
$\frac{1}{f} = (n-1) (\frac{1}{R_1}-\frac{1}{R_2})=\frac{1}{d_o}+\frac{1}{d_i}$
$(n-1)(\frac{1}{-R}-\frac{1}{-2R})=\frac{1}{2R}+\frac{1}{d_i}$
$\frac{n-1}{-2R}=\frac{1}{2R}+\frac{1}{d_i}$
$d_i=\frac{n}{-2R}$

It turned out that my attempt wasn't accurate from the start, as it deviates from the solution provided. I couldn't understand how the first equation is formed in the solution.

The result is n=2 and f=4R.

 

[drvrm]

The Attempt at a Solution

The lens is made up of to circles of radii R and 2R, using lensmaker and thin lens eq.
1f=(n−1)(1R1−1R2)=1do+1di1f=(n−1)(1R1−1R2)=1do+1di \frac{1}{f} = (n-1) (\frac{1}{R_1}-\frac{1}{R_2})=\frac{1}{d_o}+\frac{1}{d_i}
(n−1)(1−R−1−2R)=12R+1di(n−1)(1−R−1−2R)=12R+1di (n-1)(\frac{1}{-R}-\frac{1}{-2R})=\frac{1}{2R}+\frac{1}{d_i}
n−1−2R=12R+1din−1−2R=12R+1di \frac{n-1}{-2R}=\frac{1}{2R}+\frac{1}{d_i}
di=n−2Rdi=n−2R d_i=\frac{n}{-2R}

It turned out that my attempt wasn't accurate from the start, as it deviates from the solution provided. I couldn't understand how the first equation is formed in the solution.

Hello,
This question is from 1990 Turkey National Physics Olympics. I tried my best to translate it clearly.



i think you have a very thick lens system so your thin lens formula will not work.
search in the textbook for thick lens formula.