Lenses and Lasers: Focal Length of 10 cm Sphere

[PH^S!C5]

Homework Statement

A gypsy's crystal ball can act as a thick lens whose centre thickness is twice the radius of curvature of the surface. For a 10 cm sphere of glass with n=1.5 what is the focal length?

Like this? : http://postimage.org/image/niisv7bq9/

Homework Equations

I am not sure if I can simply use the thick lens equation:
1/f = (n-1)*[ (1/r1)-(1/r2)+ ((n-1)*tc)/(n*r1*r2) ]
n=1.5 for glass, and tc = diameter of ball =10cm, since 2tc=r1 then r1=5cm.

or the Gaussian formula for spherical surfaces: f=(n1*r)/(n2-n1) where n1 is the index of refraction on medium of light origin i.e. n1=1 for air; and n2 is the index on medium entered i.e. 1.5 for glass.

The Attempt at a Solution

By the thick lens equation: 1/f= (1.5-1)*[((1.5-1)*10)/(1.5*5*5)]= 0.06666
such that f=15cm.
By the Gaussian formula for spherical surfaces: f=(1*5)/(1.5-1)=10 cm

Question1: which one is correct and how can you determine this?
Question 2: I don't know if r1=-r2 as if one lens were convex (+) and the other concave (-) is taken into account by the formula(s) or if it has to be included at all! Please help! Thanks for your time.

 

[Simon Bridge]

One is 15cm and the other 10cm ... that's a big difference!

Question1: which one is correct and how can you determine this?

The way to figure that out is to look at how these equations were derived: what are the assumptions and approximations?

(Note: if desperate, you can always draw a scale diagram and plot a bunch of rays.)

Question 2: I don't know if r1=-r2 as if one lens were convex (+) and the other concave (-) is taken into account by the formula(s) or if it has to be included at all! Please help! Thanks for your time.

Work this out the same way you work out Q1 (above).

Have you tried using the matrix method directly?

 

[PH^S!C5]

Hi Simon Bridge, thanks for your reply...

I have tried using the following matrix:
[\
\begin{bmatrix}
1 & 0 \\
(n2-n1)/R2 & 1
\end{bmatrix}
x
\begin{bmatrix}
1 & d/n2 \\
0 & 1
\end{bmatrix}
x
\begin{bmatrix}
1 & 0 \\
(n2-n1)/R2 & 1
\end{bmatrix}
=
\begin{bmatrix}
0.934 & 0.67 \\
0.9 & 0.1
\end{bmatrix}\]

The final matrix doesn't make sense though...

 

[Simon Bridge]

That would be the thick lens method again.
What are the approximations used for this method?
What are the assumptions?

What is it you feel does not make sense?