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?

 

[paranormal]

I would first like to clarify that the use of the term "gypsy" can be considered derogatory and offensive to the Romani community. I would suggest using a different term or simply referring to it as a crystal ball.

Moving on to the question, the correct formula to use in this scenario would be the thick lens equation. This is because the crystal ball is acting as a thick lens, with a center thickness that is twice the radius of curvature of the surface. The Gaussian formula is used for thin lenses, where the thickness can be neglected.

In this case, the radius of curvature of the surface is equal to half the diameter of the ball (10cm/2 = 5cm). Therefore, the correct formula to use would be:

1/f = (n-1)*[(1/r1) - (1/r2) + ((n-1)*tc)/(n*r1*r2)]

Where n = 1.5 (index of refraction for glass), tc = 10cm (diameter of ball), r1 = 5cm (radius of curvature of surface), and r2 = -5cm (since the surface is concave).

Plugging in these values, we get:

1/f = (0.5)*[(1/5) - (1/-5) + ((0.5)*10)/(1.5*5*-5)]

1/f = 0.1333

Therefore, the focal length of the crystal ball would be 7.5 cm.

To answer your second question, the sign convention for the radius of curvature in the thick lens equation is as follows:

- r1 and r2 are positive for convex surfaces
- r1 and r2 are negative for concave surfaces

In this case, since the surface of the crystal ball is concave, the radius of curvature is negative. It is important to include the sign in the formula to ensure the correct calculation of the focal length.

I hope this explanation helps. Please let me know if you have any further questions.