Find index of refraction of a sphere given the beam path

[MrMoose]

Homework Statement

A beam of parallel light rays from a laser is incident on a solid transparent sphere of index of refraction n1 (see figure). (a) If a point image is produced at the back of the sphere, what is the index of refraction of the sphere? (b)What index of refraction, if any, will produce a point image at the center of the sphere.

Homework Equations

n1/p + n2/i = (n2-n1)/r

The Attempt at a Solution

n1 ~ 1.0 for air
i = 2r

Assuming that the factor n1/p -> 0 since the object is so far away that the light rays are parallel (is this assumption correct?)

(a) Substituting for n1 and i:

n2/2r = n2/r - 1/r

n2/2r - n2/r = -1/r

n2 (1/2r - 2/2r) = -1/r

n2(-1/2r) = -1/r

n2 = 2

This is the correct answer, but I want to make sure my assumption above was correct and I didn't just luck out.

(b) Using the same equation from part 2 above, but substituting i = r,

n2/r = n2/r - 1/r

0 = - 1/r

In order for this to be true, the sphere would have to be infinitely large so, realistically, there is no value of r that would place the image at the center of the sphere.

 

[haruspex]

For b, you can see that it is not possible by running the light rays in reverse. If they originate at the centre of the sphere then they will not refract at all, so cannot become parallel.