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Homework Statement
Parallel light rays travel from air towards a glass hemisphere with radius R and
index of refraction ng > nair. A top view is shown in the figure.
http://img8.imageshack.us/img8/6380/parallellightrays.jpg
(a) Determine where the light rays come to a focus relative to the point P.
(b) The index of refraction of the hemisphere is increased by 0.5 percent. Does the
focus point change from that in part (a)? If it changes, then by how much?
Homework Equations
I found an equation in the textbook for paraxial rays approaching a spherical surface that is convex towards them. In that situation, the second medium was some glass, and the original medium was air. The equation was given as follows:
n1/s + n2/s' = (n2-n1)/R
where s is object distance (from which the paraxial rays originated) from the surface, and s' is the distance of the image formed from the surface. R is the radius of the sphere.
The Attempt at a Solution
The big difference between the two scenarios is that in the actual problem, the rays are approaching a surface that is concave towards them, rather than convex. However, the textbook states that according to the sign convention for refracting surfaces, R is positive when convex toward the object, and negative when concave toward the object.
I simply put this into the equation to give
n1/s + n2/s' = (n2-n1)/-R
n1/s + n2/s' = (n1-n2)/R
where n1 is the index of refraction for the glass, and n2 is the index for air
The other issue was that there isn't really an object distance in the problem.
Since the rays are parallel, I set s to infinity (making n1/s = 0) and arranged for s'.
n2/s' = (n1-n2)/R
R(n2/(n1-n2)) = s'
I suppose this would be my final answer for part a, but I'm unsure if my method works.
If I'm correct, I figured that for part b I could simply multiply n2 by 1.005.
Any help would be appreciated.
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