Displacement of virtual image problem

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1. Jan 12, 2016

theguyoo

• Moved from a technical forum, so homework template missing.
An object is within a glass sphere of radius R with a refractive index of 1.5 . I'm trying to calculate the displacement of the virtual object relative to the actual when viewed from the side, such that the refracted ray emanating from the object becomes horizontal. I would like to know S (the image shift) as a function of d, the distance of the object from the center of the sphere.

Diagram: http://imgur.com/dCaTLvq

So far I've tried starting with Snell's law, $sin\alpha = 1.5sin\beta$, and manipulated it to get:
$\frac{\sqrt{R^{2}}-x^{2}}{R}=1.5\frac{x \cdot d}{R\sqrt{R^{2}-2d\sqrt{R^{2}-x^{2}+d^{2}}}}$

where and x is the distance form origin so $(S+d)^{2}+x^{2}=R^{2}$

The problem is it's too messy to rearrange to make S the subject, but I'm guessing there's a better way.

Thanks.

2. Jan 13, 2016

haruspex

Look at the horizontal line from the point where the ray exits the sphere to the apparent position. Can you find two expressions for that distance in terms of the angles, d and S?