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\title{Refraction at a Spherical Surface}
\author{David Harrison}
\date{March 2012}
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\section{Introduction}
\normalsize
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For refraction at a concave spherical surface with radius of curvature r, show that:

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\Large $\frac{n}{S}$\normalsize $ + $\Large $\frac{n'}{S'}$\normalsize $ = $\Large $\frac{n' - n}{r}$\normalsize

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Where n and n' are refractive indices of the media at either side of the interface, and S and S' are the object and image distances respectively. Small angle approximation is assumed to be valid.

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\section{Derivation}
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Below is a diagram of the interface:

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\includegraphics[height=4.1in,width=6in]{refraction.png}

Light leaves point O and is refracted at point P. There is a virtual image formed at point I and the centre point of the curvature of the lens is at C. Due to the sign convention used, the distance to the image is -S' and the distance to the centre of curvature is -r, both measured from point V.

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From triangle COP:

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$\alpha + \theta + 180 - \Phi = 180$

$\therefore \theta = \Phi - \alpha$

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From triangle IOP:

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$\alpha' + \theta' + 180 - \Phi = 180$

$\therefore \theta' = \Phi - \alpha'$

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Also, using the small angle approximation:

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$\Phi = $\Large $\frac{h}{-r}$\normalsize $= -$\Large $\frac{h}{r}$\normalsize

$\alpha = $\Large $\frac{h}{s}$\normalsize

$\alpha' = $\Large $\frac{h}{-S'}$\normalsize $= -$\Large $\frac{h}{S'}$\normalsize

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Now using Snell's law and the small angle approximation:

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$nsin\theta = n'sin\theta'$

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$n\theta = n'\theta'$

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$n(-$\Large $\frac{h}{r}$\normalsize $ - $\Large $\frac{h}{S}$\normalsize $) = n'(-$\Large $\frac{h}{r}$\normalsize $ + $\Large $\frac{h}{S'}$\normalsize $)$

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$n($\Large $\frac{h}{r}$\normalsize $ + $\Large $\frac{h}{S}$\normalsize $) = n'($\Large $\frac{h}{r}$\normalsize $ - $\Large $\frac{h}{S'}$\normalsize $)$

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\Large $\frac{n}{r}$\normalsize $ + $\Large $\frac{n}{S}$\normalsize $ = $\Large $\frac{n'}{r}$\normalsize $ - $\Large $\frac{n'}{S'}$\normalsize

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\Large $\frac{n}{S}$\normalsize $ + $\Large $\frac{n'}{S'}$\normalsize $ = $\Large $\frac{n'}{r}$\normalsize $ - $\Large $\frac{n}{r}$\normalsize

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\Large $\frac{n}{S}$\normalsize $ + $\Large $\frac{n'}{S'}$\normalsize $ = $\Large $\frac{n' - n}{r}$\normalsize

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Therefore it has been shown that the formula holds for a concave interface, as well as a convex interface.

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