 Solution

ID: past-exam-of-the-physics-2-paper-of-the-university-of-oxford/2017-resit/9/a/solution

\int n d s

in a stationary medium, is stationary under small variations of the path with endpoints held fixed. It need not be a global minimum.

Let a light ray join a point at height

h_{1}

above a plane interface to a point at depth

h_{2}

below it, with horizontal separation

D

. If the crossing point is horizontally

ξ

from the first endpoint, its optical path length is

L (ξ) = n_{1} h_{1}^{2} + ξ^{2} + n_{2} h_{2}^{2} + (D - ξ)^{2} .

Setting its derivative to zero gives

\frac{n _{1} ξ}{h _{1}^{2} + ξ ^{2}} = \frac{n _{2} ( D - ξ )}{h _{2}^{2} + ( D - ξ ) ^{2}} .

The two geometric ratios are the sines of the incident and refracted angles, measured from the interface normal vector. Hence Snell's law is

n_{1} sin θ_{1} = n_{2} sin θ_{2} .

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