 Solution

ID: past-exam-of-the-physics-2-paper-of-the-university-of-oxford/2013/9/1/solution

Fermat's principle says that the travel time, or equivalently the optical path length, is stationary under small path variations at fixed endpoints. For a flat interface, let the ray cross at horizontal coordinate

x

, with source and detector at heights

h_{1}, h_{2}

and horizontal separation

D

. Apart from

1/ c

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

Setting

d L / d x = 0

gives

\frac{n _{1} x}{h _{1}^{2} + x ^{2}} = \frac{n _{2} ( D - x )}{h _{2}^{2} + ( D - x ) ^{2}}, n_{1} sin i = n_{2} sin r .

The angles are measured to the normal. This is Snell's law; “least time” is more generally a stationarity condition.

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