diff --git a/_psets/7.md b/_psets/7.md index 89c0965cceae8bcbc7c153aa0459ddccc51c7791..7cfeb591c8185439bafa6239afb404d0aab7365f 100644 --- a/_psets/7.md +++ b/_psets/7.md @@ -10,6 +10,44 @@ case Fermat’s Principle: a light ray chooses the path between two points that travel between them. Apply this to two points on either side of a dielectric interface to derive Snell’s Law. +Let there be a dialectic interface along the y axis, between media with indices of refraction +$$n_1$$ (left) and $$n_2$$ (right). The time it takes for light to go from a point $$(x_1, y_1)$$ on +the left to $$(x_2, y_2)$$ on the right, passing through the interface at height $$y$$ is + +$$ +t = \frac{n_1}{c} \sqrt{x_1^2 + (y_1 - y)^2} + \frac{n_1}{c} \sqrt{x_2^2 + (y_2 - y)^2} +$$ + +The derivative of this with respect to $$y$$ is + +$$ +\begin{align*} +\frac{dt}{dy} +&= \frac{-n_1 (y_1 - y)}{c \sqrt{x_1^2 + (y_1 - y)^2}} + + \frac{-n_2 (y_2 - y)}{c \sqrt{x_2^2 + (y_2 - y)^2}} \\ +&= -n_1 \sin \theta_1 + n_2 \sin \theta_2 +\end{align*} +$$ + +since + +$$ +\sin \theta_1 = \frac{y_1 - y}{\sqrt{x_1^2 + (y_1 - y)^2}} +$$ + +and + +$$ +\sin \theta_2 = \frac{y - y_2}{\sqrt{x_2^2 + (y_2 - y)^2}} +$$ + +(by [SOHCAHTOA](http://mathworld.wolfram.com/SOHCAHTOA.html)). Hence the time of flight is minimized +when + +$$ +\frac{n_1}{n_2} = \frac{\sin \theta_2}{\sin \theta_1} +$$ + ## (9.2)