Methods | Part IB, 2003

Fermat's principle of optics states that the path of a light ray connecting two points will be such that the travel time tt is a minimum. If the speed of light varies continuously in a medium and is a function c(y)c(y) of the distance from the boundary y=0y=0, show that the path of a light ray is given by the solution to

c(y)y+c(y)(1+y2)=0c(y) y^{\prime \prime}+c^{\prime}(y)\left(1+y^{\prime 2}\right)=0

where y=dydxy^{\prime}=\frac{d y}{d x}, etc. Show that the path of a light ray in a medium where the speed of light cc is a constant is a straight line. Also find the path from (0,0)(0,0) to (1,0)(1,0) if c(y)=yc(y)=y, and sketch it.

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