# Path Followed by a Light Ray

#### Fernando Rios

Problem Statement
Find the path followed by a light ray if the index of refraction (in polar coordinates) is proportional to r^-2.
Relevant Equations
∫nds
This is an example from a textbook. They show the following:

∫nds = ∫r-2*ds = ∫r-2*√((dr)2+r2*(dθ)2)

How do you obtain the expression for ds?

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#### Abhishek11235

The arc length in Cartesian coordinate is given by :

$ds^2 = dx^2 + dy^2$
Which is Pythagoras theorem for an infinitesimal part of curve.

The transformation function to polar coordinate has form:

$x= r cos\theta$
$y=r sin\theta$

Substituting in above equation and manipulating gives desired result.

#### Fernando Rios

Got it! Thank you for your help.

"Path Followed by a Light Ray"

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