Use Fermat's principle to find the path of a light ray...

Use Fermat's principle to find the path of a light ray through a medium of index of refraction proportional to the given function.
$r^{-1} \ln r \quad$ Hint: In the last integration, let $u=\ln r$.

Key Concepts

Fermat's Principle

Fermat's Principle is the idea that light travels between two points along the path that requires the least time, or more generally, a path of stationary time. This concept is pivotal in optics as it provides the basis for determining the trajectory of light rays in various media, and it leads to the formulation of the laws of reflection and refraction.

Calculus of Variations

Calculus of Variations is a field of mathematical analysis used to find functions that optimize (minimize or maximize) a certain integral, known as a functional. In the context of optics, it is applied to determine the path of light by minimizing the optical path length, aligning with Fermat's principle.

Euler-Lagrange Equation

The Euler-Lagrange Equation is a fundamental equation in the calculus of variations that provides the necessary condition for a function to be an extremum of a given functional. It is used to derive the equations of motion or the path that light must follow, particularly in media where the refractive index varies spatially.

Index of Refraction

The index of refraction is a measure of how much the speed of light is reduced inside a medium compared to a vacuum. It plays a critical role in determining the bending of light rays when they pass from one medium to another or when the medium’s properties vary spatially.

Optical Path Length

Optical Path Length is the product of the physical distance traveled by light and the refractive index of the medium along its path. It is the quantity that is minimized or made stationary in Fermat's principle, thereby determining the actual trajectory of light.

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