Problem 5.62

A ray of light propagating in an isotropic medium with refractive index \(n\) varying gradually from point to point has a curvature radius \(\rho\) determined by the formula \[ \frac{1}{\rho}=\frac{\partial}{\partial N}(\ln n) \] where the derivative is taken with respect to the principal normal to the ray. Derive this formula, assuming that in such a medium the law of refraction \(n \sin \theta=\) const holds. Here \(\theta\) is the angle between the ray and the direction of the vector \(\boldsymbol{\nabla} n\) at a given point.