A layer of air above a hot road has a refractive index that depends only on height…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Optics & Photonics · Geometrical optics and imaging

Action and Symmetry, Ch The action, the first variation and the boundary term

Problem

A layer of air above a hot road has a refractive index that depends only on height yy above the road surface (all rays lie in a vertical plane, and the medium is at rest and non-dispersive). Take the profile nr(y)=n0(1+αy),n0=1.000, α=2.0×10−4 m−1,n_{\rm r}(y) = n_0(1 + \alpha y), \qquad n_0 = 1.000,\ \alpha = 2.0\times10^{-4}\,\mathrm{m^{-1}}, valid for y≥0y \ge 0, so that the index is lower near the road and rises with height. ((a)) Starting from Fermat's principle that a ray makes the optical path ∫nr ds\int n_{\rm r}\,\mathrm{d}s stationary, write the path as y(x)y(x) and show that nr(y)sin⁡θn_{\rm r}(y)\sin\theta is constant along the whole ray, including its lowest point, where θ\theta is the angle the ray makes with the vertical (yy) direction. ((b)) A ray leaves a point at height y0=1.7 my_0 = 1.7\,\mathrm{m} travelling at θ0=89.0∘\theta_0 = 89.0^\circ to the vertical, heading down towards the road (yy decreasing along the ray). Find the smallest height yty_t the ray reaches before it turns and heads back upward (the height at which θ=90∘\theta = 90^\circ).

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