FYKOS Year 33 Series 6 Problem 5 — golden nectar · FYKOS 33 #33.6.5

Physics · electromagnetism · optics

Problem

The magic field of Discworld is so strong that the speed of light no longer has its common meaning. This applies only close to the surface, where the refractive index of the magic field has magnitude n0=2.00⋅106n_0 = 2.00\cdot 10^{6}. The refractive index decreases with height hh as n(h)=n0e−khn(h) = n_0\mathrm{e}^{-kh}, where k=1.00⋅10−7 m−1k = 1.00\cdot 10^{-7}\,\mathrm{m^{-1}}. Calculate the optimal angle (with respect to the vertical direction) under which a light signal should be emitted from one end of the Discworld to reach the opposite end in the shortest possible time. The diameter of the Discworld is d=15 000 kmd = 15~000\,\mathrm{km} and the speed of light in vacuum is c=3.00⋅108 m⋅s−1c = 3.00\cdot 10^{8}\,\mathrm{m\cdot s^{-1}}.

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