Problem 51
When the sun is either rising or setting and appears to be just on the horizon, it is in fact below the horizon. The explanation for this seeming paradox is that light from the sun bends slightly when entering the earth’s atmosphere, as shown in Fig. \(\textbf{P33.51}\). Since our perception is based on the idea that light travels in straight lines, we perceive the light to be coming from an apparent position that is an angle \(\delta\) above the sun's true position. (a) Make the simplifying assumptions that the atmosphere has uniform density, and hence uniform index of refraction \(n\), and extends to a height \(h\) above the earth's surface, at which point it abruptly stops. Show that the angle \(\delta\) is given by $${ \delta = \mathrm{arcsin} ({{nR}\over{R + h}}}) - \mathrm{arcsin}({{{R}\over R + h}}) $$ where \(R\) = 6378 km is the radius of the earth. (b) Calculate \(\delta\) using \(n\) = 1.0003 and \(h\) = 20 km. How does this compare to the angular radius of the sun, which is about one quarter of a degree? (In actuality a light ray from the sun bends gradually, not abruptly, since the density and refractive index of the atmosphere change gradually with altitude.)
Problem 58
Given small samples of three liquids, you are asked to determine their refractive indexes. However, you do not have enough of each liquid to measure the angle of refraction for light refracting from air into the liquid. Instead, for each liquid, you take a rectangular block of glass (\(n\) = 1.52) and place a drop of the liquid on the top surface of the block. You shine a laser beam with wavelength 638 nm in vacuum at one side of the block and measure the largest angle of incidence \(\theta_a\) for which there is total internal reflection at the interface between the glass and the liquid (\(\textbf{Fig. P33.58}\)). Your results are given in the table: What is the refractive index of each liquid at this wavelength?
