Question

When the sun is either rising or setting and appears to be just in 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 figure. Assume 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 is abruptly stop. The angle $\delta$ above the Sun's true position is given by

• A. $si{n}^{-1}\left(\frac{nR}{R+h}\right)$
• B. $si{n}^{-1}\left(\frac{R}{R+h}\right)$
• C. $si{n}^{-1}\left(\frac{nR}{R+h}\right)+si{n}^{-1}\left(\frac{R}{R+h}\right)$
• D. $si{n}^{-1}\left(\frac{nR}{R+h}\right)-si{n}^{-1}\left(\frac{R}{R+h}\right)$

Answer 1

The correct option is D $si{n}^{-1}\left(\frac{nR}{R+h}\right)-si{n}^{-1}\left(\frac{R}{R+h}\right)$

From geometry, $sin\theta =\frac{h}{R+h}$ & $sinr=\frac{R}{R+h}$

Using snell's law, $nsinr=sin(\theta +r)$, as angle of incidence $i=\theta +\delta$

Thus $sin[si{n}^{-1}(\frac{h}{R+h})+\delta ]=\frac{nR}{R+h}$

$⟹\delta =si{n}^{-1}(\frac{nR}{R+h})-si{n}^{-1}(\frac{R}{R+h})$