To find the distance d over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density $ρ$ of the fog, intensity (power/area) $S$ of the light from the signal and its frequency f. The engineer finds that d is proportional to $S^{1 / n}$ .The value of $n$ is:

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answer is 3.

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Detailed Solution

$\begin{matrix} L = {(S)}^{x} {(ρ)}^{y} {(f)}^{z} \\ L = {(M^{1} L^{0} T^{- 3})}^{x} {(M^{1} L^{- 3})}^{y} {(T^{- 1})}^{z} \\ L = M^{x + y} L^{- 3 y} T^{- 3 x - z} \end{matrix}$

$x + y = 0; - 3 y = 1; - 3 x - z = 0$
$y = - \frac{1}{3} \Rightarrow z = \frac{1}{3}$

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