To find the distance $d$ over which a signal can be seen clearly foggy conditions, a railways engineer uses dimensional analysis and assume 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

The distance d can be considered to depend on $ρ, S$ and f with dimensions a,b and c. So we have $d α ρ^{a} S^{b} f^{c}$

Substituting the dimensional relation of all physical quantities in above expression, we have
$\Rightarrow [L] = {[M T^{- 3}]}^{a} {[\frac{M}{T^{3}}]}^{b} {[\frac{1}{T}]}^{c} \Rightarrow [L T^{- 1}] = M^{a + b} L^{- 3 a} T^{- 3 b - c}$
Comparing dimensions of LHS and RHS, we get
$a + b = 0, \Rightarrow b = - a - 3 a = 1 \Rightarrow a = - 1 / 3 \Rightarrow b = 1 / 3 \Rightarrow n = 3$

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