To find the distance d over which a signal can be seen clearly in...

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}^{\dfrac{1}{n}}

. The value of n is

Correct answer

Step-by-step solution

d ∝ {ρ}^{x}{S}^{y}{F}^{z}

⇒

\left[L\right] ≡ {\left[M{L}^{- 3}\right]}^{x}{\left[M{T}^{- 3}\right]}^{y}{\left[{T}^{- 1}\right]}^{z}

<br >

⇒ x + y = 0, - 3x = 1, - 3y - z = 0

<br >

⇒ x = - \dfrac{1}{3}, y = \dfrac{1}{3}, z = - 1

<br >

⇒ y = \dfrac{1}{n}

<br >

⇒ n = 3

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