To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railways eng

Question

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 $ho$ 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\

Professor PC Thomas Classes · Answer

(c) : Let $d \propto ho^x S^y f^z$ or $d=k ho^x S^y f^z$ where $k$ is a dimensionless constant and $x, y$ and $z$ are the exponents. Writing dimensions on both sides, we get \ $ \begin{aligned} & {\left[\mathrm{M}^0 \mathrm{LT}^0ight]=\left[\mathrm{ML}^{-3} \mathrm{~T}^0ight]^x\left[\mathrm{ML}^0 \mathrm{~T}^{-3}ight]^y\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}^{-1}ight]^z} \ & {\left[\mathrm{M}^0 \mathrm{LT}^0ight]=\left[\mathrm{M}^{x+y} \mathrm{~L}^{-3 x} \mathrm{~T}^{-3 y-z}ight]} \end{aligned} $ \  Applying the principle of homogeneity of dimensions, we get $$ \begin{align*} & x+y=0  	ag{i}\ & -3 x=1  	ag{ii}\ & -3 y-z=0 	ag{iii} \end{align*} $$ Solving eqns. (i), (ii) and (iii), we get $$ x=-\frac{1}{3}, y=\frac{1}{3}, z=-1, 	ext { As } d \propto S^{1 / 3} 	herefore n=3 $$