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 $\rho$ of the fog, intensity (power/area) $S$ of the light from the signal and its frequency $v$. The engineer finds that $d$ is proportional to $S^{1/n}$. The value of $n$ is

• A. $4$
• B. $2$
• C. $3$
• D. $1$

Answer 1

Answer: (C)

Let $d\propto {\rho }^x{S}^y{f}^z$ or $d=k{\rho }^x{S}^y{f}^z$
where $k$ is a dimensionless constant and $x$, $y$ and $z$ are the exponents.
Writing dimensions on both sides, we get
$\left[M^{0}L{T}^0\right]={\left[M{L}^{-3}{T}^0\right]}^x{\left[M{L}^0{T}^{-3}\right]}^y{\left[M^0{L}^0{T}^{-1}\right]}^z$
$\left[M^{0}L{T}^0\right]=\left[M^{x+y}{L}^{-3x}{T}^{-3y-z}\right]$
Applying the principle of homogeneity of dimensions, we get
$x+y=0\dots \left(i\right)$
$-3x=1\dots \left(ii\right)$
$-3y-z=0\dots \left(iii\right)$
Solving eqns. $\left(i\right)$, $\left(ii\right)$ and $\left(iii\right)$, we get
$x=-\frac{1}{3}$, $y=\frac{1}{3}$, $z=-1$, As $d\propto S^{1/3}$
$\therefore n=3$