Question

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

• A. $2$
• B. $3$
• C. $1$
• D. $4$

Answer 1

The correct option is B

$3$

Step 1: Given Data

The distance $=d$

The mass density $=\rho$

The intensity $S=\left(\frac{power}{area}\right)$

The frequency $=f$

$d\propto S^{\frac{1}{n}}$

Step 2: Formula Used

Let $x,y$ and $z$ be the exponents.

$\therefore d\propto {\rho }^x{S}^y{f}^z$

Step 3: Establish the Equation

We know that the dimension of distance is

$\left[L\right]=\left[M^0{L}^1{T}^0\right]$

Let $k$ be a dimensionless constant and $x,y$ and $z$ be the exponents.

$$\begin{gathered} \therefore d\propto {\rho }^x{S}^y{f}^z \\ \Rightarrow d=k{\rho }^x{S}^y{f}^z \\ \Rightarrow \left[M^0{L}^1{T}^0\right]=k{\left[M^1{L}^{-3}{T}^0\right]}^x{\left[M^1{L}^0{T}^{-3}\right]}^y{\left[M^0{L}^0{T}^{-1}\right]}^z \\ \Rightarrow \left[M^0{L}^1{T}^0\right]=k\left[M^{x+y}{L}^{-3x}{T}^{-3y-z}\right] \end{gathered}$$

Step 4: Calculate $n$

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 equations $\left(i\right),\left(ii\right)$ and $\left(iii\right)$, we get

$x=-\frac{1}{3},y=\frac{1}{3},z=-1$

$$\begin{gathered} \therefore d\propto S^{\frac{1}{n}} \\ \Rightarrow d\propto S^y \\ \therefore y=\frac{1}{n} \\ \Rightarrow \frac{1}{n}=\frac{1}{3} \\ \Rightarrow n=3 \end{gathered}$$

Hence, the correct answer is option (B).