Dimensional Analysis of Visibility Distance in Fog

To determine the dependence of the visibility distance $d$ on the fog mass density $\rho$, the light intensity $S$, and the frequency $f$, we use dimensional analysis.

The dimensions of the given physical quantities are:

• Distance, $d$: $[d] = \text{M}^0 \text{L}^1 \text{T}^0$
• Mass density of fog, $\rho$: $[\rho] = \frac{\text{Mass}}{\text{Volume}} = \text{M}^1 \text{L}^{-3} \text{T}^0$
• Intensity of light, $S$ (power per unit area): $[S] = \frac{\text{Power}}{\text{Area}} = \frac{[\text{Energy}]/[\text{Time}]}{\text{L}^2} = \frac{\text{M}^1 \text{L}^2 \text{T}^{-3}}{\text{L}^2} = \text{M}^1 \text{L}^0 \text{T}^{-3}$
• Frequency, $f$: $[f] = \frac{1}{[\text{Time}]} = \text{M}^0 \text{L}^0 \text{T}^{-1}$

Assume the relation: $d \propto \rho^a S^b f^c$

Equating the dimensions on both sides: $[d] = [\rho]^a [S]^b [f]^c$ $\text{M}^0 \text{L}^1 \text{T}^0 = (\text{M}^1 \text{L}^{-3} \text{T}^0)^a (\text{M}^1 \text{L}^0 \text{T}^{-3})^b (\text{M}^0 \text{L}^0 \text{T}^{-1})^c$ $\text{M}^0 \text{L}^1 \text{T}^0 = \text{M}^{a+b} \text{L}^{-3a} \text{T}^{-3b-c}$

Comparing the exponents of $\text{M}$, $\text{L}$, and $\text{T}$ on both sides:

1. For $\text{L}$: $-3a = 1 \implies a = -\frac{1}{3}$
2. For $\text{M}$: $a + b = 0 \implies b = -a = \frac{1}{3}$
3. For $\text{T}$: $-3b - c = 0 \implies c = -3b = -1$

Hence, the relation between the variables is: $d \propto \rho^{-1/3} S^{1/3} f^{-1}$

This gives: $d \propto S^{1/3}$

Comparing this with the given proportional relation $d \propto S^{1/n}$, we obtain: $\frac{1}{n} = \frac{1}{3} \implies n = 3$