Question 306: Units , Dimensions and Error Analysis

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 f. The engineer finds that d is proportional to
S
1/n
. The value of n is

Accepted Answer

Given $$d \propto {\rho ^a}{S^b}{b^c}$$
$${M^0}L{T^0} \propto {(M{L^{ - 3}})^a}{(M{T^{ - 3}})^b}{({T^{ - 1}})^c}$$
$${M^0}L{T^0} \propto {M^{(a + b)}}{L^{ - 3a}}{T^{ - 3b - c}}$$
Equating the coefficients, we get
$$a + b = 0 - 3a = 1 - 3b - c = 0$$
$$b =  - a$$ $$a =  - {1 \over 3} - c = 3b$$
$$b = {1 \over 3}c =  - 3b \Rightarrow c = 1$$
Therefore, $$b = {1 \over n} = {1 \over 3} \Rightarrow n = 3$$.

Units , Dimensions and Error Analysis

Integer / Numeric Answer Type