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.

Explanation

Density : `[p]=[MLT^(-3)]`
Intensity : `[S]=[(ML^(2)T^(-3))/(L^2)]=[MT^(-3)]`
Frequency : `[f]= [T^(-1)]`
Let us assume that
`d propto p^(a)S^(b)f^(c )`
`implies [d]=[p]^(a)[S]^(b)[f]^(c )`
`imlplies [L]=[ML^(-3)]^(a)[MT^(-3)]^(b)[T^(-1)]^(c )`
`implies [L]= [M^(a+b)L^(-3a)T^(-3b-c)]`
On comparing dimensions on both the sides we get the following :
`-3a= 1 implies a= -1"/"3`
`a+b=0 implies b = -a= 1"/"3`
We can see that `d propto S^(1/3)`
Hence value of n =3.