To find the distance over which a signal can be seen clearly in foggy conditions, a railways-engineer uses dimensions and assumes that the distance depends on the mass density of the fog, intensity (power/area) of the light from the signal and its freq

Question

To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railways-engineer uses dimensions and assumes that the distance depends on the mass density $ρ$ 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 ______. [2014]

Smart Achievers · Accepted Answer

(3)

$Let d \propto ρ^{x} s^{y} f^{z} = K ρ^{x} s^{y} f^{z} where K is a dimensionless constant.$

$M^{0} L T^{0} = M^{x} L^{- 3 x} M^{y} T^{- 3 y} T^{- z}$

Equating dimensions both sides

$M^{0} L T^{0} = M^{x + y} L^{- 3 x} T^{- 3 y - z}$

$∴$ $x + y = 0, - 3 x = 1$

$∴$ $x = - \frac{1}{3}$ and $y = \frac{1}{3}$

$∴$ $n = 3$

Solution

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Solution

Ans. (3) Let d∝ρxsyfz=Kρxsyfz where K is a dimensionless constant. M0LT0=MxL-3xMyT-3yT-z Equating dimensions both sides M0LT0=Mx+yL-3xT-3y-z ∴ x+y=0, -3x=1 ∴ x=-13 and y=13 ∴ n=3

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