Question

$P$ represents radiation pressure, $c$ represents speed of light and $S$ represents radiation energy striking per unit area per sec. The non zero integers $x$, $y$, $z$ such that $P^x S^y c^z$ is dimensionless are

• A. $x = 1$ , $y = 1$, $z = 1$
• B. $x= - 1$ , $y = 1$ , $z= 1$
• C. $x = 1$, $y = -1$ , $z = 1$
• D. $x = 1$, $y = 1$, $z = -1$

Answer 1

Answer: (C)

Let $k=P^x S^y c^z .....(i)$
$k$ is a dimensionless
Dimensions of $k=[M^0 L^0 T^0]$
$\therefore $ Dimensions of $P$
$=\frac{\text{Force}}{\text{Area}}=\frac{[MLT^{-2}]}{[L^2]}=[ML^{-1}T^{-2}]$
Dimensions of $S=$
$\frac{\text{Energy}}{\text{Area $\times$ time}}=\frac{[ML^2T^{-2}]}{[L^2][T]}=[MT^{-3}]$
Dimensions of $c=[LT^{-1}]$
Substituting these dimensions in eqn $(i)$, we get
$[M^0 L^0 T^0]=[ML^{-1}T^{-2}]^x[MT^{-3}]^y[LT^{-1}]^z.$
Applying the principle of homogeneity of dimensions, we get
$ x+y=0 ....(ii)$
$ -x+z=0 ....(iii)$
$ -2x-3y-z=0 ....(iv)$
Solving $(ii)$, $(iii)$ and $(iv)$, we get
$x=1, y=-1, z=1$