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