Question

If $p$ represents radiation pressure, $C$ represents speed of light and $q$ represents radiation energy striking a unit area per second, then non-zero integers a, b and c are such that $p^a{q}^b{C}^c$ is dimensionless, then

• A. $a=1,b=1,c=-1$
• B. $a=1,b=-1,c=1$
• C. $a=0,b=0,c=0$
• D. $a=1,b=1,c=1$

Answer 1

The correct option is

B

$a=1,b=-1,c=1$

Dimension of $\left(P\right)\to \left[M^1{L}^{-1}{T}^{-2}\right]$

Dimension of $\left(C\right)\to \left[M^0{L}^1{T}^{-1}\right]$

Dimension of $\left(Q\right)\to \left[M^1{L}^0{T}^{-3}\right]$

According to problem,

$\left[M^0{L}^0{T}^0\right]=P^x{Q}^y{C}^z$

$={\left[M^1{L}^{-1}{T}^{-2}\right]}^x{\left[M^1{L}^0{T}^{-3}\right]}^y{\left[M^0{L}^1{T}^{-1}\right]}^z$

$\left[M^0{L}^0{T}^0\right]=\left[M^{x+y}{L}^{-x+z}{T}^{-2x-3y-z}\right]$

$x=y=0\to 1$

$-x+z=0\to 2$

$2x-3y-z=0\to 3$

$x=1,y=-1,z=1$