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 x, y and z, such that $P^x{Q}^y{c}^z$ is dimensionless, may be

• A. x = 1, y =1, z = 1.
• B. x = 1, y =-1, z = 1.
• C. x = -1, y =1, z = 1.
• D. x = 0, y = 0, z = 0

Answer 1

The correct option is D x = 0, y = 0, z = 0

Radiation pressure, $P=M{L}^{-1}{T}^{-2}$

Radiation energy per unit area per second,

$Q=M{L}^0{T}^{-3}$

Speed of light, $C=M^{0}L{T}^{-3}$

${\left[P\right]}^x{\left[Q\right]}^y{\left[C\right]}^z=\left[M^0{L}^0{T}^0\right]{\left[M{L}^{-1}{T}^{-2}\right]}^x{\left[M{T}^{-3}\right]}^y{\left[L{T}^{-1}\right]}^2=\left[M^0{L}^0{T}^0\right]M^{x+y}{L}^{-x+z}{T}^{-2x-3y-z}=M^0{L}^0{T}^0\therefore x+y=0\to \left(1\right)-x+z=0\to \left(2\right)-2x-3y-z=0\Rightarrow 2x+3y+z=0\Rightarrow 2(x+y)+y+z=0\Rightarrow y+z=0\to \left(3\right)$

The only possible solution is $x=y=z=0$.