Question

If P represents radiation pressure, C represents the speed of light, and Q represents radiation energy striking a unit area per second, the non-zero integers x, y and z such that $P^x.Q^y.C^z$ is dimensionless, find the values of $x,y$ and z.

Answer 1

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

$Q=\frac{Energy/Area}{time}=\left(\frac{w}{m^2}\right)=M{T}^{-3}$

$C=$ Speed of Light $=(LT-1)$

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

Comparing powers

$M\to 0=x+y\Rightarrow x=-y...\left(1\right)$

$L\to 0=-x+z\Rightarrow x=z...\left(2\right)$

$T\to 0=-2x-3y-z...\left(3\right)$

Solving (1)(2) and (3)

$x=1$

$y=-1$

$z=1$

$\therefore \frac{P^1{c}^1}{Q}$