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, are

To determine the integers $x, y,$ and $z$, we first establish the dimensional formulae for each quantity as defined in the sources:

1. Radiation Pressure ($P$): Pressure is defined as force per unit area . Force has dimensions $[MLT^{-2}]$ and area is $[L^2]$, so pressure is $[MLT^{-2}] / [L^2] = [ML^{-1}T^{-2}]$ .
2. Speed of light ($c$): Speed is distance divided by time . Its dimensions are $[LT^{-1}]$ .
3. Radiation energy per unit area per second ($Q$): This quantity represents intensity. Its dimensions are $\text{Energy} / (\text{Area} \times \text{Time})$. Since energy is $[ML^2T^{-2}]$ , $Q$ has dimensions $[ML^2T^{-2}] / ([L^2][T]) = [MT^{-3}]$ .

The condition that $P^x Q^y c^z$ is dimensionless ($M^0 L^0 T^0$) leads to the equation: $[ML^{-1}T^{-2}]^x [MT^{-3}]^y [LT^{-1}]^z = M^0 L^0 T^0$ $[M^{x+y} L^{-x+z} T^{-2x-3y-z}] = M^0 L^0 T^0$

Equating the powers for each base unit:

• For $M: x + y = 0 \Rightarrow y = -x$
• For $L: -x + z = 0 \Rightarrow z = x$
• For $T: -2x - 3y - z = 0$

Substituting $y = -x$ and $z = x$ into the time equation: $-2x - 3(-x) - x = -2x + 3x - x = 0$, which is consistent. By testing the options, Option B ($x=1, y=-1, z=1$) is the only one that satisfies these relations ($y = -1 = -x$ and $z = 1 = x$).