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 Pˣ Sʸ cᶻ is dimensionless, are

To find the non-zero integers $x$, $y$, and $z$ such that the expression $P^x{S}^y{c}^z$ is dimensionless, we first determine the dimensional formulas for each of the physical quantities involved.

1. Radiation Pressure ($P$):
Pressure is defined as force per unit area.
$\left[P\right]=\frac{\left[\text{Force}\right]}{\left[\text{Area}\right]}=\frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]}=\left[M{L}^{-1}{T}^{-2}\right]$

2. Radiation Energy striking per unit area per second ($S$):
This quantity represents energy per unit area per unit time.
$\left[S\right]=\frac{\left[\text{Energy}\right]}{\left[\text{Area}\right]\times \left[\text{Time}\right]}=\frac{\left[M{L}^2{T}^{-2}\right]}{\left[L^2\right]\times \left[T\right]}=\left[M{T}^{-3}\right]$

3. Speed of Light ($c$):
Since $c$ is a speed, its dimensional formula is:
$\left[c\right]=\left[L{T}^{-1}\right]$

Now, we require the product $P^x{S}^y{c}^z$ to be dimensionless, which means:
$\left[P^x{S}^y{c}^z\right]=\left[M^0{L}^0{T}^0\right]$

Substitute the dimensions of $P$, $S$, and $c$ into the equation:
${\left[M{L}^{-1}{T}^{-2}\right]}^x{\left[M{T}^{-3}\right]}^y{\left[L{T}^{-1}\right]}^z=\left[M^0{L}^0{T}^0\right]$

Combining the exponents for $M$, $L$, and $T$ on the left-hand side:
$\left[M^{x+y}{L}^{-x+z}{T}^{-2x-3y-z}\right]=\left[M^0{L}^0{T}^0\right]$

By equating the powers on both sides, we obtain a system of three linear equations:
1) For $M$: $x+y=0\Rightarrow y=-x$
2) For $L$: $-x+z=0\Rightarrow z=x$
3) For $T$: $-2x-3y-z=0$

Let us check the third equation by substituting $y=-x$ and $z=x$:
$-2x-3(-x)-x=-2x+3x-x=0$
This confirms that the relations $y=-x$ and $z=x$ are consistent.

Setting $x=1$, we find:
$y=-1$
$z=1$

Thus, the non-zero integers that make the expression dimensionless are $x=1$, $y=-1$, and $z=1$.