If $P$ represents radiation pressure, $\boldsymbol{c}$ represents speed of light and $Q$ represents…

If $P$ represents radiation pressure, $\boldsymbol{c}$ represents speed of light and $Q$ represents radiation striking unit area per second, then non-zero integers $x, y$, and $z$ such that $P^{x} Q^{y} c^{z}$ is dimensionless are
  1. $x=1, y=1, z=-1$
  2. $x=1, y=-1, z=1$
  3. $x=-1, y=1, z=1$
  4. $\quad x=1, y=1, z=1$

Solution

Dimensions of
$P^{x} Q^{y} c^{z}=\left[M L^{-1} T^{-2}\right]^{x}\left[M T^{-3}\right]^{\gamma}\left[L T^{-1}\right]^{z}$
As it is dimensionless, so
$\left[M L^{-1} T^{-2}\right]^{x}\left[M T^{-3}\right]^{y}\left[L T^{-1}\right]^{z}=\left[M^{0} L^{0} T^{0}\right]$ or $\left[M^{x+y} L^{-x+2} T^{-2 x-3 y-z}\right]=\left[M^{0} L^{0} T^{0}\right]$
Comparing powers of $M, L$ and $T$, we get
$x+y=0,-x+z=0,-2 x-3 y-z=0$
Solving, $x=1, y=-1, z=1$. ,

Asked in: JEE Mains - Units and Dimensions - Test 2

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