Question

Expression for time in terms of $G$ (universal gravitational constant ), $h$ (Planck's constant) and $c$ (speed of light ) is proportional to

• A. $\sqrt{\frac{Gh}{c^3}}$
• B. $\sqrt{\frac{h{c}^5}{G}}$
• C. $\sqrt{\frac{c^3}{Gh}}$
• D. $\sqrt{\frac{Gh}{c^5}}$

Answer 1

The correct option is D $\sqrt{\frac{Gh}{c^5}}$

Let $t\propto G^x{h}^y{c}^z$
Dimensions of $G=\left[M^{-1}{L}^3{T}^{-2}\right]$
$h=\left[M{L}^2{T}^{-1}\right]$ and
$c=\left[L{T}^{-1}\right]$
$\left[T\right]=\left[M^{-1}{L}^3{T}^{-2}{]}^x\right[ML{}^2{T}^{-1}{]}^y[L{T}^{-1}{]}^z$
$\left[M^0{L}^0{T}^1\right]=\left[M^{-x+y}{L}^{3x+2y+z}{T}^{-2x-y-z}\right]$
By comparing the powers of $M,L,T$ both the sides
$-x+y=0\Rightarrow x=y$
$3x+2y=z=0\Rightarrow 5x+z=0...\left(i\right)$
$-2x-y-z=1\Rightarrow 3x+z=-1....\left(ii\right)$
Solving eqns. (i) and (ii),

$x=y=\frac{1}{2},z=-\frac{5}{2}\therefore t\propto \sqrt{\frac{Gh}{c^5}}$