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

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$\sqrt{\frac{h c^{5}}{G}}$

$\sqrt{\frac{G h}{c^{3}}}$

$\sqrt{\frac{c^{3}}{G h}}$

$\sqrt{\frac{G h}{c^{5}}}$

answer is D.

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Detailed Solution

$\begin{array}{l} F = \frac{{GM}^{2}}{R^{2}} \Rightarrow G = [M^{- 1} L^{3} T^{- 2}] \\ E = hv \Rightarrow h = [{ML}^{2} T^{- 1}] \\ C = [{LT}^{- 1}] \\ \begin{array}{l} t \propto G^{x} h^{y} C^{z} \\ [T] = {[M^{- 1} L^{3} T^{- 2}]}^{x} {[M L^{2} T^{- 1}]}^{y} {[L^{- 1}]}^{z} \\ [M^{0} L^{0} T^{1}] = [M^{- x + y} L^{3 x + 2 y + z} T^{- 2 x - y - z}] \\ o n c o m p a r i n g t h e p o w e r s o f M, L, T \end{array} \\ - x + y = 0 \Rightarrow x = y \\ 3 x + 2 y + z = 0 \Rightarrow 5 x + z = 0 \\ x = y = \frac{1}{2}, z = - \frac{5}{2} \\ t \propto \sqrt{\frac{Gh}{C^{5}}} \end{array}$