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

Question

Expression for time in terms of GG (universal gravitational constant), hh (Planck constant) and cc (speed of light) is proportional to :

A)$\sqrt{{\displaystyle \frac{Gh}{c^3}}}$ 

B)$\sqrt{{\displaystyle \frac{ch}{G^3}}}$ 

C) $\sqrt{{\displaystyle \frac{Gh}{c^5}}}$ 

D)$\sqrt{{\displaystyle \frac{G^{5}h}{c}}}$

Answer 1

The correct option is C) $\sqrt{{\displaystyle \frac{Gh}{c^5}}}$

Explaination::

$F={\displaystyle \frac{G{M}^2}{R^2}}\Rightarrow G=[M^{-1}{L}^3{T}^{-2}]$ 

$E=hv\Rightarrow h=[M{L}^2{T}^{-1}]$ 

$C=[L{T}^{-1}]$ 

$t\propto G^x{h}^y{C}^z$ 

$[M^0{L}^0{T}^1]=[M^{-x+y}{L}^{3x+2y+z}{T}^{-2x-y-z}]$ 

on comparing the powers of M, L, T

$-x+y=0\Rightarrow x=y$ 

$3x+2y+z=0\Rightarrow 5x+z=0$ 

$-2x-y-z=1\Rightarrow 3x+z=-1$ 

on solving (i) and (ii) $x=y={\displaystyle \frac{1}{2}},z=-{\displaystyle \frac{5}{2}}$ 

$t\propto \sqrt{{\displaystyle \frac{Gh}{C^5}}}$