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

Options

A. `sqrt((hc^5)/G)`
B. `sqrt((Gh)/c^3)`
C. `sqrt((c^3)/(Gh))`
D. `sqrt((Gh)/c^5)`

Answer

D

Explanation

`[t] =[G]^(alpha)[h]^(b)[c][^(d)`
`[T^(1)]=[M^(-1)L^(3)T^(-2)] [M^(1) L^(2) T^(-1)] [L^(-1)T^(-1)]^(d)`
`[T^(-1)]= [M^(-a+b)L^(3a+2b+d)T^(-2a-b-d)]`
` -a+b=0`
` a=b `
`3a +2b+d=0 " "....(i)`
`-[2a+b+d] =1 " "....(ii)`
`2a+b+d=-1 " "...(iii)`
from equation(i) and (ii)
`5a +d=0`
`d=-5a`
from equation (i) and (iii)
` 3a+d=-1`
`3a - 5a =-1 implies -2a =-1 implies a =1/2`
`b =1/2 , d =-5/2`
Time `=[G^(1/2) h^(1/2) C^(-5/2)]`