An important milestone in the evolution of the universe just after the big bang is the Planck time `t^(p),` the value of which deponds on three fundamental constants-speed `c` of light in vacuum, gravitational constant `G` and Planck's constant `h`. Then, `t_(p)prop`

Options

A. `Ghc^(5)`
B. `(c^(5))/(Gh)`
C. `(Gh)/(c^(5))`
D. `((Gh)/(c^(5)))^(1//2)`

Answer

D

Explanation

`t_(p)alphac^(x)G^(y)h^(z)`
`M^(0)L^(0)T^(1)prop[L^(1)T^(-1)]^(x)[M^(-1)L^(3)T^(-2)]^(y)xx[ML^(2)T^(-1)]^(z)propM^(-y+z)xxL^(x+3y+2z)xxT^(-x-2y-z)`
`:.-y+z=0|x+3y+2z=0|` and `-x-2y-z=1`
or `{:(y=3),("or"),(z=(1)/(2)):}|{:(x+5y=0),(x=-5y),(x=-(5)/(2)):}|{:(-x-3y=1),(5y-3y=1),(y=(1)/(2)):}`
`:.t_(p)alphaG^((1)/(2))h^((1)/(2))c((-5)/(2))=[(Gh)/(c^(5))]^((1)/(2))`
Hence `T^(2)=4pi^(2).(L)/(g)` or `g=(4pi^(2)L)/(T^(2))`.