If Plank constant (h) speed of free space light (c) and Newton gravitation constant (G) are chosen as the fundamental quantities, then dimensional formula of length will be ......

Options

A. `sqrt((hc)/(G))`
B. `sqrt((G_(c))/(h^((3)/(2))))`
C. `(sqrt(hG))/(C^((3)/(2))`
D. `(sqrt(hG))/(c^((5)/(2)))`

Explanation

Length `lprop h^(a)G^(b) c^(d` where `a,b,d in R`
`:. l=kh^(a)G^(b)c^(d)`
`M^(0)L^(1)T^(0)=(M^(2)L^(2)T^(-1))^(a)(M^(-1)L^(3)T^(-2))^(b) (M^(0)L^(1)T^(-1))^(d)`
`=M^(a-b)L^(2a+3b+d) T^(-a-2b-d)`
`a-b=0 " " ...(1)`
`2a+3b+d=1 " " ...(2)`
`-a-2b-d=0 " "...(3)`
`:.` From equation (2) and (3)
`5b+d=1 " " ...(4)`
and
`-3b-d=0 " " ...(5)`
Add by equation (4) and (5)
`2b=1`
`:.b=(1)/(2) " " :. a=(1)/(2)`
From equation (3)
`:.-d=a+2b=(1)/(2)+1=(3)/(2)`
`:.d=-(3)/(2)`
`l=h^((1)/(2))G^((1)/(2))c^(-(3)/(2))`
`:.l=((hG)^((1)/(2)))/(c^((3)/(2)))`
`= (sqrt(hG))/(c^((3)/(2)))`