Plank's constant (h), speed of light in vacuum (c) and Newton's gravitational constant (G) are taken as three fundamental constants.
Which of the following combinations of these has the dimensions of length?

Options

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

Answer

A

Explanation

`I prop h^(x)G^(y)c^(z)`
`l rArr L" "h=[ML^(2)T^(-1)]`
`G=[M^(-1)L^(3)T^(-2)]`
`c=LT^(-1)`
`therefore [L]=[ML^(2)T^(-1)]^(x)[M^(-1)L^(3)T^(-2)]^(y)[LT^(-1)]^(z)`
Comparing the combination :
`M^(0)L^(1)T^(0)=M^(x-y)L^(2x+3y+z)T^(-x-2y-z)`
`therefore x-y=0" "2x+3y+z=1`
`-x-2y-z=0`
Solving `x=(1)/(2), ly=(1)/(2), c=-(3)/(2)`
`l=(sqrt(hG))/(c^(3//2))`