Planck 's constant (h) speed of length in vaccum (C) and newton 's gravitational constant (G) are three fundamental constant .Which of the following combinations of these has the dimension of length?

Options

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

Answer

C

Explanation

` l prop h^(x) G^(y) c ^(z)`
`M^(0)L^(1)T^(0)= (ML^(2)T^(-1))^(x) (M^(-1)L^(3)T^(-2))^(y)(LT^(-1))^(z)`
`" " = M^(x-y) L ^(2x+3y+z) T^(-x-2y-z)`
Equating :
`{:(x-y=0),(2x+3y+z=1),(-x-2y-z=0):}} rArr x = (1)/(2) , y = (1)/(2), z = - (3)/(2)`
`rArr l prop (sqrt(hG))/(c^(3//2))`