Plank 's constant (h) speed of length in vacium (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

Let dimension of length `t 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 dimensions on left and right hand side
`{:(x-y=0),(2x+3y+z=1),(-x-2y-z=0):}}{:(rArrx=1/2,y=1/2,z=-3/2),(rArrlprop(sqrt(hg))/(c^(3//2))):}`