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

Answer

A

Explanation

According to question
`l prop h^(p)c^(q) G^(r ) , l =kh^(p)c^(q)G^(r)` …(i)
Writing dimensions of physical quantities on both sides,
`[M^(0)LT^(0)]= [ML^(2)T^(-1)]^(p)[LT^(-1)]^(q)[M^(-1)L^(3)T^(-2)]^(r)`
Applying the principle of homogeneity of dimensions we get
p-r= 0 ...(ii)
2p+q+3r=1 ..(iii)
-p-q-2r= 0 ....(iv)
Solving eqns. (ii), (iii) and (iv) we get
`p= r= (1)/(2), q= (3)/(2)`
From eqn. (i) `l= (sqrt(hG))/(c^(3//2))`