If velocity of light c, planck's constant h and gravitational constnat G are taken as fundamental quantities then the dimensions of the length will be

Options

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

Answer

C

Explanation

Let `lpropc^(x)h^(y)G^(z),l=kc^(x)h^(y)G^(z)`
where k is a dimensionless constant and x,y and z are the exponents.
Equating dimenions on both sides, we get
`[M^(0)LT^(0)]=[LT^(-1)]^(x)[ML^(2)T^(-1)]^(y)[M^(-1)L^(3)T^(-2)]^(z)`
`=[M^(y-z)L^(x+2y+3z)T^(-x-y-2z)]` ltBrgt Applying the principle of homogeneity of dimensions, ltBrgt we get
`y-z=0` . . . .(i)
`x+2y+3z=1` . . .(ii)
`-x-y-2z=0` . . . . (iii)
On solving Eqs. (i), (ii) and (iii), we get
`x=-(3)/(2),y=(1)/(2),z=(1)/(2)thereforel=sqrt((hG)/(c^(3)))`