If Planck's constant (h) and speed of light in vacuum (c) are taken as two fundamental quantities, which one of the following can in addition be taken to express length, mass and time in terms of the three chosen fundamental quantities?

Options

A. (A) Mass of electron `(m_(e))`
B. (B) Universal gravitational constant (G)
C. (C) Charge of electron (e)
D. (D) Mass of proton (m)

Explanation

We know that dimension of `h=[h]=[ML^(2)T^(-1)]`
`[c]=[L^(1)T^(-1)], [m_(e)]=M^(1)`
`[G]=[M^(-1)L^(3)T^(-2)]`
`[e]=[AT],[m_(p)]=[M^(1)]`
`[(hc)/(G)]=([ML^(2)-T^(-1)][LT^(-1)])/([M^(1)L^(3)T^(-2)])=[M^(2)]`
Mass, `M= sqrt((hc)/(G))`
Length, `L=(h)/(CM)=(h)/(C) sqrt((G)/(hC))= (sqrt(gh))/(C(3)/(2))`
Now `C=LT^(-1)`
`[T]=([L])/([C])=(sqrt(Gh))/(C^((3)/(2)*C))= (sqrt(Gh))/(C^((5)/(2)))`