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

Options

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

Answer

A, B, D

Explanation

We know that dimension of `h=[h]=[ML^(2)T^(-1)]`
`[c]=[LT^(-1)],[m_(e)]=M`
`[G]=[M^(-1)L^(3)T^(-2)]`
`[e]=[AT],[m_(p)]=[M]`
`[(hc)/G]=([ML^(2)T^(-1)][LT^(_1)])/([M^(-1)L^(3)T^(-2)])=[M^(2)]`
`M=sqrt((hc)/(G))`
`"Similarly,"h/c=([ML^(2)T^(-1)])/([LT^(-1)])=[ML]`
`L=h/(cM)=h/csqrt(G/(hc))=(sqrt(Gh))/(c^(3//2))`
`As,c=LT^(-1)`
`Rightarrow[T]=([L])/([c])=(sqrt(Gh))/(c^(3//2)*c)=(sqrt(Gh))/(c^(5//2))`
Hence, (a), (b) or (d) any be used to express L,M and T in terms of three chosen fundamental quantities.