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 electron `(m_0)`
B. Universal gravitational constant (G)
C. Charge of electron (e )
D. Mass of proton `(m_p)`

Answer

A, B, D

Explanation

Here, `[h]= [ML^(2)T^(-1)]`
Speed of light `[c]=[LT^(-1)]`
Mass of electron, `[m_e]= [M]`
Universal gravitational constant, `[G]=[M^(-1)L^(3)T^(-2)]`
Charge of electron, `[e]=[AT]`
Mass of proton, `[m_p]=[M]`
Now, `[(hc)/(G)]=([ML^(2)T^(-1)][LT^(-1)])/([M^(-1)L^(3)T^(-2)])=[M^2]`
`implies [M]=[(hc)/(G)]^(1/2)`
Also, `[(h)/(c )]=([ML^(2)T^(-1)])/([LT^(-1)])=[ML]`
`implies [L]=[(h)/(cm)]=[(h)/(c )][(G)/(hc)]^(1/2)= ([hG]^(1/2))/([c]^(3/2))`
As `[c]=[LT^(-1)]`
`implies [T]= [(L)/(c )]=([hG]^(1/2))/([(c )]^(5/2))`
Any of the options a, b, and d can be used to express L,M and T in terms of the given fundamental qantities.