If Planck’s constant (h) and speed of light in vacuum (c) are taken as two fundamental quantities, which of the following can, also, be taken to express length, mass, and time in terms of the three chosen fundamental quantities? a) mass of the electron (me) b) universal gravitational constant (G) c) charge of the electron (e) d) mass of proton (mp)
(h)=[T−1][ML2T−2]=[LT−1] for mass, let M=CahbGc [ML0T0]=[LT−1]a[ML2T−1]b[M−1L3T−2]c Using the principle, b−c=1⟶(1) a+2b+3c=0⟶(2) −a−b−2c=0⟶(3) So, a=2−1 also a=21 Hence, M=C1/2h1/2G−1/2⇒M=GCh Similarly, for length L=C3hG and T=C5hG ∴ Plank's length =C3Gh ; Plank's time =C5Gh and Plank's Mass =GCh
