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Question

# 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 ?i) Mass of electron (me)ii) Universal gravitational constant (G)iii) Charge of electron (e)iv) Mass of proton (mp)

A
(i), (ii) and (iii)
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B
(i) and (iii)
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C
(i) , (ii) and (iv)
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D
(i) only
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Solution

## The correct option is C (i) , (ii) and (iv)(h)=[ML2T−2][T−1]=[LT−1]for mass, let M=CahbGc[ML0T0]=[LT−1]a[ML2T−1]b[M−1L3T−2]cUsing the principle,b−c=1⟶(1)a+2b+3c=0⟶(2)−a−b−2c=0⟶(3)So, a=−12 also a=12Hence, M=C1/2h1/2G−1/2⇒M=√ChGSimilarly, for length L=√hGC3 and T=√hGC5∴ Plank's length =√GhC3 ; Plank's time =√GhC5 and Plank's Mass =√ChGHence, (a), (b), (d) are correct options.

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