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

a) mass of electron (m_e)

b) universal gravitational constant (G)

c) charge of electron (e)

d) mass of proton (m_p)

Answers (1)

The answer is the option (a) Mass of electron (m_e) and (b) Universal gravitational constant (G) and (d) Mass of proton (mp)

Explanation: Dimensions of –

h (Planck’s constant)

$= [ML^{2}T^{-1}]$

c (Speed of light in vacuum)

$= \frac{s}{t}$

$= [LT^{-1}]$

Thus, dimension of hc

$= [ML^{2}T^{-1}][LT^{-1}]$

$= [ML^{3}T^{-2}]$

$G = \frac{Fr^{2}}{M_{1}M_{2}}$

$= \frac{[ML^{3}T^{-2}]}{[M][M]}$

$= [M^{-1}L^{3}T^{-2}]$

Now,

$\frac{hc}{G} = [ML^{3}T^{-2}]/[M^{-1}L^{3}T^{-2}]$

$=[M^{2}]$

Thus,

$M= \sqrt{\frac{hc}{G}}$

$= [h^{\frac{1}{2}} c^{\frac{1}{2}}G^{\frac{1}{2}}]$

Now,

$\frac{h}{c} = [ML^{2}T^{-1}]/[LT^{-1}]$

$= [ML]$

$=\sqrt{\frac{hc}{G}}\times L$

Now,

$L=\frac{h}{c}\times \sqrt{\frac{G}{hc}}$

$=\frac{\sqrt{Gh}}{\frac{c^{3}}{2}}$

$= [G^{\frac{1}{2}} h^{\frac{1}{2}} c^{\frac{-3}{2}}]$

Also, $c = [LT^{-1}]$

$= [G^{\frac{1}{2}} h^{\frac{1}{2}} c^{\frac{-3}{2}}T^{-1}]$

& $T= [G^{\frac{1}{2}} h^{\frac{1}{2}} c^{\frac{-3}{2}-1}]$

$T= [G^{\frac{1}{2}} h^{\frac{1}{2}} c^{\frac{-5}{2}}]$

Therefore, in terms of chosen fundamental quantities, the physical quantities a, b & d can be used to represent L, M & T.

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