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 ($m_e$)
ii) Universal gravitational constant (G)
iii) Charge of electron (e)
iv) Mass of proton ($m_p$)

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

Answer 1

The correct option is C (i) , (ii) and (iv)

$\left(h\right)=\frac{\left[{ML}^2{T}^{-2}\right]}{\left[T^{-1}\right]}=\left[{LT}^{-1}\right]$

for mass, let $M=C^a{h}^b{G}^c$

$\left[{ML}^0{T}^0\right]={\left[{LT}^{-1}\right]}^a{\left[{ML}^2{T}^{-1}\right]}^b{\left[M^{-1}{L}^3{T}^{-2}\right]}^c$

Using the principle,

$b-c=1⟶\left(1\right)$

$a+2b+3c=0⟶\left(2\right)$

$-a-b-2c=0⟶\left(3\right)$

So, $a=\frac{-1}{2}$ also $a=\frac{1}{2}$

Hence, $M=C^{1/2}{h}^{1/2}{G}^{-1/2}\Rightarrow M=\sqrt{\frac{Ch}{G}}$

Similarly, for length $L=\sqrt{\frac{hG}{C^3}}$ and $T=\sqrt{\frac{hG}{C^5}}$

$\therefore$ Plank's length $=\sqrt{\frac{Gh}{C^3}}$ ; Plank's time $=\sqrt{\frac{Gh}{C^5}}$ and Plank's Mass $=\sqrt{\frac{Ch}{G}}$

Hence, $\left(a\right)$, $\left(b\right)$, $\left(d\right)$ are correct options.