Question

If Planck’s constant $\left(h\right)$ and speed of light in vacuum $\left(c\right)$ 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?

• A. Universal gravitational constant $\left(G\right)$
• B. Mass of electron $\left(m_e\right)$
• C. Charge of electron $\left(e\right)$
• D. Mass of proton $\left(m_p\right)$

Answer 1

Answer: (A), (B), (D)

Mass of proton $\left(m_p\right)$

Energy of a photon $E=hv$

$\left[h\right]=\frac{\left[M{L}^2{T}^{-2}\right]}{\left[T^{-1}\right]}=\left[M{L}^2{T}^{-1}\right]$

$\left[c\right]=\left[L{T}^{-1}\right]$

$\left[G\right]=\left[M^{-1}{L}^3{T}^{-2}\right]$

For mass, let $m=c^a{h}^b{G}^c$
where a, b and c are exponents of c,h and G respectively.

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

Using the principle of homogeneity of dimensions,

$b-c=1$ ...(i)
$a+2b+3c=0$ ...(ii)
$-a-b-2c=0$ ...(iii)
Solving,
We get a,b,c as $\frac{1}{2},\frac{1}{2},-\frac{1}{2},$ respectively
Hence, $m=c^{\frac{1}{2}}{h}^{\frac{1}{2}}{G}^{-\frac{1}{2}}$

$m=\sqrt{\frac{ch}{G}}$

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

for time, $T=\sqrt{\frac{hG}{c^5}}$

Mass can be expressed by $m_e$ and $m_p$

Final answer: (a),(b),(d)