Question

From the following combinations of physical constants (expressed through their usual symbols) the only combination, that would have the same value in different system of units, is:

• A. $\frac{ch}{2\pi {ϵ}_0^2}$
• B. $\frac{e^2}{2\pi {ϵ}_{0}G{m}_e^2}(m_e=$ mass of electron)
• C. $\frac{{\mu }_0{ϵ}_0}{c^2}\frac{G}{h{e}^2}$
• D. $\frac{2\pi \sqrt{{\mu }_0{ϵ}_0}}{c{e}^2}\frac{h}{G}$

Answer 1

Answer: (B)

$\frac{e^2}{2\pi {ϵ}_{0}G{m}_e^2}(m_e=$ mass of electron)

The combination of physical constants that will have the same value in different systems of units should be a dimensionless quantity, since dimensions are something that change the magnitude of a particular quantity.
Form Newton's law of gravitation and Coulomb's law of forces on charges,
F = $\frac{GMm}{r^2}$ = $\frac{kQq}{r^2}$
Now, Observing the above expressions suggests that,
the dimensions of $\frac{e^2}{2\pi {ϵ}_o}$ are that of $N{m}^2$.
Also, the dimensions of $G{m}^2$ are that of $N{m}^2$.
Hence, we conclude that a division of these quantities will be a dimensionless quantity and its value will not change depending on the system of units chosen.