Question

Planck’s constant $(h)$, speed of light in vacuum $(c)$ and Newton’s gravitational constant $(G)$ are three fundamental constants. Which of the following combinations of these has the dimension of length?

• A. $\frac{\sqrt{hG}}{c^{3/2}}$
• B. $\frac{\sqrt{hG}}{c^{5/2}}$
• C. $\frac{\sqrt{hc}}{G}$
• D. $\frac{\sqrt{Gc}}{h^{3/2}}$

Answer 1

Answer: (A)

Step 1: Write the SI units of given physical quantities
The SI unit of $h=$ J.s or N.m.s
SI unit of $c=m/s$
S.I unit of $G=\frac{N\cdot m^2}{K{g}^2}$
Step 2: Finding the units of all options
For A: $\sqrt{\frac{\left(N\cdot m^2/k{g}^2\right)(m/s)}{(N\cdot m\cdot s{)}^{3/2}}}=\frac{kg\cdot m^3}{N\cdot s^5}=\frac{kg\cdot m^3}{\left(kg\cdot m/s^2\right)\left(s^5\right)}=m^2/s^3$
For B: $\frac{\sqrt{(N\cdot m\cdot s)\left(N\cdot m^2/k{g}^2\right)}}{(m/s{)}^{3/2}}=\frac{Ns}{kg}=\frac{\left(kg\cdot m/s^2\right)\left(s^2\right)}{kg}=m$, it is the unit length.
For C: $\frac{\sqrt{(N\cdot m\cdot s)\left(N\cdot m^2/k{g}^2\right)}}{(m/s{)}^{5/2}}=\frac{Ns{s}^3}{m\cdot kg}=\frac{\left(kg\cdot m/s^2\right)\left(s^3\right)}{m\cdot kg}=s$
For D: $\sqrt{\frac{(N\cdot m\cdot s)(m/s)}{N\cdot m^2/k{g}^2}}=kg$