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

Answer 1

The correct option is B $\frac{\sqrt{hG}}{c^{3/2}}$

The SI unit of h is $J.s$ or $N.m.s$ and SI unit of c is $m/s$ and Si unit of G is $N.m^2/k{g}^2$

For A: $\sqrt{\frac{(N.m^2/k{g}^2)\left(m/s\right)}{(N.m.s{)}^{3/2}}}=\frac{kg.m^3}{N.s^5}$$=\frac{kg.m^3}{(kg.m/s^2)\left(s^5\right)}=m^2/s^3$

For B: $\frac{\sqrt{(N.m.s)(N.m^2/k{g}^2)}}{(m/s{)}^{3/2}}$$=\frac{N{s}^2}{kg}=\frac{(kg.m/s^2)\left(s^2\right)}{kg}=m$
It is the unit length.

For C: $\frac{\sqrt{(N.m.s)(N.m^2/k{g}^2)}}{(m/s{)}^{5/2}}=\frac{N{s}^3}{m.kg}$$=\frac{(kg.m/s^2)\left(s^3\right)}{m.kg}=s$

For D: $\sqrt{\frac{(N.m.s)\left(m/s\right)}{N.m^2/k{g}^2}}=kg$