Question

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

• A. $\frac{\sqrt{h G}}{c^{\frac{3}{2}}}$
• B. $\frac{\sqrt{h G}}{c^{\frac{5}{2}}}$
• C. $\sqrt{\frac{h c}{G}}$
• D. $\sqrt{\frac{c G _{ ⁡}}{h ⁡^{\frac{3}{2}}}}$

Answer 1

Answer: (A)

$L \, \propto h^{a}c^{b}G^{c}$
$\left[L\right]^{1}=\left[M^{1} L^{2} T^{- 1}\right]^{a}\left[L T^{- 1}\right]^{b}\left[M^{- 1} L^{3} T^{- 2}\right]^{c}$
Solving,
$a=\frac{1}{2}, \, c=\frac{1}{2}, \, b=-\frac{3}{2}$
$\Rightarrow L=\frac{\sqrt{h G}}{c^{\frac{3}{2}}}$