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)

$\frac{\sqrt{hG}}{c^{3/ 2}}$

Answer 2

$L \propto h^{x} c^{y} G^{z}$
$\left[M^{0} L^{1} T^{0}\right] = \left[ML^{2}T^{-1} \right]^{x} \left[LT^{-1} \right]^{y} \left[M^{-1}L^{3}T^{-2}\right]^{z}$
$= M^{x-z} L^{2x+y+3z}T^{-x-y-2z}$
\begin{array}{cc} x -z = 0 \, \Rightarrow \, x - z \\\ 2x + y + 3z = 1 \\\ \- x - y - 2z = 0 \end{array}\Bigg\\} \begin{array}{c} x = z = 1/2\\\ y = - 3 /2 \end{array}