Planck's constant h speed of light in vacuum c and Newton's gravitational constantG are three fundamental constants. Which of the following combinations of these has the dimension of length?

According to equation,

$\begin{array}{l} l \propto h^{p} c^{q} G^{r} \\ l = k h^{p} c^{q} G^{r} \end{array}$ ………(1)

Writting dimensions of physical quantities on both sides,

$[M^{0} {LT}^{0}] = {[{ML}^{2} T^{- 1}]}^{p} {[{LT}^{- 1}]}^{q} {[M^{- 1} L^{3} T^{- 2}]}^{r}$

Applying the principle of homogeneity of dimensions, we get

$p - r = 0$ ……..(2)

$2 p + q + 3 r = 1$ …….(3)

$- p - q - 2 r = 0$ ……….(4)

Solving eqns. (2), (3) and (4), we get

$p = r = \frac{1}{2}, q = - \frac{3}{2}$

From eq (1) $l = \frac{\sqrt{h G}}{c^{3 / 2}}$

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