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? [2016]

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Solution

Q.
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 [2016]

1
$\frac{\sqrt{h G}}{c^{3 / 2}}$

2
$\frac{\sqrt{h G}}{c^{5 / 2}}$

3
$\sqrt{\frac{h c}{G}}$

4
$\sqrt{\frac{G c}{h^{3 / 2}}}$

Ans.
(1)

According to question, $l \propto h^{p} c^{q} G^{r}$

$l = k$ $h^{p} c^{q} G^{r}$ ...(i)

Writing 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

$p - r = 0$ ...(ii), $2 p + q + 3 r = 1$ ...(iii), $- p - q - 2 r = 0$ ...(iv)

Solving eqns. (ii), (iii) and (iv), we get $p = r = \frac{1}{2}, q = - \frac{3}{2}$

From eqn. (i), we get $l = K \frac{\sqrt{h G}}{c^{3 / 2}}$

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