Q89 / 180 · NEET 2016 (Phase 2) · Physics

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

Correct answer
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
$\sqrt{\frac{{c G}_{}}{h^{\frac{3}{2}}}}$

Solution

L \propto h^{a} c^{b} G^{c}

{[L]}^{1} = {[M^{1} L^{2} T^{- 1}]}^{a} {[L T^{- 1}]}^{b} {[M^{- 1} L^{3} T^{- 2}]}^{c}

Solving,

a = \frac{1}{2}, c = \frac{1}{2}, b = - \frac{3}{2}

\Rightarrow L = \frac{\sqrt{h G}}{c^{\frac{3}{2}}}

Hence, the correct answer is option: 'A'.

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