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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 $?$
$\sqrt {\frac{{hc}}{G}} $
$\;\sqrt {\frac{{Gc}}{{{h^{\frac{3}{2}}}}}} $
$\frac{{\sqrt {hG} }}{{{c^{\frac{3}{2}}}}}$
$\;\frac{{\sqrt {hG} }}{{{c^{\frac{5}{2}}}}}$
Solution
According to questions
$l \propto \,{h^p}{c^q}{G^r}$
$l = k\,\,{h^p}$ …………..($i$)
Writting dimensions of physical quantities on both sides
$\left[ {{M^0}L{T^0}} \right] = {\left[ {M{L^2}{T^{ – 1}}} \right]^p}{\left[ {L{T^{ – 1}}} \right]^q}{\left[ {{M^{ – 1}}{L^3}{T^{ – 2}}} \right]^r}$
Applying the principle of homogeneity of dimensions we get
$P – r = 0$ ………($ii$)
${2_p} + q + 3r = 1$ …………($iii$)
$ – P – q – 2r = 0$ …………….($iv$)
Solving eqns. ($ii$), ($iii$), and ($iv$), we get
$P = r = \frac{1}{2},q = – \frac{3}{2}$
From eqn.$\left( i \right)\,l = \frac{{\sqrt {hG} }}{{_c3/2}}$
