Question

If velocity of light $c$, Planck’s constant $h$ and gravitational constant $G$ are taken as fundamental quantities then the dimensions of length will be

• A. $\sqrt{\frac{ch}{G}}$
• B. $\sqrt{\frac{hG}{c^5}}$
• C. $\sqrt{\frac{hG}{c^3}}$
• D. $\sqrt{\frac{h{c}^3}{G}}$

Answer 1

Answer: (C)

Let $l\propto c^x{h}^y{G}^z$ ;
$l=k{c}^x{h}^y{G}^z$
where $k$ is a dimensionless constant and $x$, $y$ and $z$ are the exponents.
Equating dimensions on both sides, we get
$\left[M^{0}L{T}^0\right]={\left[L{T}^{-1}\right]}^x{\left[M{L}^2{T}^{-1}\right]}^y{\left[M^{-1}{L}^3{T}^{-2}\right]}^z$
$=\left[M^{y-z}{L}^{x+2y+3z}{T}^{-x-y-2z}\right]$
Applying the principle of homogeneity of dimensions,we get
$y-z=0\dots \left(i\right)$
$x+2y+3z=1\dots \left(ii\right)$
$-x-y-2z=0\dots \left(iii\right)$
On solving Eqs. $\left(i\right)$, $\left(ii\right)$ and $\left(iii\right)$, we get
$x=-\frac{3}{2}$, $y=\frac{1}{2}$, $z=\frac{1}{2}$
$\therefore l=\sqrt{\frac{hG}{c^3}}$