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. $\sqrt{\frac{hc^{3}}{G}}$

Answer 1

Answer: (C)

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