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

The correct option is C $\sqrt{\frac{hG}{c^3}}$

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
$\left[M^{0}L{T}^0\right]=\left[L{T}^{-1}{]}^x\right[M{L}^2{T}^{-1}{]}^y[M^{-1}{L}^3{T}^{-2}{]}^z$
$=\left[M^{y-z}{L}^{x+2y+3z}{T}^{-x-y-2z}\right]$
Applying the principle of homogeneity of dimensions, we get
$y-z=0$ ....(i)
$x+2y+3z=1$ ....(ii)
$-x-y-2z=0$ ...(iii)
On solving Eqs. (i), (ii) and (iii), we get
$x=\frac{-3}{2},y=\frac{1}{2}z=\frac{1}{2}\therefore l=\sqrt{\frac{hG}{c^3}}$