Question

If the velocity of light c, gravitational constant G and Planck's constant $h,$ are chosen as fundamental units, the dimensional formula of length L in the new system is:

• A. $[{{h}^{1}}{{c}^{1}}{{G}^{-1}}]$
• B. $[{{h}^{1/2}}{{c}^{1/2}}{{G}^{-1/2}}]$
• C. $[{{h}^{1}}{{c}^{-3}}{{G}^{-1}}]$
• D. $[{{h}^{1/2}}{{c}^{-3/2}}{{G}^{1/2}}]$

Answer 1

Answer: (D)

$[{{h}^{1/2}}{{c}^{-3/2}}{{G}^{1/2}}]$

Answer 2

Key Idea: Every equation relating physical quantities should be in dimensional balance. In order to establish relation among various physical quantities, let a, b, c be the powers to which h, c and G are raised, then
$[L]=[h{{\,}^{a}}{{c}^{b}}{{G}^{c}}]$
Putting the dimensions on RHS of above equation, we get
$[L]=[M{{L}^{2}}{{T}^{-1}}]{{\,}^{a}}{{[L{{T}^{-1}}]}^{b}}{{[M{{L}^{-1}}{{L}^{3}}{{T}^{-2}}]}^{c}}$
$[L]=[{{M}^{a-c}}{{L}^{2a+b+3c}}{{T}^{-a-b-2c}}]$
Comparing the power, we get
$a-c=0$ ..(i) $2a+b+3c=1$ ..(ii)
$-a-b-2c=0$ ..(iii)
Solving Eqs. (i), (ii) and (iii), we get
$a=\frac{1}{2},b=\frac{-3}{2},c=\frac{1}{2}$
Hence, $[L]=[{{h}^{1/2}}{{c}^{-3/2}}{{G}^{1/2}}]$