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

The Correct Option is D

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}}] $