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

Answer 1

Answer: (D)

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 $[L]=\left[h^{a} c^{b} G^{c}\right]$
Putting the dimensions on RHS of above equation, we get
${[L]=\left[M L^{2} T^{-1}\right]^{a}\left[L T^{-1}\right]^{b}\left[M L^{-1} L^{3} T^{-2}\right]^{c}}$
$[L]=\left[M^{a-c} L^{2 a+b+3 c} T^{-a-b-2 c}\right]$
Comparing the power, we get
$a-c=0 ...$(i)
$2 a+b+3 c=1...$ (ii)
Solving Eqs. (i), (ii) and (iii), we get
$a=\frac{1}{2}, b=\frac{-3}{2}, c=\frac{1}{2}$
Hence, $[L]=\left[h^{1 / 2} c^{-3 / 2} G^{1 / 2}\right]$