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