Question

The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be determined from a suitable combination of the fundamental physical constants $G, h$ and $c$. Which of the following correctly gives the Planck length ?

• A. $G \hbar^2 \, c^3 $
• B. $G^2 \hbar \, c $
• C. $G^{1/2} \hbar^2 \, c$
• D. $\left( \frac{G \hbar}{c^3} \right)^{1/2}$

Answer 1

Answer: (D)

Given Planck length $l$ can be determined from suitable combination of $G, \hbar$ and $c$. Therefore,
$l=f(G, h, c)=\left[ M ^{-1} \,L ^{3}\, T ^{-2}\right]^{ x }\left[ M\,L ^{2}\, T ^{-1}\right]^{y}\left[ LT ^{-1}\right]^{2}= M ^{-2}\, M ^{y}\, L ^{3 x} \,L ^{2 y}\, L ^{z}\, T ^{-2 x} \,T ^{-y}\, T ^{z}$
$\Rightarrow M ^{\circ} L ^{1} T ^{0}= M ^{x+y}\, L ^{3 x+2 y+z} \,T ^{-2 x-y-z}$
Equating the exponents, we have
$-x+y =0 \Rightarrow x=y$
$-2 x-y-z =0 $
$\Rightarrow -2 x-x-z=0 $
$\Rightarrow z=-3 x $
$3 x+2 y+z =1$
$ \Rightarrow 3 x+2 x-3 x=1 $
$\Rightarrow x=\frac{1}{2} $
$ \Rightarrow x =\frac{1}{2}, y=\frac{1}{2}, z=\frac{-3}{2} $
Therefore, $ l =G^{1 / 2}\,h^{1 / 2}\, c^{-3 / 2}=\sqrt{\frac{G \hbar}{c^{3}}} $