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. ${G^2}\hbar c$
B. ${(\dfrac{{{G^2}\hbar }}{{{c^3}}})^{1/2}}$
C. ${G^{1/2}}{\hbar ^2}c$
D. $G{\hbar ^2}{c^3}$

Answer 1

Hint: Taking the dimensional value of gravitational constant, Planck constant and velocity and combining them, we will get the dimensional formula for Planck length.

Formula used:
Dimensional constants: $L,T,M$
Where $L$ is the dimensional constant of length and is expressed in meter $(m)$, $M$ is the dimensional constant of mass and is expressed kilograms \[(kg)\] and $R$ is the dimensional constant of time and is expressed in seconds $(s)$.
Resistance applied to the wire: $R = \dfrac{{\rho l}}{A} = \dfrac{{\rho l}}{{\pi {r^2}}}$
Where $l$ is the length of the wire and is expressed in meter $(m)$, $A$ is the area of the wire and is expressed in meter squares $({m^2})$ and $r$ is the radius of the wire and is expressed in meter $(m)$.

Complete step by step answer:
Every quantity and unit in this universe can be simplified into dimensional constants of mass, length, time, temperature, etc. raised to the power of their base values. In our given question we have to find which relationship among the given between gravitational constant, Planck constant and velocity will give us the dimensional constants of Planck length.
We know the following formula of dimensional constants for Planck length, gravitational constant, Planck constant and velocity:
Planck length: ${l_P} = [{L^1}]$
Gravitational constant: $G = [{L^1}{T^{ - 1}}]$
Planck constant: $\hbar = [M{L^2}{T^{ - 1}}]$
Velocity of light: $c = [{M^{ - 1}}{L^3}{T^{ - 1}}]$
When we apply these dimensional constants in place of the physical quantities given in the options, we will be able to determine the correct relation where the resultant will be the dimensional formula of Planck length.
By trial and error method, we arrive at option B.
To check, we will substitute the dimensional values in the given relation.
So,
$
{(\dfrac{{{G^2}\hbar }}{{{c^3}}})^{1/2}} = {(\dfrac{{{{[{L^1}{T^{ - 1}}]}^2} \times [M{L^2}{T^{ - 1}}]}}{{[{M^{ - 1}}{L^3}{T^{ - 1}}]}})^{1/2}} \\
\Rightarrow {(\dfrac{{{G^2}\hbar }}{{{c^3}}})^{1/2}} = [{L^1}] \\
 $
But $[{L^1}]$ is the dimensional constant of Planck length.
Therefore,
${l_P} = {(\dfrac{{{G^2}\hbar }}{{{c^3}}})^{1/2}} = [{L^1}]$

So, the correct answer is “Option B”.

Note:
If we take real units instead of dimensional units, we will not be able to conclude the answer.
Therefore, this question must be solved using dimensional formulae and not physical units.