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 ?

Units and Measurements PYQ — JEE Main 2018 · 15 Apr · Shift 2 · Q66 with Solution

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Correct answer: D

We want a combination of $G$ , $\hbar$ and $c$ that has the dimensions of length.
Write dimensions of each constant:

$[G] = M^{-1}L^3T^{-2}$ $[\hbar] = ML^2T^{-1}$ $[c] = LT^{-1}$
Assume Planck length is of the form

$l_P = G^a\hbar^b c^d$

Then its dimensions are

$[l_P] = (M^{-1}L^3T^{-2})^a (ML^2T^{-1})^b (LT^{-1})^d$

$= M^{-a+b}L^{3a+2b+d}T^{-2a-b-d}$
Since this must be a length, we require

$M^0L^1T^0$

So equating powers:
- For $M$ : $-a+b=0 \Rightarrow b=a$
- For $T$ : $-2a-b-d=0$ Using $b=a$ : $-2a-a-d=0 \Rightarrow d=-3a$
- For $L$ : $3a+2b+d=1$ Using $b=a$ , $d=-3a$ : $3a+2a-3a=1$ $2a=1 \Rightarrow a=\frac12$
Hence,

$b=\frac12, \quad d=-\frac32$
Therefore,

$l_P = G^{1/2}\hbar^{1/2}c^{-3/2}$

$l_P = \left(\frac{G\hbar}{c^3}\right)^{1/2}$
Now compare with the options:
- A: incorrect
- B: incorrect
- C: incorrect
- D: $\left(\frac{G\hbar}{c^3}\right)^{1/2}$ correct

Therefore, the correct option is D.