Q 11-01-230JEE MainJEE Main 2018 (15 Apr, Shift 2)Easy
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?
Answer: (B) $\left(\dfrac{Gh}{c^3}\right)^{1/2}$
Dimensions: $[G] = M^{-1}L^3T^{-2}$, $[h] = ML^2T^{-1}$, $[c] = LT^{-1}$.
Let $\ell = G^a h^b c^d$:
$$M:\ -a + b = 0, \qquad L:\ 3a + 2b + d = 1, \qquad T:\ -2a - b - d = 0$$
From the first, $b = a$. Then $5a + d = 1$ and $-3a - d = 0$, so $a = \frac12$, $b = \frac12$, $d = -\frac32$:
$$\ell = \left(\frac{Gh}{c^3}\right)^{1/2}$$
Solution by Sreeraj P, M.Sc Physics