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NEET Physics QUESTION #7231
Question 1

Planck's constant ($h$), speed of light in vacuum ($c$), and Newton's gravitational constant ($G$) are taken as fundamental constants. Which combination of these has the dimension of length?

  • $\sqrt{\dfrac{hc}{G}}$
  • $\sqrt{\dfrac{Gc}{h^{1/2}}}$
  • $\sqrt{\dfrac{hG}{c^3}}$✔️
  • $\sqrt{\dfrac{hG}{c^{5/2}}}$
Correct Answer Explanation

This is the Planck length $\ell_P = \sqrt{\dfrac{\hbar G}{c^3}}$ (with $h$ in place of $\hbar$ up to a constant).

Dimensions: $[h] = ML^2T^{-1}$, $[G] = M^{-1}L^3T^{-2}$, $[c] = LT^{-1}$

$\left[\dfrac{hG}{c^3}\right] = \dfrac{ML^2T^{-1} \cdot M^{-1}L^3T^{-2}}{L^3T^{-3}} = \dfrac{L^5T^{-3}}{L^3T^{-3}} = L^2$

$\Rightarrow \sqrt{\dfrac{hG}{c^3}} \sim [L]$ ✓