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(B) To determine the Planck length $l_p$,we use dimensional analysis with the constants $G$ (gravitational constant),$h$ (Planck's constant),and $c$ (

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$\left( \frac{Gh}{c^3} \right)^{1/2}$

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(B) To determine the Planck length $l_p$,we use dimensional analysis with the constants $G$ (gravitational constant),$h$ (Planck's constant),and $c$ (speed of light).Dimensions are:$[G] = M^{-1}L^3T^{-2}$$[h] = ML^2T^{-1}$$[c] = LT^{-1}$Let $l_p \propto G^a h^b c^d$.Substituting dimensions: $L^1 = (M^{-1}L^3T^{-2})^a (ML^2T^{-1})^b (LT^{-1})^d$.Equating powers of $M, L, T$:$M: -a + b = 0 \implies a = b$$T: -2a - b - d = 0 \implies -3a = d$$L: 3a + 2b + d = 1 \implies 3a + 2a - 3a = 1 \implies 2a = 1 \implies a = 1/2$.Thus,$a = 1/2, b = 1/2, d = -3/2$.Therefore,$l_p = \sqrt{\frac{Gh}{c^3}}$.

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?

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