The characteristic distance at which quantum | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

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

Vedclass · Accepted Answer

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

Vedclass · Answer

(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?

Similar Questions

If the dimensions of $A$ and $B$ are different,which of the following operations is physically meaningful?

If the dimensional formula of $(\text{Energy} \times \text{speed})$ is $[M^{a} L^{b} T^{c}]$,then $a, b,$ and $c$ are:

The speed of light $(c)$,gravitational constant $(G)$,and Planck's constant $(h)$ are taken as fundamental units in a system. The dimensions of time in this new system should be

In an electromagnetic system,the quantity representing the ratio of electric flux and magnetic flux has dimensions of $M^{P} L^{Q} T^{R} A^{S}$. The values of $Q$ and $R$ are:

In an experiment,four quantities $a, b, c$ and $d$ are measured with percentage errors of $1\%, 2\%, 3\%$ and $4\%$ respectively. Quantity $w$ is calculated as follows: $w = \frac{a^4 b^3}{c^2 \sqrt{d}}$. The percentage error in the measurement of $w$ is .......... $\%$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module