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

The force $F$ is given in terms of time $t$ and displacement $x$ by the equation $F = A \cos(Bx) + C \sin(Dt).$ The dimensional formula of $D/B$ is

If velocity $[V]$,time $[T]$,and force $[F]$ are chosen as the base quantities,the dimensions of mass will be:

If energy $(E)$, velocity $(v)$, and force $(F)$ are taken as fundamental quantities, what are the dimensions of mass?

Which of the following expressions can be deduced on the basis of dimensional analysis? (All symbols have their usual meanings)

In a new system of units,the unit of mass is $a \ kg$,and the units of length and time are $b \ m$ and $c \ s$,respectively. What is the magnitude of $6 \ W$ of power in this system?

Vedclass Test Series

Exam Paper Generator

Online Exam Module