Planck's constant (.h.) , speed of light in a vacuum (. c .) and Newton's gravitational constant (. G .) are three fundamental constants. Which of the following combinations of these has the dimension of length?

Question

Planck's constant (.h.) , speed of light in a vacuum (. c .) and Newton's gravitational constant (. G .) are three fundamental constants. Which of the following combinations of these has the dimension of length? (A) (√h G/c(3)2) (B) (√h G/c(5)2) (C) √(h c/G) (D) √(c G â¡/h â¡(3)2)

Tardigrade Admin · Accepted Answer

L propto hacbGc [L]1=[M1 L2 T- 1]a[L T- 1]b[M- 1 L3 T- 2]c Solving, a=(1/2), c=(1/2), b=-(3/2) ⇒ L=(√h G/c(3)2)

Q. Planck's constant $(h)$ , speed of light in a vacuum $(c)$ and Newton's gravitational constant $(G)$ are three fundamental constants. Which of the following combinations of these has the dimension of length?

$\frac{h G}{c ^{\frac{3}{2}}}$

$\frac{h G}{c ^{\frac{5}{2}}}$

$\frac{h c}{G}$

$\frac{c G}{h ^{\frac{3}{2}}}$

$L \propto h^{a} c^{b} G^{c}$
$[L]^{1} = [M^{1} L^{2} T^{- 1}]^{a} [L T^{- 1}]^{b} [M^{- 1} L^{3} T^{- 2}]^{c}$
Solving,
$a = \frac{1}{2}, c = \frac{1}{2}, b = - \frac{3}{2}$
$\Rightarrow L = \frac{h G}{c ^{\frac{3}{2}}}$

Q. Planck's constant (h) , speed of light in a vacuum (c) and Newton's gravitational constant (G) are three fundamental constants. Which of the following combinations of these has the dimension of length?

c23​hG​​

c25​hG​​

Ghc​​

h⁡23​cG⁡​​​