Question

Plancks Constant, Speed Of Light In Vacuum And Newtons Gravitational Constant Are Three Fundamental Constants. Which Of The Following Combinations Of These Has The Dimension Of Length.

Answer 1

Answer:

Explanation:

Dimension of plancks constant D(h) = [ML²T⁻¹]

D(c) = [LT⁻¹]

D(G) = [M⁻¹L³T⁻²]

Dimnesion of length ,

D(l) = [L] = D(h)ᵃD(c)ᵇD(G)ⁿ = [ML²T⁻¹]ᵃ[LT⁻¹]ᵇ[M⁻¹L³T⁻²]ⁿ

$=[M^{a-n} L^{2a+b+3n} T^{-a-b-2n}]$

by comparing the powers

we get,

a-n = 0  ⇒ a = n .............................................(1)

2a+b+3n = 1 ⇒ 5a+b = 1 ...............................(2)

-a-b-2n = 0 ⇒ b = -3a ....................................(3)

Using (2) & (3), we get

2a = 1 ⇒ a = 1/2

∵ b = -3a ⇒ b = -3/2

∵ n = a ⇒ n = 1/2

∴ The combination that gives dimension of length is

$h^{\frac{1}{2}} c^{-\frac{3}{2}}G^{\frac{1}{2}} = \sqrt{\frac{hG}{c^{3}}}$