Question

Expression for time in terms of G (universal gravitational constant), h (Planck constant) and c (speed of light) is proportional to:
(A) √(hc⁵/G)
(B) √(c³/Gh)
(C) √(Gh/c⁵)
(D) √(Gh/c³)

Answer 1

Answer:

option C

Explanation:

may it helps you

Answer 2

The expression for time in terms of G (universal gravitational constant), h (Planck constant) and c (speed of light) is proportional to:

(C) √(Gh/c⁵)

We know,

F = GM²/R², according to Newton's law of gravitation.

Expressing G in M,L,T forrmat,

G = [M⁻¹L³T⁻²]

Again, we know by Planck's relation,

E = hν

So, h in M,L,T format is:

h = [ML²T⁻¹]

Again, c is nothing but velocity.

c = [LT⁻¹]

Let us consider that

t ∝ Gˣhⁿcᵇ

[T] = [M⁻¹L³T⁻²]ˣ[ML²T⁻¹]ⁿ[LT⁻¹]ᵇ

[M⁰L⁰T] = [M⁻ˣ⁺ⁿ L³ˣ⁺²ⁿ⁺ᵇ T⁻²ˣ⁻ⁿ⁻ᵇ]

On comparing both sides the powers of M, L, T , we get

−x + n = 0

⇒ x = y

And, 3x + 2n + b = 0

⇒ 5x + z = 0                                   [As x = y]

⇒ z = -5x               ...(1)

And, −2x − n − b = 1                      

⇒ 3x + z = −1                                 [x = y]

⇒ 3x - 5x = -1                                 [From 1]

⇒ -2x = -1

⇒ x = 1/2

y = x = 1/2

z = -5x = -5(1/2) = -5/2

So, t ∝ √(Gh/c⁵)

Option (C) is correct.