Question

Plancks constant $(h)$, speed of light in 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. $\frac{\sqrt{hG}}{c^{3/ 2}}$
• B. $\frac{\sqrt{hG}}{c^{5/ 2}}$
• C. $\frac{\sqrt{hc}}{G}$
• D. $\frac{\sqrt{Gc}}{h^{3/ 2}}$

Answer 1

The Correct Option is A

Dimensional Analysis for Length 

The question asks for the combination of the fundamental constants: Planck's constant (h), the speed of light (c), and Newton's gravitational constant (G) that gives the dimension of length.

Dimensional Formulae of Constants:

• \(h\) (Planck's constant): \([h] = M L² T⁻¹\)
• \(c\) (speed of light): \([c] = L T⁻¹\)
• \(G\) (Newton's gravitational constant): \([G] = M⁻¹ L³ T⁻²\)

Finding the Combination of Constants with Dimension of Length:

We want to find the correct combination that results in the dimension of length (L).

Consider the following combination:

Combination: \(\frac{\sqrt{hG}}{c^{3/2}}\)

Let's calculate the dimensional formula for this combination:

The dimensional formula for each constant is:

• \([h] = M L² T⁻¹\)
• \([G] = M⁻¹ L³ T⁻²\)
• \([c] = L T⁻¹\)

Now calculate the dimensional formula for \(\frac{\sqrt{hG}}{c^{3/2}}\):

For \(\sqrt{hG}\): \([hG] = (M L² T⁻¹) (M⁻¹ L³ T⁻²) = M⁰ L⁵ T⁻³\) \(\sqrt{hG}\) = \(M⁰ L².5 T⁻1.5\)

For \(c^{3/2}\): \([c^{3/2}] = (L T⁻¹)^{3/2} = L^1.5 T⁻1.5\)

Now, combining the two parts:

\(\frac{\sqrt{hG}}{c^{3/2}}$ = $\frac{M⁰ L².5 T⁻1.5}{L^1.5 T⁻1.5}$ =\)

Therefore, the combination \(\frac{\sqrt{hG}}{c^{3/2}}\) gives the dimension of length.

Conclusion:

The correct combination of constants that has the dimension of length is: \(\frac{\sqrt{hG}}{c^{3/2}}\).