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

 To find which combination of Planck's constant \(h\), speed of light \(c\), and the gravitational constant \(G\) has the dimension of length, we need to analyze the dimensional formula of each option.

The fundamental dimensions involved are:

• Planck's constant (h): Dimension = \([ML^2T^{-1}]\)
• Speed of light (c): Dimension = \([LT^{-1}]\)
• Gravitational constant (G): Dimension = \([M^{-1}L^3T^{-2}]\)

Let's analyze each option:

1. Option A: \(\frac{\sqrt{hG}}{c^{3/2}}\)
   • Dimensions of \(hG\): \([ML^2T^{-1}][M^{-1}L^3T^{-2}] = [L^5T^{-3}]\)
   • Dimensions of \(\sqrt{hG}\): \([L^{5/2}T^{-3/2}]\)
   • Dimensions of \(c^{3/2}\): \([L^{3/2}T^{-3/2}]\)
   • Dimensions of \(\frac{\sqrt{hG}}{c^{3/2}}\): \([L^{5/2}T^{-3/2}] \over [L^{3/2}T^{-3/2}] = [L]\)
2. Option B: \(\frac{\sqrt{hG}}{c^{5/2}}\)
   • Dimensions of \(c^{5/2}\): \([L^{5/2}T^{-5/2}]\)
   • Dimensions of \(\frac{\sqrt{hG}}{c^{5/2}}\): \([L^{5/2}T^{-3/2}] \over [L^{5/2}T^{-5/2}] = [T]\)\)
3. Option C: \(\frac{\sqrt{hc}}{G}\)
   • Dimensions of \(hc\): \([ML^2T^{-1}][LT^{-1}] = [ML^3T^{-2}]\)
   • Dimensions of \(\sqrt{hc}\): \([M^{1/2}L^{3/2}T^{-1}]\)\)
   • Dimensions of \(\frac{\sqrt{hc}}{G}\): \([M^{1/2}L^{3/2}T^{-1}] \over [M^{-1}L^3T^{-2}] = [M^{3/2}L^{-3/2}T]\)\)
4. Option D: \(\frac{\sqrt{Gc}}{h^{3/2}}\)
   • Dimensions of \(Gc\): \([M^{-1}L^3T^{-2}][LT^{-1}] = [M^{-1}L^4T^{-3}]\)
   • Dimensions of \(\sqrt{Gc}\): \([M^{-1/2}L^2T^{-3/2}]\)\)
   • Dimensions of \(h^{3/2}\): \([M^{3/2}L^3T^{-3/2}]\)
   • Dimensions of \(\frac{\sqrt{Gc}}{h^{3/2}}\): \([M^{-1/2}L^2T^{-3/2}] \over [M^{3/2}L^3T^{-3/2}] = [M^{-2}L^{-1}]\)\)

Therefore, the correct choice is option A: \(\frac{\sqrt{hG}}{c^{3/2}}\), which has the dimension of length \([L]\).