Dimensions of Length (h, c, G) – Rankers Physics
Topic: Unit And Dimensions
Subtopic: Dimensions

Dimensions of Length (h, c, G)

Planck's 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?

[2016-II]

\(\sqrt{\frac{hc}{G}}\)
\(\sqrt{\frac{Gc}{h^{3/2}}}\)
\(\frac{\sqrt{hG}}{c^{3/2}}\)
\(\frac{\sqrt{hG}}{c^{5/2}}\)

Solution:

The Planck length formula is \(l_P = \sqrt{\frac{hG}{c^3}}\). Checking option c: \(\frac{\sqrt{hG}}{c^{3/2}} = h^{1/2}G^{1/2}c^{-3/2}\). \(h=[ML^2T^{-1}]\), \(G=[M^{-1}L^3T^{-2}]\), \(c=[LT^{-1}]\). Thus, \([M^{1/2}L^1T^{-1/2}] [M^{-1/2}L^{3/2}T^{-1}] [L^{-3/2}T^{3/2}] = [M^0L^{1+3/2-3/2}T^{-1/2-1+3/2}] = [L]\).

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