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Units and Measurements
Dimensional analysis – Planck length
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2016 Phase II 1 question
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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?
Match powers of M, L and T.
$l \propto h^xG^yc^z$
$M^0L^1T^0 = (ML^2T^{-1})^x(M^{-1}L^3T^{-2})^y(LT^{-1})^z = M^{x-y}L^{2x+3y+z}T^{-x-2y-z}$
Equating: $x - y = 0$, $2x + 3y + z = 1$, $-x - 2y - z = 0$
$\Rightarrow x = \frac{1}{2}$, $y = \frac{1}{2}$, $z = -\frac{3}{2}$
$\Rightarrow l \propto \frac{\sqrt{hG}}{c^{3/2}}$