2016 Phase II Units and MeasurementsDimensional analysis – Planck length Medium Question 35 of 53 in Units and Measurements

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?

A $\sqrt{\frac{hc}{G}}$
B $\sqrt{\frac{Gc}{h^{3/2}}}$
C $\frac{\sqrt{hG}}{c^{3/2}}$
D $\frac{\sqrt{hG}}{c^{5/2}}$

The answer is C.

Solution

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: $x - y = 0$, $2x + 3y + z = 1$, $-x - 2y - z = 0$
$\Rightarrow x = \frac{1}{2}$, $y = \frac{1}{2}$, $z = -\frac{3}{2}$
$\Rightarrow l \propto \frac{\sqrt{hG}}{c^{3/2}}$

Concept planck-length-dimensions

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53 questions from this chapter have appeared across 25 exam years.