Applications Of Dimensional Analysis

Question

A fundamental length scale is built purely from the gravitational constant G, the Planck constant h and the speed of light c through L = G^x h^y c^z; what are the exponents x, y and z?

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Constructing a length from fundamental constants is a powerful application of dimensional analysis, yielding the famous Planck length. The dimensions are [G] = [M^-1 L^3 T^-2], [h] = [M L^2 T^-1], and [c] = [L T^-1]. Writing L = G^x h^y c^z and demanding the result equal [M^0 L^1 T^0] gives three equations. For mass: -x + y = 0, so x = y. For time: -2x - y - z = 0. For length: 3x + 2y + z = 1. Substituting y = x into the time equation gives z = -3x, and inserting both into the length equation gives 3x + 2x - 3x = 2x = 1, so x = 1/2, hence y = 1/2 and z = -3/2. The choice z = -5/2 violates the length equation. The choice x=1, y=1, z=-3 fails the mass balance, since -1 + 1 = 0 holds but the length equation gives 3+2-3 = 2, not 1. The choice with negative x breaks the mass condition. This is the standard JEE Advanced Planck-length derivation. A check confirms √(hG/c^3) indeed reduces to a length.

Applications Of Dimensional Analysis

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