Applications Of Dimensional Analysis
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?
Select the correct option:
Solution
x=1/2,y=1/2,z=−3/2
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.
🔒 Solution Hidden from View
Submit your answer to unlock the detailed step-by-step solution.
More applications of dimensional analysis Practice Questions
About This Question
- Subject
- physics
- Chapter
- physics and measurement
- Topic
- applications of dimensional analysis
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
x=1/2,y=1/2,z=−3/2
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.
This hard difficulty physics question is from the chapter physics and measurement, covering the topic of applications of dimensional analysis. It appeared in the 2025 exam.
Looking for more practice? Explore all physics questions or browse physics and measurement questions on RankGuru.