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Question:

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?

√Gch3/2

√hGc5/2

√hGc3/2

√hcG

Solution:

The correct option is B √hGc5/2
The SI unit of h is J.s or N.m.s and SI unit of c is m/s and Si unit of G is N.m²/kg²
For A: √(N.m²/kg²)(m/s)(N.m.s)³/² = kg.m³N.s⁵ = kg.m³(kg.m/s²)(s⁵) = m²/s³
For B: √(N.m.s)(N.m²/kg²)(m/s)⁵/² = Ns²/kg = (kg.m/s²)(s²)kg = m, it is the unit length.
For C: √(N.m.s)(N.m²/kg²)(m/s)³/² = Ns³m.kg = (kg.m/s²)(s³)m.kg = s
For D: √(N.m.s)(m/s)N.m²/kg² = kg