1. frac√(hG)c^(5/2) 2. √(hc/G) 3. sqrtfracGch^(3/2) 4. frac√(hG)c^(3/2)

Answer: 4. frac√(hG)c^(3/2) Write the dimensions: [h]=ML²T⁻¹ , [c]=LT⁻¹ , [G]=M⁻¹L³T⁻² . Length has the dimension of √(Gh/c³) (the Planck length). Check: Gh/c³=fracM⁻¹L³T⁻²· ML²T⁻¹L³T⁻³=fracL⁵T⁻³L³T⁻³=L² So dfrac√(hG)c^(3/2) has dimension L . Option (4). Option (1) is dimensionally a different quantity (it has an extra factor of c⁻¹ ), so it is not a length.

Topic: Dimensions and dimensional analysis· Physics and Measurement

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?

1. $\frac{\sqrt{hG}}{c^{5/2}}$

2. $\sqrt{\frac{hc}{G}}$

3. $\sqrt{\frac{Gc}{h^{3/2}}}$

4. $\frac{\sqrt{hG}}{c^{3/2}}$

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Answer: 4. $\frac{\sqrt{hG}}{c^{3/2}}$

Write the dimensions: $[h]=ML^2T^{-1}$, $[c]=LT^{-1}$, $[G]=M^{-1}L^3T^{-2}$.

Length has the dimension of $\sqrt{\dfrac{Gh}{c^3}}$ (the Planck length). Check:

$$\frac{Gh}{c^3}=\frac{M^{-1}L^3T^{-2}\cdot ML^2T^{-1}}{L^3T^{-3}}=\frac{L^5T^{-3}}{L^3T^{-3}}=L^2$$

So $\dfrac{\sqrt{hG}}{c^{3/2}}$ has dimension $L$. Option (4). Option (1) is dimensionally a different quantity (it has an extra factor of $c^{-1}$), so it is not a length.

Doubtly AI · 30 Oct 2026 AI-generated · not yet verified Report

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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?

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