Dimensional formula of Bohr radius

How to get the dimensional formula of Bohr radius

1. Start from the definition: Bohr radius = permittivity of free space × plancks constant² / (electron mass × elementary charge²).
2. electric charge = electric current × time, so electric charge is [A] × [T] = [T A].
3. velocity = length / time, so velocity is [L] / [T] = [L T^-1].
4. acceleration = velocity / time, so acceleration is [L T^-1] / [T] = [L T^-2].
5. force = mass × acceleration, so force is [M] × [L T^-2] = [M L T^-2].
6. permittivity of free space = electric charge² / (force × length²), so permittivity of free space is [T A]² / ([M L T^-2] × [L]²) = [M^-1 L^-3 T⁴ A²].
7. energy = mass × velocity², so energy is [M] × [L T^-1]² = [M L² T^-2].
8. Put in the dimension of each part: [M^-1 L^-3 T⁴ A²] × [M L² T^-1]² / ([M] × [T A]²).
9. Add the powers of each base quantity: [L].

Quantities with the dimensions [L]

• Length
• Displacement
• Radius of gyration
• Mean free path
• Wavelength
• Amplitude
• Focal length
• Fringe width
• de Broglie wavelength

Equal dimensions do not make them the same quantity. Compare Bohr radius with the others in the full grid.

Three questions on Bohr radius

Related quantities

• G Gravitational constant [M⁻¹ L³ T⁻²]
• c Speed of light [L T⁻¹]
• h Planck's constant [M L² T⁻¹]
• ħ Reduced Planck's constant (h-bar) [M L² T⁻¹]
• k Boltzmann constant [M L² T⁻² K⁻¹]
• R Universal gas constant [M L² T⁻² K⁻¹ mol⁻¹]