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Formulas

Radius of nth Bohr Orbit

Radius of electron orbit in hydrogen atom according to Bohr model.

Physics
Subject
Quantum Mechanics
Sub Subject
Atomic Structure
Chapter
Bohr Model
Topic

Radius of nth Bohr Orbit

📊 intermediate
$$r_n = \frac{n^2 h^2 \epsilon_0}{\pi m e^2} = n^2 a_0$$
Radius of electron orbit in hydrogen atom according to Bohr model.
Variables & Units
$r_n$
Orbit radius
Unit: $\text{m}$
$n$
Principal quantum number
Unit: $\text{unitless}$
$h$
Planck's constant
Unit: $\text{J·s}$
$\epsilon_0$
Vacuum permittivity
Unit: $\text{F/m}$
$m$
Electron mass
Unit: $\text{kg}$
$e$
Elementary charge
Unit: $\text{C}$
$a_0$
Bohr radius
Unit: $\text{m}$
Key Information
📐 Derived From
Quantization of angular momentum $mvr = n\hbar$ and Coulomb force
💡 Example
Ground state $n=1$: $r_1 = a_0 = 5.29\times10^{-11}\,\text{m}$ (0.0529 nm).
🔧 Applications
Atomic physics, quantum chemistry, semiconductor devices
⚠️ Constraints
Hydrogen-like atoms, non-relativistic

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