Problem — Fermi estimate: force between two electrons

Problem

Fermi estimate: force between two electrons in an atom. Estimate the Coulomb force between two electrons at a distance $r \sim 1\ \text{Å} = 10^{-10}\ \text{m}$. Compare the order of magnitude with the weight of a single atom to get a sense of how intense the repulsion is at microscopic scale.

Solution

Coulomb force. With $k = 8{,}99\cdot 10^9\ \text{N·m}^2/\text{C}^2$, $e = 1{,}6\cdot 10^{-19}\ \text{C}$, $r = 10^{-10}\ \text{m}$:

$F = k\,\frac{e^2}{r^2} = 8{,}99\cdot 10^9 \cdot \frac{(1{,}6\cdot 10^{-19})^2}{(10^{-10})^2} = 8{,}99\cdot 10^9 \cdot \frac{2{,}56\cdot 10^{-38}}{10^{-20}} \approx 2{,}3\cdot 10^{-8}\ \text{N}$

The order of magnitude is $\sim 10^{-8}\ \text{N}$.

Comparison with the weight of an atom. An atom has a mass of order $m_{\text{atomo}} \sim 10^{-26}\ \text{kg}$, hence weight

$P = m_{\text{atomo}}\,g \sim 10^{-26} \cdot 10 \sim 10^{-25}\ \text{N}$

Ratio.

$\frac{F}{P} \sim \frac{10^{-8}}{10^{-25}} \sim 10^{17}$

The Coulomb repulsion between two electrons is about $10^{17}$ times more intense than the weight of an atom: at the microscopic scale the electric force is enormously dominant.

$\ev{F \sim 2{,}3\cdot 10^{-8}\ \text{N}}$

Linked atoms

Topics: Elettrostatica Concepts: Legge di Coulomb Skills: Stime di Fermi