Problem — Ratio between gravitational and Coulomb force

Problem

Calculate the ratio between the gravitational force and the Coulomb force between two electrons ($m_e = 9{,}11\cdot 10^{-31}\ \text{kg}$, $e = 1{,}60\cdot 10^{-19}\ \text{C}$).

Solution

Both forces go as $1/r^2$, so the distance cancels in the ratio: $r$ does not appear in the result.

$\frac{F_g}{F_e} = \frac{G\,m_e^2/r^2}{k\,e^2/r^2} = \frac{G\,m_e^2}{k\,e^2}$

With $G = 6{,}67\cdot 10^{-11}$, $k = 8{,}99\cdot 10^9$:

$\frac{F_g}{F_e} = \frac{6{,}67\cdot 10^{-11} \cdot (9{,}11\cdot 10^{-31})^2}{8{,}99\cdot 10^9 \cdot (1{,}60\cdot 10^{-19})^2} = \frac{6{,}67\cdot 10^{-11} \cdot 8{,}3\cdot 10^{-61}}{8{,}99\cdot 10^9 \cdot 2{,}56\cdot 10^{-38}} = \frac{5{,}53\cdot 10^{-71}}{2{,}30\cdot 10^{-28}} \approx 2{,}4\cdot 10^{-43}$

Gravity between two electrons is about $43$ orders of magnitude weaker than the electric repulsion: entirely negligible.

$\ev{F_g/F_e \approx 2{,}4\cdot 10^{-43}}$

Links

Topics: Elettrostatica Concepts: Legge di Coulomb Skills: Analisi dimensionale