Daily Physicist

Image courtesy of American Institute of Physics (AIP)

Today is the birthday of

• Herman Feshbach (1917-2000).

Today’s Problem

What is the ratio of the electric to gravitational force between two electrons?

Answer

The gravitational force between two electrons is \[F_g=\frac{G{m_e}^2}{r^2}\]

while the electric force is \[F_q=\frac{1}{4\pi\epsilon_0}\frac{{q_e}^2}{r^2}\]

Taking the ratio \(F_q/F_g\), the distance between the electrons cancels, giving

\[\begin{align} \frac{F_q}{F_g}&=\frac{1}{4\pi\epsilon_0}\cdot\frac{{q_e}^2}{G{m_e}^2}\\ &=\frac{1}{4\pi(8.854\times 10^{-12}\mathrm{F}/\mathrm{m})}\cdot\frac{(1.602\times 10^{-19}\mathrm{C})^2}{(6.674\times 10^{-11}\mathrm{m^3}/\mathrm{kg}\cdot\mathrm{s^2})(9.109\times 10^{-31}\mathrm{kg})^2}\\ &=4.2\times 10^{42} \end{align}\] which you can verify to be a unitless number, and is dependent only on the charge and mass of the electrons.

So the electric force between the electrons is on the order of \(10^{42}\) times stronger than the gravitational force.