How far apart must two protons be if the magnitude of the electrostatic force acting on either one due to the other is equal to the magnitude of the gravitational force on a proton at Earth’s surface?

1. Electrostatic force on either of the proton is equal to the magnitude of the gravitation force on a proton.
2. Mass of the proton,$\mathrm{m}=1.67\times {10}^{-27}\mathrm{kg}$
3. Acceleration due to gravity,$\mathrm{g}=9.8\mathrm{m}/{\mathrm{s}}^2$

Using the concept of gravitational force and Force from Coulomb's law, we can get the value of the distance between the two protons.

Formulae:

The magnitude of the electrostatic force between any two protons of charge q,

${\mathrm{F}}_{\mathrm{e}}={\mathrm{kq}}^2/{\mathrm{r}}^2$ (i)

The magnitude of gravitational force, ${\mathrm{F}}_{\mathrm{g}}=\mathrm{mg}$ (ii)

The problem compares the electrostatic force between two protons and the gravitational force by Earth on a proton. Thus, equating equations (i) and (ii) and the given data, we can get the distance between the two protons as follows:

$\begin{array}{c}{\mathrm{kq}}^2/{\mathrm{r}}^2=\mathrm{mg}\\ \mathrm{r}=\mathrm{q}\sqrt{\frac{\mathrm{k}}{\mathrm{mg}}}\\ =(1.60\times {10}^{-19}\mathrm{C})\sqrt{\frac{9.00\times {10}^9{\mathrm{Nm}}^2/{\mathrm{C}}^2}{(1.67\times {10}^{-27}\mathrm{kg})(9.8\mathrm{m}/{\mathrm{s}}^2)}}\\ =0.119\mathrm{m}\end{array}$

Hence, the value of the distance between the protons is $0.119\mathrm{m}.$