Question

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?

Short answer

The distance between the two protons for electrostatic force to be equal to the gravitational force of the Earth is$0.119m$.

Step-by-step solution

The given data 

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,$\mathrm{g}=9.8\mathrm{m}/{\mathrm{s}}^2$

Understanding the concept of gravitational force and electric force  

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)

Calculation of the distance between the two protons

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 $0.119\mathrm{m}.$