Chapter 20, Problem 60A

To determine

The ratio of the electrostatic force to the gravitational force between the electron and the proton.

Answer to Problem 60A

  $2.3\times {10}^{39}$

Explanation of Solution

Given:

It is known that,

The mass of electron, $m_e=9.10\times {10}^{-31}\text{kg}$

The mass of proton, $m_e=1.67\times {10}^{-27}\text{kg}$

The charge on proton, $q_p=1.6\times {10}^{-19}\text{C}$

The charge on electron, $q_e=-1.6\times {10}^{-19}\text{C}$

Formula used:

The electrostatic force between two charges, $q_1$ and $q_2$ , $r$ distance apart is given by,

  $F=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r^2}$

Where, ${\epsilon }_0=8.85\times {10}^{-12}\text{F/m}$ is permittivity of free space.

And, Law of Universal Gravitation,

  $F=\frac{G{m}_1{m}_2}{r^2}$

Where, $G$ is the gravitational constant, $m_1$ and $m_2$ are the masses, and $r$ is the distance between the masses.

It is known that the value of gravitational constant is,

  $G=6.67\times {10}^{-11}\text{N}{\text{.m}}^{\text{2}}{\text{/kg}}^{\text{2}}$

Calculation:

Let the distance between the electron and proton be $r$ .

Then, the magnitude of the electrostatic force between the electron and proton is,

  $\begin{array}{c}{F}_E=\left|\frac{1}{4\pi {\epsilon }_0}\frac{q_p{q}_e}{r^2}\right|\\ =\frac{1}{4\pi {\epsilon }_0}\frac{q_e^2}{r^2}\end{array}$

And, the gravitational force between them is,

  $F_G=\frac{G{m}_p{m}_e}{r^2}$

Then, the ratio of the electrostatic force to the gravitational force between electron and proton will be,

  $\begin{array}{c}\frac{F_E}{F_G}=\frac{\left(\frac{1}{4\pi {\epsilon }_0}\frac{q_e^2}{r^2}\right)}{\left(\frac{G{m}_p{m}_e}{r^2}\right)}\\ \frac{F_E}{F_G}=\frac{\left(q_e^2\right)}{\left(4\pi {\epsilon }_{0}G{m}_p{m}_e\right)}\\ =\frac{{\left(1.6\times {10}^{-19}\right)}^2}{\left(4\pi \times 8.85\times {10}^{-12}\times 6.67\times {10}^{-11}\times 1.67\times {10}^{-27}\times 9.10\times {10}^{-31}\right)}\\ =2.3\times {10}^{39}\end{array}$

Conclusion:

Hence, the ratio of the electrostatic force to the gravitational force between electron and proton is $2.3\times {10}^{39}$ .