Chapter 18, Problem 30TP

To determine

(a)

The ratio of the electrostatic force to thegravitational force between two electrons.

Answer to Problem 30TP

The ratio of the electrostatic force to thegravitational force between two electrons is $4.25\times {10}^{42}$.

Explanation of Solution

Formula used:

The magnitude of the electrostatic force between two electrons isgiven by: $F_C=k\frac{q^2}{r^2}$

Here: $k=8.99\times {10}^9\text{N}\cdot {\text{m}}^{\text{2}}{\text{/C}}^2$ is the Coulomb's constant

  $q=1.6\times {10}^{-19}\text{C}$, magnitude of the charge of an electron.

r is thedistance between them.

The gravitational force between two electrons is given by $F_G=\frac{G{m}^2}{r^2}$.

Here: $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}{\text{/kg}}^2$ is the gravitational constant

  $m=9.11\times {10}^{-31}\text{ kg}$ is themass of an electron

r isdistance between them

Calculation:

The ratio of the electrostatic force to thegravitational force between two electrons is

  $\begin{array}{c}\frac{F_C}{F_G}=\frac{\frac{k{q}^2}{r^2}}{\frac{G{m}^2}{r^2}}=\frac{k{q}^2}{G{m}^2}=\frac{8.99\times {10}^9\times {\left(1.6\times {10}^{-19}\right)}^2}{6.67\times {10}^{-11}\times {\left(9.11\times {10}^{-31}\right)}^2}\\ =4.25\times {10}^{42}\end{array}$

Conclusion:

The ratio of the electrostatic force to thegravitational force between two electrons is $4.25\times {10}^{42}$.

To determine

(b)

Whether or not, the ratio would change if the two electrons are replacedby protons.

The value of new ratio.

Answer to Problem 30TP

Yes.

The ratio of the electrostatic force to thegravitational force between two protons is $1.24\times {10}^{36}$.

Explanation of Solution

Formula used:

The magnitude of the electrostatic force between two protons isgiven by: $F_C=k\frac{q^2}{r^2}$

Here: $k=8.99\times {10}^9\text{N}\cdot {\text{m}}^{\text{2}}{\text{/C}}^2$ is theCoulomb's constant,

  $q=1.6\times {10}^{-19}\text{C}$ is the magnitude of the charge of a proton,

r isthe distance between them.

The gravitational force between the two protons is given by:

  $F_G=\frac{G{m}^2}{r^2}$.

Here: $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}{\text{/kg}}^2$ is the gravitational constant,

  $m=1.67\times {10}^{-27}\text{ kg}$ is the mass of a proton,

r isthe distance between them.

Calculation:

The ratio of the electrostatic force to the gravitational force between two protons is

  $\begin{array}{c}\frac{F_C}{F_G}=\frac{\frac{k{q}^2}{r^2}}{\frac{G{m}^2}{r^2}}=\frac{k{q}^2}{G{m}^2}=\frac{8.99\times {10}^9\times {\left(1.6\times {10}^{-19}\right)}^2}{6.67\times {10}^{-11}\times {\left(1.67\times {10}^{-27}\right)}^2}\\ =1.24\times {10}^{36}\end{array}$

Conclusion:

The ratio of the electrostatic force to the gravitational force between two protons is $1.24\times {10}^{36}$.