Chapter 6, Problem 6MC

The electric force between a proton and an electron

1. a. is weaker than the gravitational force between them
2. b. is equal in strength to the gravitational force between them
3. c. is stronger than the gravitational force between them
4. d. is any of these, depending on the distance between the proton and the electron

To determine

To choose: The correct option from the given options.

Answer to Problem 6MC

The correct option is “c. is stronger than the gravitational force between them”.

Explanation of Solution

Given data:

From the given data, it is required to compare the electric force acts between a proton and an electron with respect to the gravitational force acts between them.

Formula used:

Refer to the equation (2-20) in the textbook, and modify the expression for gravitation force acts between the proton and the electron by Newton’s law of gravity as follows.

$F_{\text{g}}=\frac{G{m}_{\text{p}}{m}_{\text{e}}}{R^2}$ (1)

Here,

$G$ is a constant of nature, which is $6.670\times {10}^{-11}\frac{\text{N}\cdot {\text{m}}^2}{{\text{kg}}^2}$,

$m_{\text{p}}$ is the proton’s mass,

$m_{\text{e}}$ is the electron’s mass, and

$R$ is the distance between the electron and the proton.

Refer to the equation (6-1) in the textbook, and modify the expression for electric force acts between the proton and the electron by Coulomb’s law as follows.

$F_{\text{ele}}=\frac{K{Q}_{\text{p}}{Q}_{\text{e}}}{R^2}$ (2)

Here,

$K$ is the electric force constant, which is $9\times {10}^9\frac{\text{N}\cdot {\text{m}}^2}{{\text{C}}^2}$,

$Q_{\text{p}}$ is the quantity (magnitude) of the charge of the proton, which is $1.6\times {10}^{-19}\text{C}$, and

$Q_{\text{e}}$ is quantity of the charge of the electron, which is $1.6\times {10}^{-19}\text{C}$.

Consider the distance $R$ as $R\text{m}$.

Consider the proton’s mass $\left(m_{\text{p}}\right)$ as $1.673\times {10}^{-27}\text{kg}$, and the electron’s mass $\left(m_{\text{e}}\right)$ as $9.11\times {10}^{-31}\text{kg}$.

Calculation of gravitational force acts between the proton and the electron $\left(F_{\text{g}}\right)$:

Substitute $6.670\times {10}^{-11}\frac{\text{N}\cdot {\text{m}}^2}{{\text{kg}}^2}$ for $G$, $1.673\times {10}^{-27}\text{kg}$ for $m_{\text{p}}$, $9.11\times {10}^{-31}\text{kg}$ for $m_{\text{e}}$, and $R\text{m}$ for $R$ in equation (1),

$\begin{array}{c}{F}_{\text{g}}=\frac{\left(6.670\times {10}^{-11}\frac{\text{N}\cdot {\text{m}}^2}{{\text{kg}}^2}\right)\left(1.673\times {10}^{-27}\text{kg}\right)\left(9.11\times {10}^{-31}\text{kg}\right)}{{\left(R\text{m}\right)}^2}\\ =\frac{\left(6.670\times {10}^{-11}\frac{\text{N}\cdot {\text{m}}^2}{{\text{kg}}^2}\right)\left(15.2410\times {10}^{-58}{\text{kg}}^2\right)}{R^2{\text{m}}^2}\end{array}$

Simplify the expression as follows.

$F_{\text{g}}=\frac{101.6574\times {10}^{-69}}{R^2}\text{N}$ (3)

Calculation of electric force acts between the proton and the electron $\left(F_{\text{ele}}\right)$:

Substitute $9\times {10}^9\frac{\text{N}\cdot {\text{m}}^2}{{\text{C}}^2}$ for $K$, $1.6\times {10}^{-19}\text{C}$ for $Q_{\text{p}}$, $1.6\times {10}^{-19}\text{C}$ for $Q_{\text{e}}$, and $R\text{m}$ for $R$ in equation (2),

$\begin{array}{c}{F}_{\text{ele}}=\frac{\left(9\times {10}^9\frac{\text{N}\cdot {\text{m}}^2}{{\text{C}}^2}\right)\left(1.6\times {10}^{-19}\text{C}\right)\left(1.6\times {10}^{-19}\text{C}\right)}{{\left(R\text{m}\right)}^2}\\ =\frac{\left(9\times {10}^9\frac{\text{N}\cdot {\text{m}}^2}{{\text{C}}^2}\right)\left(2.56\times {10}^{-38}{\text{kg}}^2\right)}{R^2{\text{m}}^2}\end{array}$

Simplify the expression as follows.

$F_{\text{ele}}=\frac{23.04\times {10}^{-29}}{R^2}\text{N}$ (4)

From equation (3), and (4), the force in equation (3) is very much less than the force in equation (4).

Therefore, the electric force acts between the electron and the proton is stronger than the gravitation force acts between the electron and the proton. Thus, the option c is an adequate option.

Since, the electric force acts between the electron and the proton is given as weaker than the gravitational force acts between them in option a, the option a is incorrect answer.

Since, the electric force acts between the electron and the proton is given as equal to the gravitational force acts between them in option b, the option b is incorrect answer.

From the analysis, the only option c is the correct choice. Therefore, the option d is absolutely incorrect.

Conclusion:

Hence, the correct option is “c. is stronger than the gravitational force between them”.