Chapter 6, Problem 4MC

Protons and electrons have different masses. When they are the same distance apart, the electric force between two electrons

1. a. is less than the force between two protons
2. b. is the same as the magnitude of the force between two protons
3. c. is greater than the force between two protons
4. d. Any of these choices could be correct, depending on the distance between each pair of particles.

To determine

To choose: The correct option from the given options.

Answer to Problem 4MC

The correct option is “b. is same as the magnitude of the force between two protons”.

Explanation of Solution

Given data:

Protons and electrons have different masses.

From the given data, it is required to compare the force between two electrons with respect to the force between two protons when the distance between two electrons is equal to the distance between two protons.

Formula used:

Refer to the equation (6-1) in the textbook, and write the expression for force between two electric charges as follows.

$F=\frac{K{Q}_1{Q}_2}{R^2}$ (1)

Here,

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

$Q_1$ is the value of charge 1,

$Q_2$ is the value of charge 2, and

$R$ is the distance between two charges.

From equation (1), it is clear that, the force between two electric charges does not depend on the mass of the charges.

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

As the distance between two protons is same as distance between two electrons, the distance between two electrons is $R\text{m}$ (since, the distance between protons is assumed as $R\text{m}$).

In nature, the electric charge on a proton is $+1.6\times {10}^{-19}\text{C}$, and the electric charge on an electron is $-1.6\times {10}^{-19}\text{C}$.

Calculation of force between two protons $\left(F_{\text{protons}}\right)$:

Modify the expression in equation (1) for the force between two protons as follows.

$F_{\text{protons}}=\frac{K{Q}_{\text{p-}1}{Q}_{\text{p-2}}}{R^2}$ (2)

Here,

$Q_{\text{p-}1}$ is the charge of proton 1, which is $+1.6\times {10}^{-19}\text{C}$, and

$Q_{\text{p-2}}$ is the charge of proton 2, which is $+1.6\times {10}^{-19}\text{C}$.

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 $Q_{\text{p-}1}$, $+1.6\times {10}^{-19}\text{C}$ for $Q_{\text{p-2}}$, and $R\text{m}$ for $R$ in equation (2),

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

Simplify the expression as follows.

$F_{\text{protons}}=\frac{23.04\times {10}^{-29}}{R^2}\text{N}$ (3)

Calculation of force between two electrons $\left(F_{\text{electrons}}\right)$:

Modify the expression in equation (1) for the force between two electrons as follows.

$F_{\text{electrons}}=\frac{K{Q}_{\text{e-}1}{Q}_{\text{e-2}}}{R^2}$ (4)

Here,

$Q_{\text{e-}1}$ is the charge of electron 1, which is $-1.6\times {10}^{-19}\text{C}$, and

$Q_{\text{e-2}}$ is the charge of electron 2, which is $-1.6\times {10}^{-19}\text{C}$.

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 $Q_{\text{e-}1}$, $-1.6\times {10}^{-19}\text{C}$ for $Q_{\text{e-2}}$, and $R\text{m}$ for $R$ in equation (4),

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

Simplify the expression as follows.

$F_{\text{electrons}}=\frac{23.04\times {10}^{-29}}{R^2}\text{N}$ (5)

From equation (3), and (5), the force between two electrons is equal to the force between two protons. Therefore, the option b is an adequate option.

Since, force between two electrons is given as less than the force between two protons, the option a is incorrect answer.

Since, force between two electrons is given as greater than the force between two protons, the option c is incorrect answer.

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

Conclusion:

Hence, the correct option is “b. is same as the magnitude of the force between two protons”.