The ratio of magnitude of electrostatic force and gravitational force for an electron and a proton is

Options

A. `6.6xx10^(39)`
B. `2.4xx10^(39)`
C. `6.6xx10^(29)`
D. `2.4xx10^(29)`

Explanation

To find the ratio of the magnitude of the electrostatic force to the gravitational force between an electron and a proton, we can follow these steps:

### Step 1: Write the formulas for electrostatic and gravitational forces.
The electrostatic force (\(F_e\)) between two charges is given by Coulomb's law:
\[ F_e = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]
where:
- \(k\) is Coulomb's constant (\(9 \times 10^9 \, \text{N m}^2/\text{C}^2\)),
- \(q_1\) and \(q_2\) are the magnitudes of the charges (for an electron and a proton, both have a charge of \(1.6 \times 10^{-19} \, \text{C}\)),
- \(r\) is the distance between the charges.

The gravitational force (\(F_g\)) between two masses is given by Newton's law of gravitation:
\[ F_g = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
where:
- \(G\) is the gravitational constant (\(6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\)),
- \(m_1\) and \(m_2\) are the masses of the electron and proton, respectively.

### Step 2: Set up the ratio of the forces.
We want to find the ratio of the electrostatic force to the gravitational force:
\[ \frac{F_e}{F_g} = \frac{\frac{k \cdot |q_1 \cdot q_2|}{r^2}}{\frac{G \cdot m_1 \cdot m_2}{r^2}} \]
The \(r^2\) terms cancel out:
\[ \frac{F_e}{F_g} = \frac{k \cdot |q_1 \cdot q_2|}{G \cdot m_1 \cdot m_2} \]

### Step 3: Substitute the known values.
Now we will substitute the known values into the equation:
- \(k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2\)
- \(q_1 = q_2 = 1.6 \times 10^{-19} \, \text{C}\)
- \(G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\)
- Mass of electron (\(m_1\)) = \(9 \times 10^{-31} \, \text{kg}\)
- Mass of proton (\(m_2\)) = \(1.66 \times 10^{-27} \, \text{kg}\)

Substituting these values gives:
\[ \frac{F_e}{F_g} = \frac{(9 \times 10^9) \cdot (1.6 \times 10^{-19})^2}{(6.67 \times 10^{-11}) \cdot (9 \times 10^{-31}) \cdot (1.66 \times 10^{-27})} \]

### Step 4: Calculate the numerator and denominator.
Calculating the numerator:
\[ 9 \times 10^9 \cdot (1.6 \times 10^{-19})^2 = 9 \times 10^9 \cdot 2.56 \times 10^{-38} = 2.304 \times 10^{-28} \]

Calculating the denominator:
\[ (6.67 \times 10^{-11}) \cdot (9 \times 10^{-31}) \cdot (1.66 \times 10^{-27}) = 6.67 \times 10^{-11} \cdot 1.494 \times 10^{-57} = 9.95 \times 10^{-68} \]

### Step 5: Find the ratio.
Now we can find the ratio:
\[ \frac{F_e}{F_g} = \frac{2.304 \times 10^{-28}}{9.95 \times 10^{-68}} \approx 2.32 \times 10^{39} \]

### Final Answer:
Thus, the ratio of the magnitude of the electrostatic force to the gravitational force for an electron and a proton is approximately:
\[ \frac{F_e}{F_g} \approx 2.4 \times 10^{39} \]