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 electrostatic force (F_e) and gravitational force (F_g) between an electron and a proton, we will 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:
\[ F_e = \frac{1}{4\pi \epsilon_0} \cdot \frac{q_e \cdot q_p}{R^2} \]

The gravitational force (F_g) between two masses is given by Newton's law of gravitation:
\[ F_g = G \cdot \frac{m_e \cdot m_p}{R^2} \]

Where:
- \( q_e \) and \( q_p \) are the charges of the electron and proton, respectively.
- \( m_e \) and \( m_p \) are the masses of the electron and proton, respectively.
- \( R \) is the distance between the electron and proton.
- \( \epsilon_0 \) is the permittivity of free space.
- \( G \) is the gravitational constant.

### Step 2: Substitute the known values
Given:
- Charge of electron and proton, \( q_e = q_p = 1.6 \times 10^{-19} \, \text{C} \)
- Mass of electron, \( m_e = 9.1 \times 10^{-31} \, \text{kg} \)
- Mass of proton, \( m_p = 1.66 \times 10^{-27} \, \text{kg} \)
- Gravitational constant, \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)
- Coulomb's constant, \( \frac{1}{4\pi \epsilon_0} \approx 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)

### Step 3: Calculate the electrostatic force (F_e)
Substituting the values into the electrostatic force formula:
\[ F_e = 9 \times 10^9 \cdot \frac{(1.6 \times 10^{-19})^2}{R^2} \]

### Step 4: Calculate the gravitational force (F_g)
Substituting the values into the gravitational force formula:
\[ F_g = 6.67 \times 10^{-11} \cdot \frac{(9.1 \times 10^{-31}) \cdot (1.66 \times 10^{-27})}{R^2} \]

### Step 5: Find the ratio of electrostatic force to gravitational force
Now, we can find the ratio \( \frac{F_e}{F_g} \):
\[ \frac{F_e}{F_g} = \frac{9 \times 10^9 \cdot \frac{(1.6 \times 10^{-19})^2}{R^2}}{6.67 \times 10^{-11} \cdot \frac{(9.1 \times 10^{-31}) \cdot (1.66 \times 10^{-27})}{R^2}} \]
The \( R^2 \) cancels out:
\[ \frac{F_e}{F_g} = \frac{9 \times 10^9 \cdot (1.6 \times 10^{-19})^2}{6.67 \times 10^{-11} \cdot (9.1 \times 10^{-31}) \cdot (1.66 \times 10^{-27})} \]

### Step 6: Calculate the numerical value
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:
\[ 6.67 \times 10^{-11} \cdot (9.1 \times 10^{-31}) \cdot (1.66 \times 10^{-27}) = 6.67 \times 10^{-11} \cdot 1.51 \times 10^{-57} = 1.007 \times 10^{-67} \]

Now, substituting these values into the ratio:
\[ \frac{F_e}{F_g} = \frac{2.304 \times 10^{-28}}{1.007 \times 10^{-67}} \approx 2.28 \times 10^{39} \]

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

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