Calculate the ratio of electrostatic to gravitational force between two electrons placed at certain distance in air. Given that `m_(e)= 9.1xx10^(-31) kg,e= 1.6xx10^(-19)C` and `G=6.6xx10^(-11)Nm^(-2)`.

Options

A. `8.4xx10^(42)`
B. `3.2xx10^(41)`
C. `4.2xx10^(42)`
D. `1.2xx10^(42)`

Explanation

To calculate the ratio of the electrostatic force to the gravitational force between two electrons, we will follow these steps:

### Step 1: Calculate the Electrostatic Force (F_e)
The formula for the electrostatic force between two point 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, approximately \( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)
- \( q_1 \) and \( q_2 \) are the charges of the electrons, both equal to \( e = 1.6 \times 10^{-19} \, \text{C} \)
- \( r \) is the distance between the charges (we will keep it as \( r \) for now)

Substituting the values:

\[ F_e = \frac{9 \times 10^9 \cdot (1.6 \times 10^{-19})^2}{r^2} \]

Calculating \( (1.6 \times 10^{-19})^2 \):

\[ (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \]

Thus, we have:

\[ F_e = \frac{9 \times 10^9 \cdot 2.56 \times 10^{-38}}{r^2} \]

### Step 2: Calculate the Gravitational Force (F_g)
The formula for the gravitational force 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, approximately \( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)
- \( m_1 \) and \( m_2 \) are the masses of the electrons, both equal to \( m_e = 9.1 \times 10^{-31} \, \text{kg} \)

Substituting the values:

\[ F_g = \frac{6.67 \times 10^{-11} \cdot (9.1 \times 10^{-31})^2}{r^2} \]

Calculating \( (9.1 \times 10^{-31})^2 \):

\[ (9.1 \times 10^{-31})^2 = 8.27 \times 10^{-61} \]

Thus, we have:

\[ F_g = \frac{6.67 \times 10^{-11} \cdot 8.27 \times 10^{-61}}{r^2} \]

### Step 3: Calculate the Ratio of Electrostatic to Gravitational Force
Now we can find the ratio \( \frac{F_e}{F_g} \):

\[ \frac{F_e}{F_g} = \frac{\frac{9 \times 10^9 \cdot 2.56 \times 10^{-38}}{r^2}}{\frac{6.67 \times 10^{-11} \cdot 8.27 \times 10^{-61}}{r^2}} \]

The \( r^2 \) cancels out:

\[ \frac{F_e}{F_g} = \frac{9 \times 10^9 \cdot 2.56 \times 10^{-38}}{6.67 \times 10^{-11} \cdot 8.27 \times 10^{-61}} \]

Calculating the denominator:

\[ 6.67 \times 10^{-11} \cdot 8.27 \times 10^{-61} = 5.52 \times 10^{-71} \]

Now substituting back into the ratio:

\[ \frac{F_e}{F_g} = \frac{9 \times 10^9 \cdot 2.56 \times 10^{-38}}{5.52 \times 10^{-71}} \]

Calculating the numerator:

\[ 9 \times 10^9 \cdot 2.56 \times 10^{-38} = 2.304 \times 10^{-28} \]

Thus, we have:

\[ \frac{F_e}{F_g} = \frac{2.304 \times 10^{-28}}{5.52 \times 10^{-71}} \approx 4.17 \times 10^{42} \]

### Final Result
The ratio of the electrostatic force to the gravitational force between two electrons is approximately:

\[ \frac{F_e}{F_g} \approx 4.17 \times 10^{42} \]