The ratio of gravitational force and electrostatic repulsive force between two electrons is approximately (gravitational constant
`=6.7xx10^(-11)Nm^2//kg^2`, mass of an electron `=9.1xx10^(-31)kg`, charge on an electron `= 1.6 xx 10^(–19) C)`

Options

A. `24 xx 10^(–24)`
B. `24xx10^(-36)`
C. `24xx10^(-44)`
D. `24xx10^(-54)`

Explanation

To find the ratio of the gravitational force to the electrostatic repulsive force between two electrons, we can follow these steps:

### Step 1: Write the formulas for gravitational and electrostatic forces.

The gravitational force \( F_g \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:

\[ F_g = \frac{G m_1 m_2}{r^2} \]

The electrostatic force \( F_e \) between two charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by:

\[ F_e = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2} \]

### Step 2: Set up the ratio of gravitational force to electrostatic force.

To find the ratio \( \frac{F_g}{F_e} \), we can divide the gravitational force by the electrostatic force:

\[ \frac{F_g}{F_e} = \frac{\frac{G m_1 m_2}{r^2}}{\frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2}} \]

### Step 3: Simplify the expression.

The \( r^2 \) terms cancel out:

\[ \frac{F_g}{F_e} = \frac{G m_1 m_2}{\frac{1}{4 \pi \epsilon_0} q_1 q_2} \]

This can be rewritten as:

\[ \frac{F_g}{F_e} = \frac{G m_1 m_2 \cdot 4 \pi \epsilon_0}{q_1 q_2} \]

### Step 4: Substitute the values.

Now, we substitute the known values:

- Gravitational constant \( G = 6.7 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \)
- Mass of an electron \( m_1 = m_2 = 9.1 \times 10^{-31} \, \text{kg} \)
- Charge of an electron \( q_1 = q_2 = 1.6 \times 10^{-19} \, \text{C} \)
- The value of \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)

Substituting these values into the equation:

\[ \frac{F_g}{F_e} = \frac{(6.7 \times 10^{-11}) (9.1 \times 10^{-31})^2 \cdot 4 \pi (8.85 \times 10^{-12})}{(1.6 \times 10^{-19})^2} \]

### Step 5: Calculate the values.

Calculating the numerator:

\[ G m_1 m_2 = 6.7 \times 10^{-11} \times (9.1 \times 10^{-31})^2 \approx 6.7 \times 10^{-11} \times 8.27 \times 10^{-61} \approx 5.55 \times 10^{-71} \]

Now multiply by \( 4 \pi \epsilon_0 \):

\[ 4 \pi \epsilon_0 \approx 4 \times 3.14 \times 8.85 \times 10^{-12} \approx 1.11 \times 10^{-10} \]

So, the numerator becomes:

\[ 5.55 \times 10^{-71} \times 1.11 \times 10^{-10} \approx 6.17 \times 10^{-81} \]

Now, calculate the denominator:

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

### Step 6: Final calculation of the ratio.

Now, we can find the ratio:

\[ \frac{F_g}{F_e} = \frac{6.17 \times 10^{-81}}{2.56 \times 10^{-38}} \approx 2.41 \times 10^{-43} \]

### Conclusion

Thus, the ratio of the gravitational force to the electrostatic force between two electrons is approximately:

\[ \frac{F_g}{F_e} \approx 2.41 \times 10^{-43} \]