`F_(g)` and `F_(e)` represent gravitational and electrostatic force respectively between electrons situated at a distance 10 cm. The ratio of `F_(g)//F_(e)` is of the order of

Options

A. `10^(42)`
B. `10^(-21)`
C. `10^(24)`
D. `10^(-43)`

Answer

To find the ratio of the gravitational force \( F_g \) to the electrostatic force \( F_e \) between two electrons situated at a distance of 10 cm, we can follow these steps:

### Step 1: Write the formula for gravitational force
The gravitational force \( F_g \) between two masses is given by Newton's law of gravitation:
\[ F_g = \frac{G m_1 m_2}{r^2} \]
For two electrons, \( m_1 = m_2 = m_e \) (mass of an electron), and the distance \( r = 0.1 \, \text{m} \) (10 cm converted to meters). Therefore:
\[ F_g = \frac{G m_e^2}{r^2} \]

### Step 2: Substitute the values for gravitational force
Using the universal gravitational constant \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \) and the mass of an electron \( m_e = 9.11 \times 10^{-31} \, \text{kg} \):
\[ F_g = \frac{(6.67 \times 10^{-11}) (9.11 \times 10^{-31})^2}{(0.1)^2} \]
Calculating \( (9.11 \times 10^{-31})^2 \):
\[ (9.11 \times 10^{-31})^2 = 8.27 \times 10^{-61} \]
Now substituting this back:
\[ F_g = \frac{(6.67 \times 10^{-11}) (8.27 \times 10^{-61})}{0.01} = \frac{5.52 \times 10^{-71}}{0.01} = 5.52 \times 10^{-69} \, \text{N} \]

### Step 3: Write the formula for electrostatic force
The electrostatic force \( F_e \) between two charges is given by Coulomb's law:
\[ F_e = \frac{k |q_1 q_2|}{r^2} \]
For two electrons, \( q_1 = q_2 = e \) (charge of an electron), and the distance \( r = 0.1 \, \text{m} \):
\[ F_e = \frac{k e^2}{r^2} \]
Where \( k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \) and \( e = 1.6 \times 10^{-19} \, \text{C} \).

### Step 4: Substitute the values for electrostatic force
Calculating \( e^2 \):
\[ e^2 = (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \]
Now substituting this back:
\[ F_e = \frac{(9 \times 10^9) (2.56 \times 10^{-38})}{(0.1)^2} = \frac{(9 \times 10^9) (2.56 \times 10^{-38})}{0.01} = 2.304 \times 10^{-26} \, \text{N} \]

### Step 5: Calculate the ratio \( \frac{F_g}{F_e} \)
Now we can find the ratio:
\[ \frac{F_g}{F_e} = \frac{5.52 \times 10^{-69}}{2.304 \times 10^{-26}} \]
Calculating this gives:
\[ \frac{F_g}{F_e} = 2.40 \times 10^{-43} \]

### Conclusion
Thus, the ratio of the gravitational force to the electrostatic force between two electrons is:
\[ \frac{F_g}{F_e} \approx 2.4 \times 10^{-43} \]