`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 will use the formulas for both forces and then calculate the ratio step by step.

### 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:
\[ F_g = \frac{G m_1 m_2}{r^2} \]
where \( G \) is the gravitational constant.

The electrostatic force \( F_e \) between two charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by:
\[ F_e = \frac{k q_1 q_2}{r^2} \]
where \( k \) is Coulomb's constant.

### Step 2: Set up the ratio \( \frac{F_g}{F_e} \)

Now, we can set up the ratio of the gravitational force to the electrostatic force:
\[ \frac{F_g}{F_e} = \frac{\frac{G m_1 m_2}{r^2}}{\frac{k q_1 q_2}{r^2}} = \frac{G m_1 m_2}{k q_1 q_2} \]
The \( r^2 \) terms cancel out.

### Step 3: Substitute the values for electrons

For electrons:
- Mass of an electron \( m_e = 9.1 \times 10^{-31} \, \text{kg} \)
- Charge of an electron \( q_e = 1.6 \times 10^{-19} \, \text{C} \)

Now substituting these values into the ratio:
\[ \frac{F_g}{F_e} = \frac{G m_e m_e}{k q_e q_e} \]

### Step 4: Substitute the constants

Using the values:
- Gravitational constant \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)
- Coulomb's constant \( k = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)

We can now write:
\[ \frac{F_g}{F_e} = \frac{(6.67 \times 10^{-11}) (9.1 \times 10^{-31})^2}{(8.99 \times 10^9) (1.6 \times 10^{-19})^2} \]

### Step 5: Calculate the numerator and denominator

Calculating the numerator:
\[ \text{Numerator} = 6.67 \times 10^{-11} \times (9.1 \times 10^{-31})^2 = 6.67 \times 10^{-11} \times 8.27 \times 10^{-61} = 5.53 \times 10^{-71} \]

Calculating the denominator:
\[ \text{Denominator} = 8.99 \times 10^9 \times (1.6 \times 10^{-19})^2 = 8.99 \times 10^9 \times 2.56 \times 10^{-38} = 2.30 \times 10^{-28} \]

### Step 6: Calculate the ratio

Now, substituting the values back into the ratio:
\[ \frac{F_g}{F_e} = \frac{5.53 \times 10^{-71}}{2.30 \times 10^{-28}} \approx 2.40 \times 10^{-43} \]

### Conclusion

Thus, the ratio of the gravitational force to the electrostatic force between two electrons situated at a distance of 10 cm is of the order of:
\[ \frac{F_g}{F_e} \approx 10^{-43} \]