The force between two electrons when placed in air is equal to 0.5 times the weight of an electron. Find the distance between two electrons (mass of electron `= 9.1 xx 10^(-31)kg`)

Options

A. 7.2m
B. 72m
C. 72m.
D. 720m

Answer

To solve the problem, we need to find the distance between two electrons given that the electrostatic force between them is equal to 0.5 times the weight of an electron.

### Step-by-Step Solution:

1. **Identify the weight of an electron**:
The weight (W) of an electron can be calculated using the formula:
\[ W = m \cdot g \]
where \( m \) is the mass of the electron and \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)).
Given \( m = 9.1 \times 10^{-31} \, \text{kg} \):
\[ W = 9.1 \times 10^{-31} \, \text{kg} \times 10 \, \text{m/s}^2 = 9.1 \times 10^{-30} \, \text{N} \]

2. **Calculate the electrostatic force**:
According to the problem, the electrostatic force (F) between the two electrons is:
\[ F = 0.5 \times W = 0.5 \times 9.1 \times 10^{-30} \, \text{N} = 4.55 \times 10^{-30} \, \text{N} \]

3. **Use Coulomb's Law**:
The electrostatic force between two point charges (in this case, two electrons) is given by Coulomb's Law:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{r^2} \]
where:
- \( e \) is the charge of an electron, approximately \( 1.6 \times 10^{-19} \, \text{C} \)
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)

4. **Substituting values into Coulomb's Law**:
Rearranging the equation to solve for \( r \):
\[ r^2 = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{F} \]
Substituting in the known values:
\[ r^2 = \frac{1}{4 \pi (8.85 \times 10^{-12})} \frac{(1.6 \times 10^{-19})^2}{4.55 \times 10^{-30}} \]

5. **Calculating \( r^2 \)**:
First, calculate \( \frac{(1.6 \times 10^{-19})^2}{4.55 \times 10^{-30}} \):
\[ (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \]
\[ \frac{2.56 \times 10^{-38}}{4.55 \times 10^{-30}} \approx 5.62 \times 10^{-9} \]
Now calculate \( 4 \pi (8.85 \times 10^{-12}) \):
\[ 4 \pi (8.85 \times 10^{-12}) \approx 1.11 \times 10^{-10} \]
Therefore,
\[ r^2 = \frac{5.62 \times 10^{-9}}{1.11 \times 10^{-10}} \approx 50.63 \]

6. **Taking the square root**:
\[ r = \sqrt{50.63} \approx 7.115 \, \text{m} \]

7. **Final answer**:
Rounding this to one decimal place gives:
\[ r \approx 7.2 \, \text{m} \]

### Conclusion:
The distance between the two electrons is approximately **7.2 meters**.