The force between two electrons separated by a distance r is proportional to

Options

A. `r^(3)`
B. `r^(1//2)`
C. `r^(-1//3)`
D. `r^(-2)`

Answer

To determine the relationship of the force between two electrons separated by a distance \( r \), we can use Coulomb's law, which describes the electrostatic force between two charged particles. Here are the steps to derive the proportionality:

### Step-by-Step Solution:

1. **Identify the Charges**:
Each electron has a charge of \( -e \), where \( e \) is the elementary charge (approximately \( 1.6 \times 10^{-19} \) coulombs).

2. **Apply Coulomb's Law**:
According to Coulomb's law, the electrostatic force \( F \) between two point charges is given by the formula:
\[ F = k \frac{|Q_1 Q_2|}{r^2} \]
where \( k \) is Coulomb's constant (\( k \approx 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)), \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the charges.

3. **Substitute the Charges**:
For two electrons, we substitute \( Q_1 = -e \) and \( Q_2 = -e \):
\[ F = k \frac{|-e \cdot -e|}{r^2} = k \frac{e^2}{r^2} \]

4. **Determine Proportionality**:
From the equation \( F = k \frac{e^2}{r^2} \), we can see that the force \( F \) is proportional to \( \frac{1}{r^2} \):
\[ F \propto \frac{1}{r^2} \]

5. **Conclusion**:
Therefore, the force between two electrons separated by a distance \( r \) is proportional to \( \frac{1}{r^2} \).

### Final Answer:
The force between two electrons separated by a distance \( r \) is proportional to \( \frac{1}{r^2} \).

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