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

Options

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

Answer

To determine how the force between two electrons separated by a distance \( r \) is proportional, we can use Coulomb's Law. Here’s a step-by-step solution:

### Step 1: Understand Coulomb's Law
Coulomb's Law states that the electric force \( F \) between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, it can be expressed as:

\[ F = k \frac{q_1 q_2}{r^2} \]

where:
- \( F \) is the force between the charges,
- \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),
- \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
- \( r \) is the distance between the charges.

### Step 2: Substitute the Charges
In this case, both charges are electrons, and the charge of an electron \( e \) is approximately \( 1.6 \times 10^{-19} \, \text{C} \). Thus, we can substitute \( q_1 \) and \( q_2 \) with \( e \):

\[ F = k \frac{e \cdot e}{r^2} = k \frac{e^2}{r^2} \]

### Step 3: Identify the Proportionality
From the equation \( F = k \frac{e^2}{r^2} \), we can see that the force \( F \) is proportional to \( \frac{1}{r^2} \). This means:

\[ F \propto \frac{1}{r^2} \]

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