How far two electrons should be placed so that the force of repulsion between them is equal to the weight of an electron ?

Answer

To solve the problem of how far two electrons should be placed so that the force of repulsion between them equals the weight of an electron, we will follow these steps:

### Step 1: Understand the Forces Involved
The force of repulsion between two electrons is given by Coulomb's law, which states:
\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]
where:
- \( F \) is the electrostatic force,
- \( k \) is Coulomb's constant (\( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),
- \( q_1 \) and \( q_2 \) are the charges of the electrons (both equal to \( -1.6 \times 10^{-19} \, \text{C} \)),
- \( r \) is the distance between the charges.

The weight of an electron is given by:
\[ W = m \cdot g \]
where:
- \( m \) is the mass of the electron (\( 9.1 \times 10^{-31} \, \text{kg} \)),
- \( g \) is the acceleration due to gravity (\( 9.8 \, \text{m/s}^2 \)).

### Step 2: Set the Forces Equal
We need to set the electrostatic force equal to the weight of the electron:
\[ \frac{k \cdot |q_1 \cdot q_2|}{r^2} = m \cdot g \]

### Step 3: Substitute the Values
Substituting the known values into the equation:
- \( |q_1| = |q_2| = 1.6 \times 10^{-19} \, \text{C} \)
- \( m = 9.1 \times 10^{-31} \, \text{kg} \)
- \( g = 9.8 \, \text{m/s}^2 \)

The equation becomes:
\[ \frac{9 \times 10^9 \cdot (1.6 \times 10^{-19})^2}{r^2} = 9.1 \times 10^{-31} \cdot 9.8 \]

### Step 4: Calculate the Right Side
Calculating the weight of the electron:
\[ W = 9.1 \times 10^{-31} \cdot 9.8 = 8.918 \times 10^{-30} \, \text{N} \]

### Step 5: Calculate the Left Side
Calculating \( (1.6 \times 10^{-19})^2 \):
\[ (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \]

Now substitute this back into the equation:
\[ \frac{9 \times 10^9 \cdot 2.56 \times 10^{-38}}{r^2} = 8.918 \times 10^{-30} \]

### Step 6: Rearranging for \( r^2 \)
Rearranging gives:
\[ r^2 = \frac{9 \times 10^9 \cdot 2.56 \times 10^{-38}}{8.918 \times 10^{-30}} \]

### Step 7: Calculate \( r^2 \)
Calculating the numerator:
\[ 9 \times 10^9 \cdot 2.56 \times 10^{-38} = 2.304 \times 10^{-28} \]

Now divide by the weight:
\[ r^2 = \frac{2.304 \times 10^{-28}}{8.918 \times 10^{-30}} \approx 25.8 \]

### Step 8: Calculate \( r \)
Taking the square root gives:
\[ r = \sqrt{25.8} \approx 5.08 \, \text{m} \]

### Final Answer
Thus, the distance \( r \) that two electrons should be placed apart so that the force of repulsion between them equals the weight of an electron is approximately:
\[ r \approx 5.08 \, \text{m} \]

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