When the distance between two charged particles is doubled, the force between them becomes

Options

A. one-fourth
B. half
C. double
D. four times

Answer

To solve the problem of how the force between two charged particles changes when the distance between them is doubled, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Electrostatic Force**:
The force \( F \) between two charged particles is given by Coulomb's Law:
\[ F = k \frac{Q_1 Q_2}{r^2} \]
where \( k \) is the electrostatic constant, \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the charges.

2. **Initial Conditions**:
Let's denote the initial distance between the charges as \( r \). Therefore, the initial force \( F_1 \) can be expressed as:
\[ F_1 = k \frac{Q_1 Q_2}{r^2} \]

3. **Change the Distance**:
If the distance between the two charges is doubled, the new distance becomes \( 2r \).

4. **Calculate the New Force**:
Using the new distance, the new force \( F_2 \) can be calculated as:
\[ F_2 = k \frac{Q_1 Q_2}{(2r)^2} \]

5. **Simplify the New Force**:
Simplifying \( F_2 \):
\[ F_2 = k \frac{Q_1 Q_2}{4r^2} \]

6. **Relate the New Force to the Initial Force**:
We can express \( F_2 \) in terms of \( F_1 \):
\[ F_2 = \frac{1}{4} \left( k \frac{Q_1 Q_2}{r^2} \right) = \frac{1}{4} F_1 \]

7. **Conclusion**:
Therefore, when the distance between the two charged particles is doubled, the force between them becomes:
\[ F_2 = \frac{1}{4} F_1 \]

### Final Answer:
The force between the two charged particles becomes one-fourth of the initial force.

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