An electric charged q exerts a force F on a similar electric charge q swparated by a distance r. A third charge `q//4` is placed midway between the two charges. Now the force F will

Options

A. `become F//3`
B. `become F//9`
C. `become F//27`
D. remain F

Explanation

To solve the problem, we need to analyze the forces acting on the third charge \( \frac{q}{4} \) placed midway between two identical charges \( q \) separated by a distance \( r \).

### Step-by-Step Solution:

1. **Identify the Setup**:
We have two charges \( q \) and \( q \) separated by a distance \( r \). A third charge \( \frac{q}{4} \) is placed exactly in the middle of these two charges.

2. **Determine the Distance**:
The distance from each charge \( q \) to the charge \( \frac{q}{4} \) is \( \frac{r}{2} \).

3. **Calculate the Force Between Charges**:
The force \( F \) between the two charges \( q \) is given by Coulomb's law:
\[ F = k \frac{q^2}{r^2} \]
where \( k \) is Coulomb's constant.

4. **Calculate the Forces Acting on \( \frac{q}{4} \)**:
- The force exerted on \( \frac{q}{4} \) by the left charge \( q \):
\[ F_1 = k \frac{q \cdot \frac{q}{4}}{\left(\frac{r}{2}\right)^2} = k \frac{q \cdot \frac{q}{4}}{\frac{r^2}{4}} = k \frac{q^2}{r^2} \]
- The force exerted on \( \frac{q}{4} \) by the right charge \( q \):
\[ F_2 = k \frac{q \cdot \frac{q}{4}}{\left(\frac{r}{2}\right)^2} = k \frac{q \cdot \frac{q}{4}}{\frac{r^2}{4}} = k \frac{q^2}{r^2} \]

5. **Direction of Forces**:
Both forces \( F_1 \) and \( F_2 \) will act in opposite directions due to the repulsive nature of like charges.

6. **Net Force on \( \frac{q}{4} \)**:
Since both forces are equal in magnitude but opposite in direction, the net force \( F_{net} \) acting on the charge \( \frac{q}{4} \) is:
\[ F_{net} = F_1 - F_2 = 0 \]
However, the original force \( F \) between the two charges \( q \) remains unchanged.

7. **Conclusion**:
The introduction of the charge \( \frac{q}{4} \) does not affect the force \( F \) between the two original charges \( q \) and \( q \). Therefore, the force \( F \) remains the same.

### Final Answer:
The force \( F \) will remain \( F \).

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