Two point charges `+q` and `-q` are placed a distance x apart. A third charge is so placed that al the three charges are in equilibrium. Then

Options

A. uknknown charge is `-4q//9`
B. unknown charge is `-9q//4`
C. it should be at `(x//3) `from smaller chargge between them
D. none of these

Explanation

To solve the problem of placing a third charge such that all three charges are in equilibrium, we need to analyze the forces acting on the third charge due to the other two charges.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two point charges: \( +q \) and \( -q \) placed a distance \( x \) apart.
- We need to place a third charge \( Q \) such that all three charges are in equilibrium.

2. **Identifying Possible Positions for the Third Charge**:
- The third charge cannot be placed between the two charges because:
- If placed between \( +q \) and \( -q \), the repulsive force from \( +q \) and the attractive force from \( -q \) will not balance out.
- Therefore, the third charge must be placed outside the segment defined by \( +q \) and \( -q \), either to the left of \( +q \) or to the right of \( -q \).

3. **Analyzing Forces**:
- Let’s denote the position of \( +q \) as point A, \( -q \) as point B, and the third charge \( Q \) can be placed at point C either to the left of A or to the right of B.
- If \( Q \) is placed to the left of \( +q \), it will experience:
- An attractive force towards \( +q \) (if \( Q \) is negative).
- A repulsive force away from \( -q \) (if \( Q \) is negative).
- If \( Q \) is placed to the right of \( -q \), it will experience:
- An attractive force towards \( -q \) (if \( Q \) is negative).
- A repulsive force away from \( +q \) (if \( Q \) is negative).

4. **Setting Up the Equilibrium Condition**:
- For equilibrium, the net force acting on charge \( Q \) must be zero.
- This means the attractive force from one charge must equal the repulsive force from the other charge.

5. **Mathematical Expression for Forces**:
- The force between two charges is given by Coulomb's Law:
\[ F = k \frac{|q_1 \cdot q_2|}{r^2} \]
- Where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them.

6. **Conclusion**:
- After analyzing the forces, it is clear that no matter how we position the third charge \( Q \) (either to the left of \( +q \) or to the right of \( -q \)), the forces will not balance out due to the nature of the charges involved.
- Therefore, it is concluded that it is impossible to achieve equilibrium with the given configuration of charges.

### Final Answer:
None of the options provided will allow for equilibrium; hence, the correct answer is **D: None of these**.