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

Options

A. unknown 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 the three charges \( +q \), \( -q \), and the third charge are in equilibrium, we can follow these steps:

### Step-by-Step Solution

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

2. **Identifying Forces**:
- The charge \( +q \) will exert a repulsive force on \( Q \) if \( Q \) is positive and an attractive force if \( Q \) is negative.
- The charge \( -q \) will exert an attractive force on \( Q \) if \( Q \) is positive and a repulsive force if \( Q \) is negative.

3. **Considering Possible Positions for \( Q \)**:
- **Case 1**: Placing \( Q \) to the left of \( +q \).
- **Case 2**: Placing \( Q \) to the right of \( -q \).
- **Case 3**: Placing \( Q \) between \( +q \) and \( -q \).

4. **Analyzing Case 1** (Left of \( +q \)):
- If \( Q \) is placed to the left of \( +q \), it will experience a repulsive force from \( +q \) and an attractive force from \( -q \). The forces will not balance because they act in opposite directions, leading to no equilibrium.

5. **Analyzing Case 2** (Right of \( -q \)):
- If \( Q \) is placed to the right of \( -q \), it will experience a repulsive force from \( -q \) and an attractive force from \( +q \). Again, the forces will not balance as they act in opposite directions, leading to no equilibrium.

6. **Analyzing Case 3** (Between \( +q \) and \( -q \)):
- If \( Q \) is placed between \( +q \) and \( -q \), \( +q \) will exert a repulsive force on \( Q \) and \( -q \) will exert an attractive force on \( Q \). The forces will act in the same direction, which means they cannot balance each other out, leading to no equilibrium.

7. **Conclusion**:
In all cases analyzed, it is impossible to place the third charge \( Q \) such that all three charges are in equilibrium. Therefore, the answer is that it is not possible to achieve equilibrium with the given configuration of charges.

### Final Answer:
The third charge cannot be placed in any position to achieve equilibrium with the two charges \( +q \) and \( -q \).