A charge q is placed at the centre of the line joining two equal charges Q. The system of the three charges will be in equilibrium if q is equal to:

Options

A. (a) `-Q/2`
B. (b) `-Q/4`
C. (c) `+Q/4`
D. (d) `+Q/2`

Explanation

To solve the problem, we need to find the charge \( q \) that will allow the system of three charges (two equal charges \( Q \) and one charge \( q \) at the center) to be in equilibrium.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have two equal charges \( Q \) placed at points A and B, and a charge \( q \) placed at the midpoint (let's call it point C) between A and B.
- The distance between charges A and B is \( d \). Therefore, the distance from each charge \( Q \) to charge \( q \) is \( \frac{d}{2} \).

2. **Forces Acting on Charge \( q \)**:
- The charge \( q \) experiences forces due to both charges \( Q \).
- The force exerted by charge \( Q \) at point A on charge \( q \) is given by Coulomb's law:
\[ F_A = k \frac{Q \cdot q}{\left(\frac{d}{2}\right)^2} = k \frac{4Qq}{d^2} \]
- Similarly, the force exerted by charge \( Q \) at point B on charge \( q \) is:
\[ F_B = k \frac{Q \cdot q}{\left(\frac{d}{2}\right)^2} = k \frac{4Qq}{d^2} \]

3. **Direction of Forces**:
- Both forces \( F_A \) and \( F_B \) will act in opposite directions.
- If \( q \) is positive, both forces will repel \( q \) away from the charges \( Q \). If \( q \) is negative, both forces will attract \( q \) towards the charges \( Q \).

4. **Condition for Equilibrium**:
- For the system to be in equilibrium, the net force acting on charge \( q \) must be zero.
- Since both forces \( F_A \) and \( F_B \) are equal in magnitude and opposite in direction, we can set their magnitudes equal:
\[ F_A = F_B \]
- This condition is satisfied when the charge \( q \) is such that the forces balance each other out.

5. **Finding the Value of \( q \)**:
- For the forces to balance, we need to consider the nature of the charges. If both \( Q \) are positive, \( q \) must be negative to attract towards them. Conversely, if both \( Q \) are negative, \( q \) must be positive.
- The equilibrium condition can be expressed as:
\[ k \frac{4Qq}{d^2} = k \frac{4Qq}{d^2} \]
- This simplifies to \( q = -Q \) if \( Q \) is positive, or \( q = Q \) if \( Q \) is negative.

### Final Result:
Thus, the charge \( q \) must be equal to \( -Q \) for the system to be in equilibrium.