Three charged particles are collinear and are in equilibrium, then

Options

A. all the charged particles have the same polarity
B. the equilibrium is unstable
C. all the charged particles cannot have the same polarity
D. Both (b) and (c ) are correct

Answer

To solve the problem of three charged particles being collinear and in equilibrium, we will analyze the conditions under which this equilibrium can occur.

### Step-by-Step Solution:

1. **Understanding Equilibrium**: For three charged particles to be in equilibrium, the net force acting on each particle must be zero. This means that the forces acting on each particle due to the other two must balance out.

2. **Analyzing Forces**: Consider three charged particles \( q_1, q_2, \) and \( q_3 \) placed at points A, B, and C respectively. The forces acting on each charge due to the other two charges must be equal in magnitude and opposite in direction.

3. **Charge Polarity Consideration**: If all three charges were of the same polarity (for example, all positive), they would repel each other. This would mean that no matter how they are positioned, they cannot be in equilibrium because the forces would not balance out. Therefore, not all charges can have the same polarity.

4. **Possible Charge Configurations**: The only way to achieve equilibrium is if at least one charge is of opposite polarity to the others. For example, if \( q_1 \) and \( q_3 \) are positive and \( q_2 \) is negative, the repulsive forces between \( q_1 \) and \( q_3 \) can be balanced by the attractive force between \( q_2 \) and either \( q_1 \) or \( q_3 \).

5. **Stable vs. Unstable Equilibrium**: In electrostatics, if a charged particle is in equilibrium, it is typically in an unstable equilibrium. This means that if the particle is slightly displaced, it will not return to its original position. Instead, it will move away from the equilibrium point.

6. **Conclusion**: Based on the analysis:
- **Option A** (All charges have the same polarity) is incorrect.
- **Option B** (Equilibrium is unstable) is correct.
- **Option C** (Not all charges have the same polarity) is correct.
- **Option D** (Both B and C are correct) is the correct answer.

### Final Answer:
The correct option is D: Both B and C are correct.