In which of the following cases the electric field at the centre is not zero ?

Options

A.
B.
C.
D.

Explanation

To determine in which case the electric field at the center is not zero, we will analyze the four given cases of charge arrangements step by step.

### Step-by-Step Solution:

1. **Understanding Electric Field Contribution**:
- The electric field \( E \) at a point due to a charge \( Q \) is given by the formula:
\[ E = \frac{k \cdot |Q|}{r^2} \]
where \( k \) is Coulomb's constant and \( r \) is the distance from the charge to the point where the field is being calculated.

2. **Analyzing Each Case**:
- We will analyze each of the four cases to see if the contributions to the electric field at the center cancel out or not.

3. **Case A**:
- Charges: \( +q \) and \( -q \) at equal distances from the center.
- The electric fields due to \( +q \) and \( -q \) will be equal in magnitude but opposite in direction, leading to:
\[ E_{\text{net}} = E_{+q} + E_{-q} = 0 \]
- **Conclusion**: Electric field at the center is zero.

4. **Case B**:
- Arrangement: A combination of positive and negative charges.
- The electric field contributions from the charges will not be symmetric. Some charges will attract while others will repel.
- The net electric field will not cancel out completely due to the asymmetrical arrangement of charges.
- **Conclusion**: Electric field at the center is not zero.

5. **Case C**:
- Charges: All negative charges \( -q, -q, -q \) and one positive charge \( +q_0 \).
- The electric fields due to the negative charges will attract the positive charge towards them. Since they are symmetrically placed, their contributions will cancel out.
- **Conclusion**: Electric field at the center is zero.

6. **Case D**:
- Charges: All positive charges \( +q \).
- The electric fields due to each charge will repel the test charge \( +q_0 \) in the same direction. Since they are symmetrically placed, their contributions will also cancel out.
- **Conclusion**: Electric field at the center is zero.

### Final Answer:
- The case in which the electric field at the center is not zero is **Case B**.