A fixed ring of radius R is given a charge q. A charge `q_0` is brought slowly from infinity to the centre of the ring. The wrong statement is-

Options

A. work done by external agent in bringing `q_(0)` is `1/(4pi epsilon_(0)). (q q_(0))/R` when charge distribution is uniform
B. work done by external agent in bringing `q_(c)` is `1/(4piepsilon_(0)) (q q_(0))/R`
C. force on `q_(0)` at the centre is zero, when charge distribution is uniform
D. force on `q_(0)` at the central is zero, when one half of the ring has positive charge and other half has negative charge

Explanation

To solve the problem, we need to analyze the statements regarding the work done by an external agent when bringing a charge \( q_0 \) to the center of a charged ring and the forces acting on \( q_0 \) at that position.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a fixed ring of radius \( R \) with a total charge \( q \) distributed uniformly along its circumference.
- A charge \( q_0 \) is being brought from infinity to the center of the ring.

2. **Statement 1 Analysis**:
- The work done by an external agent in bringing \( q_0 \) from infinity to the center of the ring can be calculated using the formula for electric potential energy.
- The electric field \( E \) at the center of the ring due to the charge \( q \) is zero because the contributions from all parts of the ring cancel out.
- Therefore, the work done \( W \) is given by:
\[ W = \int_{\infty}^{0} F \cdot dr \]
- Since \( E = 0 \) at the center, the force \( F \) on \( q_0 \) is also zero, leading to:
\[ W = 0 \]
- Thus, Statement 1 is **true**.

3. **Statement 2 Analysis**:
- This statement suggests that the work done is \( \frac{1}{4 \pi \epsilon_0} \cdot \frac{q q_0}{R} \).
- This expression is incorrect because, as established earlier, the work done in bringing \( q_0 \) to the center is zero.
- Hence, Statement 2 is **false**.

4. **Statement 3 Analysis**:
- This statement claims that the force on \( q_0 \) at the center is zero when the charge distribution is uniform.
- As previously discussed, the electric field at the center of a uniformly charged ring is indeed zero, leading to zero force on \( q_0 \).
- Therefore, Statement 3 is **true**.

5. **Statement 4 Analysis**:
- This statement claims that the force on \( q_0 \) at the center is zero when one half of the ring has positive charge and the other half has negative charge.
- If one half of the ring has positive charge and the other half has negative charge, the forces exerted by the positive and negative charges on \( q_0 \) will also cancel each other out.
- Thus, the force on \( q_0 \) at the center will still be zero, making Statement 4 **true**.

### Conclusion:
The wrong statement among the options provided is **Statement 2**.