Which of the following statement(s) is(are) correct ?

Options

A. If the electric field due to a point charge varies as `r^(-2 5)` instead of `r^(-2)` , then the Gauss' law will still be vaild
B. The Gauss' law can be used to calculate the field distribution around an electric dipole .
C. If the electric field between two point charges is zero somewhere , then the sign of the two charges is the same
D. The work done by the extermal force is moving a unit positive chagre form potential `V_(A)` to point B at potential `V_(B) ` is `(V_(B) - V_(A))`

Explanation

To determine which of the statements is correct, we will analyze each statement one by one based on the principles of electrostatics and Gauss's law.

### Step-by-Step Solution:

**Statement 1:** "If there is an electric field due to some point charge, it will vary with r to the power of -25 instead of r to the power of -2."

1. **Understanding Electric Field of a Point Charge:**
The electric field \( E \) due to a point charge \( Q \) is given by the formula:
\[ E = \frac{kQ}{r^2} \]
where \( k \) is Coulomb's constant and \( r \) is the distance from the charge.

2. **Analyzing the Statement:**
The statement suggests that the electric field varies as \( r^{-25} \). This is incorrect because the electric field due to a point charge does not vary as \( r^{-25} \); it varies as \( r^{-2} \).

3. **Conclusion for Statement 1:**
Since the electric field does not follow the proposed relationship, **Statement 1 is incorrect**.

---

**Statement 2:** "Gauss's law can be used to calculate the field distribution around an electric dipole."

1. **Understanding Gauss's Law:**
Gauss's law states that the electric flux through a closed surface is proportional to the charge enclosed by that surface:
\[ \Phi_E = \frac{Q_{\text{enc}}}{\epsilon_0} \]

2. **Analyzing the Electric Field of a Dipole:**
An electric dipole consists of two equal and opposite charges. The electric field produced by a dipole is not symmetric and varies with direction, making it complex.

3. **Conclusion for Statement 2:**
Gauss's law is typically applied to systems with high symmetry (spherical, cylindrical). Since the electric field around a dipole is not symmetric, **Statement 2 is incorrect**.

---

**Statement 3:** "If the electric field between two charges is zero somewhere, then the sign of the two charges is the same."

1. **Understanding Electric Field Zero Condition:**
For two charges \( Q_1 \) and \( Q_2 \), the electric field can be zero at a point if the magnitudes of the electric fields due to both charges at that point are equal and opposite.

2. **Analyzing Charge Signs:**
- If \( Q_1 \) is positive and \( Q_2 \) is negative, there can be a point where the electric fields cancel each other out.
- If both charges are of the same sign (both positive or both negative), the electric field will never be zero between them.

3. **Conclusion for Statement 3:**
The statement is incorrect; the electric field can be zero if one charge is positive and the other is negative, thus **Statement 3 is incorrect**.

---

**Statement 4:** "Work done by the external force in moving a unit positive charge from a potential point \( V_A \) to a point \( V_B \)."

1. **Understanding Work Done in Electric Field:**
The work done \( W \) in moving a charge \( q \) in an electric field from a potential \( V_A \) to \( V_B \) is given by:
\[ W = q(V_B - V_A) \]
For a unit positive charge (\( q = 1 \)):
\[ W = V_B - V_A \]

2. **Conclusion for Statement 4:**
This statement is valid as it correctly describes the work done in terms of electric potential. Therefore, **Statement 4 is correct**.

### Final Conclusion:
- **Correct Statement:** Statement 4
- **Incorrect Statements:** Statements 1, 2, and 3

---