A and B are two points on the axis and the perpendicular bisector, reapectively, of and electric dipole. A and B are far away from the dipole and at equal distance from it. The fields at A and B are `vecE_(A)` and `vecE_(B)`. Then

Options

A. `vecE_(A) = vecE_(B)`
B. `vecE_(A) = 2vecE_(B)`
C. `vecE_(A)= -2vecE_(B)`
D. `|vecE_(B)=(1)/(2)|vecE_(A)|`, and `vecE_(A)` is perpendicular to `vecE_(B)`.

Explanation

To solve the problem, we need to analyze the electric fields at points A and B due to an electric dipole. Let's break down the solution step by step.

### Step 1: Understanding Electric Dipole and Points A and B
An electric dipole consists of two equal and opposite charges separated by a small distance. The electric field due to a dipole varies with distance and direction.

- Point A is located on the axis of the dipole.
- Point B is located on the perpendicular bisector of the dipole.
- Both points A and B are at equal distances from the dipole.

### Step 2: Electric Field at Point A
The electric field \( \vec{E}_A \) at point A, which is on the axis of the dipole, can be expressed as:
\[ \vec{E}_A = \frac{2kp}{r^3} \hat{r} \]
where:
- \( k \) is the Coulomb's constant,
- \( p \) is the dipole moment,
- \( r \) is the distance from the center of the dipole to point A,
- \( \hat{r} \) is the unit vector in the direction from the dipole to point A.

### Step 3: Electric Field at Point B
The electric field \( \vec{E}_B \) at point B, which is on the perpendicular bisector of the dipole, can be expressed as:
\[ \vec{E}_B = \frac{kp}{r^3} (-\hat{r}) \]
where:
- The negative sign indicates that the direction of the electric field at point B is opposite to the direction of the dipole moment.

### Step 4: Comparing the Magnitudes and Directions
- The magnitude of \( \vec{E}_A \) is greater than that of \( \vec{E}_B \) because:
\[ |\vec{E}_A| = \frac{2kp}{r^3}, \quad |\vec{E}_B| = \frac{kp}{r^3} \]
- The direction of \( \vec{E}_A \) is along the dipole moment, while \( \vec{E}_B \) is in the opposite direction.

### Step 5: Final Relationship Between \( \vec{E}_A \) and \( \vec{E}_B \)
Since \( \vec{E}_B \) is in the opposite direction of \( \vec{E}_A \), we can express the relationship as:
\[ \vec{E}_A = -2 \vec{E}_B \]

### Conclusion
Thus, the relationship between the electric fields at points A and B is:
\[ \vec{E}_A = -2 \vec{E}_B \]

### Final Answer
The correct answer is \( \vec{E}_A = -2 \vec{E}_B \).

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