What is the angle between the directions of electric field due to an electric dipole and its dipole moment at any :
(i) Axial p oint . (ii) Equatorial point .

Explanation

To solve the problem of finding the angle between the directions of the electric field due to an electric dipole and its dipole moment at axial and equatorial points, we can follow these steps:

### Step-by-Step Solution:

**Step 1: Understand the Configuration of the Dipole**
- An electric dipole consists of two equal and opposite charges, +q and -q, separated by a distance (2a). The dipole moment (p) is defined as \( p = q \cdot 2a \) and points from the negative charge to the positive charge.

**Step 2: Analyze the Axial Point**
- An axial point is located along the line extending from the positive charge to the negative charge. Let's denote this point as point A.
- At this point, the electric field due to the dipole can be calculated using the formula:
\[ E_{\text{axial}} = \frac{1}{4\pi \epsilon_0} \cdot \frac{2p}{r^3} \]
where \( r \) is the distance from the center of the dipole to the axial point.

**Step 3: Determine the Direction of the Electric Field at the Axial Point**
- The electric field at the axial point points in the same direction as the dipole moment, which is from the negative charge (-q) to the positive charge (+q).

**Step 4: Calculate the Angle at the Axial Point**
- Since both the electric field and the dipole moment are in the same direction, the angle between them is:
\[ \theta_{\text{axial}} = 0^\circ \]

**Step 5: Analyze the Equatorial Point**
- An equatorial point is located on the perpendicular bisector of the dipole. Let's denote this point as point B.
- At this point, the electric field due to the dipole can be calculated using the formula:
\[ E_{\text{equatorial}} = \frac{1}{4\pi \epsilon_0} \cdot \frac{p}{r^3} \]
where \( r \) is the distance from the center of the dipole to the equatorial point.

**Step 6: Determine the Direction of the Electric Field at the Equatorial Point**
- The electric field at the equatorial point points in the opposite direction to the dipole moment. The electric field due to the positive charge points away from it, and the electric field due to the negative charge points towards it. The net electric field at this point is directed opposite to the dipole moment.

**Step 7: Calculate the Angle at the Equatorial Point**
- Since the electric field is directed opposite to the dipole moment, the angle between them is:
\[ \theta_{\text{equatorial}} = 180^\circ \]

### Final Answers:
- (i) The angle between the electric field and the dipole moment at the axial point is \( 0^\circ \).
- (ii) The angle between the electric field and the dipole moment at the equatorial point is \( 180^\circ \).