The electric field intensity `vec(E)` , , due to an electric dipole of dipole moment `vec(p)` , at a point on the equatorial line is :

Options

A. parallel to the axis of the dipole and opposite to the direction of dipole moment
B. perpendicular to the axis of the dipole and is directed away from it
C. parallel to the dipole moment
D. perpendicular to the axis of the dipole and is directed towards it

Explanation

To find the electric field intensity \( \vec{E} \) due to an electric dipole of dipole moment \( \vec{p} \) at a point on the equatorial line, we can follow these steps:

### 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 \( \vec{p} \) is defined as:
\[ \vec{p} = q \cdot 2a \]
The dipole moment points from the negative charge to the positive charge.

### Step 2: Identify the Equatorial Line
The equatorial line is the line that is perpendicular to the dipole axis and passes through the midpoint of the dipole. At any point on this line, the electric field contributions from both charges must be considered.

### Step 3: Calculate the Electric Field Contributions
At a point on the equatorial line, the electric field \( \vec{E} \) due to the positive charge \( +q \) points away from the charge, while the electric field due to the negative charge \( -q \) points towards the charge.

Let’s denote the distance from the midpoint of the dipole to the point on the equatorial line as \( r \). The electric field due to each charge at this point can be expressed as:
\[ E_{+} = \frac{k \cdot q}{(r^2 + a^2)} \quad \text{(due to +q)} \]
\[ E_{-} = \frac{k \cdot q}{(r^2 + a^2)} \quad \text{(due to -q)} \]
where \( k \) is Coulomb's constant.

### Step 4: Analyze the Direction of the Electric Fields
The electric field \( E_{+} \) due to the positive charge points away from it, while the electric field \( E_{-} \) due to the negative charge points towards it. The vertical components of these fields will cancel out, while the horizontal components will add up.

### Step 5: Determine the Resultant Electric Field
The resultant electric field \( \vec{E} \) on the equatorial line is directed opposite to the dipole moment \( \vec{p} \) and is parallel to the axis of the dipole. Therefore, we can conclude:
\[ \vec{E} \text{ is parallel to the axis of the dipole and opposite to the direction of } \vec{p}. \]

### Conclusion
Thus, the electric field intensity \( \vec{E} \) at a point on the equatorial line of the dipole is:
\[ \vec{E} \text{ is parallel to the axis of dipole and opposite to the direction of dipole moment.} \]