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 \(d\). The dipole moment \(\vec{p}\) is defined as:
\[ \vec{p} = Q \cdot \vec{d} \]
where \(\vec{d}\) is the vector pointing from the negative charge to the positive charge.

**Hint:** Remember that the dipole moment points from the negative charge to the positive charge.

### Step 2: Identify the Point on the Equatorial Line
The equatorial line of the dipole is the line that is perpendicular to the dipole axis and bisects the dipole. Let’s denote a point \(P\) on this equatorial line, which is equidistant from both charges.

**Hint:** The equatorial line is perpendicular to the line joining the two charges.

### Step 3: Calculate the Electric Field Due to Each Charge
The electric field due to a point charge \(Q\) at a distance \(r\) is given by:
\[ E = \frac{k \cdot |Q|}{r^2} \]
where \(k\) is Coulomb's constant. For the dipole, the distance from point \(P\) to each charge is the same, and we can denote this distance as \(r\).

**Hint:** Use the formula for electric field due to a point charge to find the contributions from both charges.

### Step 4: Determine the Direction of the Electric Fields
The electric field due to the positive charge \(+Q\) at point \(P\) will point away from the charge, while the electric field due to the negative charge \(-Q\) will point towards the charge.

- The electric field due to \(+Q\) at point \(P\) will be directed away from \(+Q\).
- The electric field due to \(-Q\) at point \(P\) will be directed towards \(-Q\).

**Hint:** Visualize the direction of the electric fields due to both charges to find the net electric field.

### Step 5: Calculate the Net Electric Field at Point \(P\)
Since the electric fields due to both charges are in opposite directions, we can find the net electric field \(\vec{E}\) at point \(P\) by subtracting the magnitudes of the electric fields:
\[ \vec{E} = E_{+Q} - E_{-Q} \]
However, since we are on the equatorial line, the resultant electric field will be directed along the axis of the dipole but in the opposite direction to the dipole moment \(\vec{p}\).

**Hint:** Remember that the net electric field on the equatorial line is directed opposite to the dipole moment.

### Step 6: Conclusion
Thus, the electric field intensity \(\vec{E}\) at a point on the equatorial line of the dipole is:
\[ \vec{E} \propto -\frac{1}{r^3} \quad \text{(in the direction opposite to } \vec{p}\text{)} \]
This indicates that the electric field is parallel to the axis of the dipole but opposite to the direction of the dipole moment.

**Final Answer:** The electric field intensity \(\vec{E}\) due to an electric dipole at a point on the equatorial line is directed parallel to the axis of the dipole and opposite to the direction of the dipole moment \(\vec{p}\).