The electric field at a point on the equatorial plane at a distance r from the centre of a dipole having dipole moment `vectp` is given by (r >> seperation of two charges forming dipole , `epsilon_0` = permittivity of free space

Options

A. `vecE = vecP /4pi epsilon_0 r^3`
B. `vecE = 2 vecP /4pi epsilon_0 r^3`
C. `vecE = - vecP /4pi epsilon_0 r^2`
D. `vecE = - vecP /4pi epsilon_0 r^3`

Answer

To find the electric field at a point on the equatorial plane of a dipole, we can follow these steps:

### Step 1: Understand the Dipole Moment
The dipole moment \( \vec{p} \) is defined as:
\[ \vec{p} = q \cdot \vec{d} \]
where \( q \) is the magnitude of one of the charges and \( \vec{d} \) is the vector pointing from the negative charge to the positive charge.

### Step 2: Identify the Position on the Equatorial Plane
In the equatorial plane of a dipole, the angle \( \theta \) is \( 90^\circ \). Thus, the electric field will be symmetrical about the dipole axis.

### Step 3: Electric Field Expression
The electric field \( \vec{E} \) at a distance \( r \) from the center of the dipole on the equatorial plane can be derived from the general expression for the electric field due to a dipole:
\[ \vec{E} = \frac{1}{4\pi \epsilon_0} \cdot \frac{2\vec{p}}{r^3} \]
This expression is valid for points where \( r \) is much greater than the separation of the charges in the dipole.

### Step 4: Substitute Values
Substituting the dipole moment \( \vec{p} \) into the equation, we get:
\[ \vec{E} = \frac{1}{4\pi \epsilon_0} \cdot \frac{2p}{r^3} \]

### Step 5: Final Expression
Thus, the electric field at a point on the equatorial plane at a distance \( r \) from the center of the dipole is given by:
\[ \vec{E} = \frac{p}{2\pi \epsilon_0 r^3} \]