Electric field on the axis of a small electric dipole at a distance r is `vec(E)_(1)` and `vec(E)_(2)` at a distance of `2r` on a line of perpendicular bisector is

Options

A. `vec(E)_(2)=-vec(E)_(1)//8`
B. `vec(E_(2))=-vec(E)_(1)//16`
C. `vec(E_(2))=-vec(E)_(1)//14`
D. `vec(E_(2))=-vec(E)_(1)//8`

Explanation

To solve the problem of finding the electric fields \( \vec{E}_1 \) and \( \vec{E}_2 \) at specified distances from a small electric dipole, we can follow these steps:

### Step 1: Understand the Electric Dipole
An electric dipole consists of two equal and opposite charges separated by a small distance. The dipole moment \( \vec{p} \) is defined as \( \vec{p} = q \cdot \vec{d} \), where \( q \) is the charge and \( \vec{d} \) is the separation vector.

### Step 2: Electric Field on the Axis of the Dipole
The electric field \( \vec{E}_1 \) on the axis of a dipole at a distance \( r \) from the center of the dipole is given by the formula:
\[ \vec{E}_1 = \frac{2kp}{r^3} \hat{i} \]
where \( k \) is Coulomb's constant, \( p \) is the dipole moment, and \( \hat{i} \) indicates the direction along the axis of the dipole.

### Step 3: Electric Field on the Perpendicular Bisector
The electric field \( \vec{E}_2 \) at a distance \( 2r \) on the perpendicular bisector of the dipole is given by:
\[ \vec{E}_2 = \frac{kp}{(2r)^3} (-\hat{i}) = \frac{kp}{8r^3} (-\hat{i}) \]
Here, the negative sign indicates that the direction of the electric field is opposite to the direction of the dipole moment.

### Step 4: Summary of Results
Thus, we have:
- The electric field on the axis at distance \( r \):
\[ \vec{E}_1 = \frac{2kp}{r^3} \hat{i} \]

- The electric field on the perpendicular bisector at distance \( 2r \):
\[ \vec{E}_2 = -\frac{kp}{8r^3} \hat{i} \]