At a far away distance `r` along the axis from a electric dipole electric field is E. Find the electric field at distance `2r` along the perpendicular bisector.

Explanation

To find the electric field at a distance \(2r\) along the perpendicular bisector of an electric dipole, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Electric Dipole:**
An electric dipole consists of two equal and opposite charges, +q and -q, separated by a distance \(d\). The dipole moment \(p\) is defined as:
\[ p = q \cdot d \]

2. **Electric Field along the Axis:**
The electric field \(E\) at a distance \(r\) along the axis of the dipole is given by the formula:
\[ E = \frac{1}{4 \pi \epsilon_0} \cdot \frac{2p}{r^3} \]

3. **Electric Field along the Perpendicular Bisector:**
The electric field \(E'\) at a distance \(r\) along the perpendicular bisector of the dipole is given by:
\[ E' = \frac{1}{4 \pi \epsilon_0} \cdot \frac{p}{2r^3} \]

4. **Finding the Electric Field at Distance \(2r\):**
To find the electric field at a distance \(2r\) along the perpendicular bisector, we substitute \(2r\) into the formula for the electric field along the perpendicular bisector:
\[ E'' = \frac{1}{4 \pi \epsilon_0} \cdot \frac{p}{2(2r)^3} \]

5. **Simplifying the Expression:**
Simplifying the expression:
\[ E'' = \frac{1}{4 \pi \epsilon_0} \cdot \frac{p}{2 \cdot 8r^3} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{p}{16r^3} \]
\[ E'' = \frac{1}{16} \cdot \frac{1}{4 \pi \epsilon_0} \cdot \frac{p}{r^3} \]

6. **Relating it to \(E\):**
Since we know that \(E = \frac{1}{4 \pi \epsilon_0} \cdot \frac{2p}{r^3}\), we can express \(E''\) in terms of \(E\):
\[ E'' = \frac{1}{16} \cdot \frac{1}{2} E = \frac{1}{32} E \]

### Final Result:
The electric field at a distance \(2r\) along the perpendicular bisector of the dipole is:
\[ E'' = \frac{1}{32} E \]