Electric field intensity (E) due to an electric dipole varies with distance (r ) from the point of the center of dipole as :

Options

A. `(1)/rand 1/(r^(2))`
B. `(1)/(r^(2))and 1/r`
C. `(1)/(r^(3))and (1)/(r^(3))`
D. `(1)/(r^(3))and (1)/(r^(2))`

Answer

To determine how the electric field intensity (E) due to an electric dipole varies with distance (r) from the center of the dipole, we need to analyze two specific points: the axial point and the equatorial point.

### 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 (2L). The dipole moment (P) is defined as \( P = Q \cdot 2L \).

2. **Electric Field at an Axial Point**:
- Consider a point P located at a distance r from the center of the dipole along the axis.
- The electric field due to the positive charge (+Q) at point P is directed away from the charge, while the field due to the negative charge (-Q) is directed towards it.
- The distances from point P to the charges are \( r - L \) for the positive charge and \( r + L \) for the negative charge.
- The electric field contributions from both charges can be expressed as:
\[ E_1 = \frac{kQ}{(r - L)^2} \quad \text{(from +Q)} \]
\[ E_2 = \frac{kQ}{(r + L)^2} \quad \text{(from -Q)} \]
- The net electric field at point P (E_axial) is given by:
\[ E_{\text{axial}} = E_2 - E_1 = \frac{kQ}{(r + L)^2} - \frac{kQ}{(r - L)^2} \]
- Simplifying this expression leads to:
\[ E_{\text{axial}} \approx \frac{2kP}{r^3} \quad \text{(for large r, where \( r >> L \))} \]

3. **Electric Field at an Equatorial Point**:
- Now consider a point P located at a distance r from the center of the dipole along the perpendicular bisector (equatorial point).
- The electric fields due to both charges at this point are equal in magnitude but opposite in direction.
- The electric field contributions can be expressed as:
\[ E_1 = \frac{kQ}{(r^2 + L^2)} \quad \text{(from +Q)} \]
\[ E_2 = \frac{kQ}{(r^2 + L^2)} \quad \text{(from -Q)} \]
- The net electric field at point P (E_equatorial) is given by:
\[ E_{\text{equatorial}} = E_1 + E_2 = 0 \quad \text{(along the axis)} \]
- However, the vertical components add up, leading to:
\[ E_{\text{equatorial}} \approx \frac{kP}{r^3} \quad \text{(for large r, where \( r >> L \))} \]

4. **Conclusion**:
- From the above analysis, we can conclude:
- The electric field intensity at an axial point varies as \( E_{\text{axial}} \propto \frac{1}{r^3} \).
- The electric field intensity at an equatorial point also varies as \( E_{\text{equatorial}} \propto \frac{1}{r^3} \).
- Therefore, both electric field intensities vary inversely with the cube of the distance from the center of the dipole.

### Final Result:
The electric field intensity (E) due to an electric dipole varies with distance (r) from the center of the dipole as:
\[ E \propto \frac{1}{r^3} \]