A point `Q` lies on the perpendicular bisector of an electrical dipole of dipole moment `p`. If the distance of `Q` from the dipole is `r` (much larger than the size of the dipole), then electric field at `Q` is proportional to

Options

A. `p^(-1)` and `r^(-2)`
B. `p` and `r^(-2)`
C. `p^(2)` and `r^(-3)`
D. `p` and `r^(-3)`

Explanation

To solve the problem, we need to determine how the electric field at point Q, which lies on the perpendicular bisector of an electric dipole, is related to the dipole moment \( p \) and the distance \( r \) from the dipole.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have an electric dipole consisting of two charges \( +q \) and \( -q \) separated by a distance \( d \).
- The dipole moment \( p \) is defined as \( p = q \cdot d \).
- Point \( Q \) is located on the perpendicular bisector of the dipole.

2. **Distance Consideration**:
- The distance \( r \) from point \( Q \) to the center of the dipole is much larger than the size of the dipole. Thus, we can consider \( r \) to be significantly larger than \( d \) (i.e., \( r \gg d \)).

3. **Electric Field of a Dipole**:
- The electric field \( E \) at a point on the perpendicular bisector of a dipole at a distance \( r \) is given by the formula:
\[ E = \frac{K \cdot p}{r^3} \]
where \( K \) is a constant that depends on the medium.

4. **Proportionality Analysis**:
- From the formula \( E = \frac{K \cdot p}{r^3} \), we can see that:
- The electric field \( E \) is directly proportional to the dipole moment \( p \) (i.e., \( E \propto p \)).
- The electric field \( E \) is inversely proportional to the cube of the distance \( r \) (i.e., \( E \propto \frac{1}{r^3} \)).

5. **Conclusion**:
- Therefore, we can conclude that the electric field at point \( Q \) is proportional to the dipole moment \( p \) and inversely proportional to \( r^3 \).

### Final Answer:
The electric field at point \( Q \) is proportional to \( \frac{p}{r^3} \).