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 the electric field at a point Q that lies on the perpendicular bisector of an electric dipole with dipole moment \( p \). The distance from the dipole to point Q is \( r \), which is much larger than the size of the dipole.

### Step-by-Step Solution:

1. **Understanding the Dipole**:
- An electric dipole consists of two equal and opposite charges separated by a distance \( 2a \). The dipole moment \( p \) is defined as:
\[ p = q \cdot d \]
where \( q \) is the magnitude of one of the charges and \( d \) is the separation between the charges.

2. **Position of Point Q**:
- Point Q is located on the perpendicular bisector of the dipole. This means that it is equidistant from both charges of the dipole.

3. **Distance Consideration**:
- Since \( r \) (the distance from the dipole to point Q) is much larger than the size of the dipole, we can consider \( r \) to be the effective distance when calculating the electric field.

4. **Electric Field on the Perpendicular Bisector**:
- The electric field \( E \) at a point on the perpendicular bisector of a dipole is given by the formula:
\[ E = \frac{1}{4\pi \epsilon_0} \cdot \frac{2p}{r^3} \]
- Here, \( \epsilon_0 \) is the permittivity of free space.

5. **Proportionality**:
- From the formula, we can see that the electric field \( E \) is directly proportional to the dipole moment \( p \) and inversely proportional to the cube of the distance \( r \):
\[ E \propto \frac{p}{r^3} \]

6. **Conclusion**:
- Therefore, 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 \( p \) and \( \frac{1}{r^3} \).