The electric field due to an electric dipole at a distance `r` from its centre in axial position is `E`. If the dipole is rotated through an angle of `90^(@)` about its perpendicular axis, the electric field at the same point will be

Options

A. E
B. `E//4`
C. E/2
D. 2E

Answer

To solve the problem, we need to analyze the electric field due to an electric dipole at different orientations.

### 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 given by \( p = q \cdot d \).

2. **Electric Field at Axial Position**:
The electric field \( E \) at a point along the axial line (the line extending from the positive charge through the negative charge) at a distance \( r \) from the center of the dipole is given by:
\[ E = \frac{2kp}{r^3} \]
where \( k \) is Coulomb's constant and \( p \) is the dipole moment.

3. **Initial Configuration**:
Initially, the dipole is oriented such that the angle \( \theta = 0^\circ \) with respect to the line connecting the dipole to the point where the electric field is measured. Thus, the electric field at this position is:
\[ E = \frac{2kp}{r^3} \]

4. **Rotating the Dipole**:
When the dipole is rotated through an angle of \( 90^\circ \) about its perpendicular axis, it now lies in a plane perpendicular to the line connecting the dipole to the point of interest. In this new position, the angle \( \theta = 90^\circ \).

5. **Electric Field at New Position**:
For the new orientation, we need to calculate the electric field components:
- The radial component (along the line connecting the dipole to the point) becomes:
\[ E_r = \frac{kp \cos(90^\circ)}{r^3} = 0 \]
- The tangential component (perpendicular to the radial direction) becomes:
\[ E_t = \frac{kp \sin(90^\circ)}{r^3} = \frac{kp}{r^3} \]

6. **Relating the New Electric Field to the Original**:
Since we know from the initial configuration that:
\[ E = \frac{2kp}{r^3} \]
We can express the tangential electric field as:
\[ E_t = \frac{kp}{r^3} = \frac{E}{2} \]

7. **Final Result**:
Therefore, after rotating the dipole through \( 90^\circ \), the electric field at the same point becomes:
\[ E_t = \frac{E}{2} \]

### Conclusion:
The electric field at the same point after rotating the dipole through \( 90^\circ \) about its perpendicular axis is \( \frac{E}{2} \).

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