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`

Explanation

To solve the problem of finding the electric field at a point due to an electric dipole after it has been rotated, we can follow these steps:

### Step 1: Understand the Initial Condition
Initially, we have an electric dipole consisting of two charges, +q and -q, separated by a distance `d`. The dipole moment `p` is defined as:
\[ p = q \cdot d \]
The electric field `E` at a point on the axial line (along the line extending from the positive charge through the negative charge) at a distance `r` from the center of the dipole is given by:
\[ E = \frac{k \cdot 2p}{r^3} \]
where `k` is the Coulomb's constant.

### Step 2: Rotate the Dipole
When the dipole is rotated by `90 degrees`, it is now oriented perpendicular to the line connecting the point of interest and the dipole. In this new orientation, the point of interest is now on the equatorial line of the dipole.

### Step 3: Calculate the Electric Field on the Equatorial Line
The electric field `E_eq` at a point on the equatorial line of the dipole is given by:
\[ E_{eq} = \frac{k \cdot p}{r^3} \]
This is because the contributions from the two charges are equal in magnitude but opposite in direction, resulting in a net field that is half of that on the axial line.

### Step 4: Relate the Two Electric Fields
Now, we can relate the electric field on the axial line to that on the equatorial line:
1. From the axial position, we have:
\[ E = \frac{k \cdot 2p}{r^3} \]
2. From the equatorial position, we have:
\[ E_{eq} = \frac{k \cdot p}{r^3} \]

### Step 5: Find the Ratio of the Electric Fields
To find the relationship between `E` and `E_eq`, we can take the ratio:
\[ \frac{E_{eq}}{E} = \frac{\frac{k \cdot p}{r^3}}{\frac{k \cdot 2p}{r^3}} = \frac{1}{2} \]

### Step 6: Conclusion
Thus, the electric field at the same point after the dipole has been rotated by `90 degrees` is:
\[ E_{eq} = \frac{E}{2} \]

### Final Answer
The electric field at the same point after the dipole is rotated through an angle of `90 degrees` will be:
\[ E_{eq} = \frac{E}{2} \]

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