If the intensity of electric field at a distance x from the centre in axial position of small electric dipole is equal to the intensity at a distance y in equatorial position, then

Options

A. `x=y`
B. `x=y//2`
C. `y=x//2^(2//3)`
D. `y=x//2^(1//3)`

Answer

To solve the problem, we need to establish the relationship between the distances \( x \) and \( y \) where the electric field intensity at distance \( x \) from the center of a small electric dipole in the axial position is equal to the intensity at distance \( y \) in the equatorial position.

### Step-by-Step Solution:

1. **Understand the Configuration**:
- We have a small electric dipole consisting of two charges: \( +q \) and \( -q \) separated by a distance \( 2a \).
- The axial position is along the line extending from the dipole, while the equatorial position is perpendicular to the dipole at its center.

2. **Electric Field Intensity in Axial Position**:
- The electric field intensity \( E_x \) at a distance \( x \) from the center of the dipole in the axial position is given by the formula:
\[ E_x = \frac{k \cdot 2P}{x^3} \]
- Here, \( P \) is the dipole moment, and \( k \) is the electrostatic constant \( \left( k = \frac{1}{4\pi \epsilon_0} \right) \).

3. **Electric Field Intensity in Equatorial Position**:
- The electric field intensity \( E_y \) at a distance \( y \) from the center of the dipole in the equatorial position is given by:
\[ E_y = \frac{k \cdot P}{y^3} \]

4. **Set the Electric Fields Equal**:
- According to the problem, the electric field intensities are equal:
\[ E_x = E_y \]
- Substituting the expressions for \( E_x \) and \( E_y \):
\[ \frac{k \cdot 2P}{x^3} = \frac{k \cdot P}{y^3} \]

5. **Cancel Common Terms**:
- We can cancel \( k \) and \( P \) from both sides (assuming \( P \neq 0 \)):
\[ \frac{2}{x^3} = \frac{1}{y^3} \]

6. **Cross-Multiply to Solve for y**:
- Cross-multiplying gives:
\[ 2y^3 = x^3 \]

7. **Solve for y**:
- Dividing both sides by 2:
\[ y^3 = \frac{x^3}{2} \]
- Taking the cube root of both sides:
\[ y = \frac{x}{\sqrt[3]{2}} \]

### Final Result:
The relationship between \( x \) and \( y \) is:
\[ y = \frac{x}{2^{1/3}} \]