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 \( y \) based on the electric field intensities at these points due to a small electric dipole.

### 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 \( 2L \). The dipole moment \( p \) is defined as \( p = q \cdot 2L \).

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

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

4. **Setting the Electric Fields Equal**:
- According to the problem, the electric field at the axial position is equal to the electric field at the equatorial position:
\[ E_x = E_y \]
- Substituting the expressions for \( E_x \) and \( E_y \):
\[ \frac{k \cdot 2p}{x^3} = \frac{k \cdot p}{y^3} \]

5. **Canceling 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-Multiplying**:
- Cross-multiplying gives:
\[ 2y^3 = x^3 \]

7. **Finding the Relationship**:
- To express \( y \) in terms of \( x \), we rearrange the equation:
\[ y^3 = \frac{x^3}{2} \]
- Taking the cube root of both sides:
\[ y = \frac{x}{\sqrt[3]{2}} \]

8. **Final Relation**:
- Thus, the relationship between \( x \) and \( y \) is:
\[ y = \frac{x}{\sqrt[3]{2}} \]