If the magnitude of intensity of electric field at a distance `x` on axial line and at a distance `y` on equatorial line on a given dipole are equal, then `x:y` is

Options

A. `1:1`
B. `1:sqrt(2)`
C. `1:2`
D. `2^(1/3):1`

Explanation

To solve the problem, we need to find the ratio \( x:y \) when the magnitudes of the electric field intensity at a distance \( x \) on the axial line and at a distance \( y \) on the equatorial line of a dipole are equal.

### Step-by-Step Solution:

1. **Understanding the Electric Field of a Dipole**:
- The electric field \( E_A \) at a distance \( x \) on the axial line of a dipole is given by the formula:
\[ E_A = \frac{1}{4\pi \epsilon_0} \cdot \frac{2p}{x^3} \]
- The electric field \( E_E \) at a distance \( y \) on the equatorial line of a dipole is given by the formula:
\[ E_E = \frac{1}{4\pi \epsilon_0} \cdot \frac{p}{y^3} \]

2. **Setting the Electric Fields Equal**:
- According to the problem, we have:
\[ E_A = E_E \]
- Substituting the expressions for \( E_A \) and \( E_E \):
\[ \frac{1}{4\pi \epsilon_0} \cdot \frac{2p}{x^3} = \frac{1}{4\pi \epsilon_0} \cdot \frac{p}{y^3} \]

3. **Canceling Common Terms**:
- We can cancel \( \frac{1}{4\pi \epsilon_0} \) and \( p \) from both sides (assuming \( p \neq 0 \)):
\[ \frac{2}{x^3} = \frac{1}{y^3} \]

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

5. **Finding the Ratio \( x:y \)**:
- Taking the cube root of both sides:
\[ \frac{x}{y} = \sqrt[3]{2} \]
- Therefore, we can express the ratio \( x:y \) as:
\[ x:y = \sqrt[3]{2}:1 \]

### Final Result:
Thus, the ratio \( x:y \) is \( \sqrt[3]{2}:1 \).