An electric dipole coincides on Z-axis and its mid-point is on origin of the coordinates system. The electric field at an axial point at a distance z from origin is `E_((z))` and electric field at an equatorial point at a distance y from origin is `E_((y))`. Here, `z = y gt gt a`, so `|(E_((z)))/(E_((y)))| =`... 1 4 3 2

Options

A. 1
B. 4
C. 3
D. 2

Answer

To solve the problem, we need to find the ratio of the electric fields at an axial point and an equatorial point of an 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 small distance '2a'. The dipole moment \( P \) is defined as \( P = Q \cdot 2a \).

2. **Electric Field at an Axial Point**:
The electric field \( E_z \) at a point on the axial line (along the dipole moment) at a distance \( z \) from the origin is given by the formula:
\[ E_z = \frac{2P}{4\pi \epsilon_0 z^3} \]
where \( \epsilon_0 \) is the permittivity of free space.

3. **Electric Field at an Equatorial Point**:
The electric field \( E_y \) at a point on the equatorial line (perpendicular to the dipole moment) at a distance \( y \) from the origin is given by the formula:
\[ E_y = \frac{P}{4\pi \epsilon_0 y^3} \]

4. **Given Condition**:
We are given that \( z = y \) and both are much greater than \( a \) (i.e., \( z \gg a \) and \( y \gg a \)).

5. **Finding the Ratio of Electric Fields**:
To find the ratio \( \left| \frac{E_z}{E_y} \right| \):
\[ \frac{E_z}{E_y} = \frac{\frac{2P}{4\pi \epsilon_0 z^3}}{\frac{P}{4\pi \epsilon_0 y^3}} = \frac{2P}{4\pi \epsilon_0 z^3} \cdot \frac{4\pi \epsilon_0 y^3}{P} \]
Simplifying this expression:
\[ \frac{E_z}{E_y} = \frac{2y^3}{z^3} \]

6. **Substituting \( z = y \)**:
Since \( z = y \):
\[ \frac{E_z}{E_y} = \frac{2y^3}{y^3} = 2 \]

7. **Final Result**:
Therefore, the ratio \( \left| \frac{E_z}{E_y} \right| = 2 \).

### Conclusion:
The answer is \( 2 \).