If `E_(a)` be the electric field strength of a short dipole at a point on its axial line and `E_(e)` that on the equatorial line at the same distance, then

Options

A. `E_(e) = 2 E_(a)`
B. `E_(a) = 2E_(e)`
C. `E_(a) = E_(e)`
D. None of these

Answer

To solve the problem of comparing the electric field strengths \( E_a \) (at the axial line of a dipole) and \( E_e \) (at the equatorial line of a dipole) at the same distance from the dipole, we can follow these steps:

### Step 1: Understand the Configuration of the Dipole
A dipole consists of two equal and opposite charges \( +q \) and \( -q \) separated by a distance \( d \). The dipole moment \( p \) is given by:
\[ p = q \cdot d \]

### Step 2: Electric Field on the Axial Line
The electric field \( E_a \) 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 the formula:
\[ E_a = \frac{2kp}{r^3} \]
where \( k \) is Coulomb's constant.

### Step 3: Electric Field on the Equatorial Line
The electric field \( E_e \) at a point on the equatorial line (perpendicular bisector of the dipole) at the same distance \( r \) from the center of the dipole is given by:
\[ E_e = \frac{kp}{r^3} \]

### Step 4: Compare the Two Electric Fields
Now, we can compare \( E_a \) and \( E_e \):
\[ \frac{E_a}{E_e} = \frac{\frac{2kp}{r^3}}{\frac{kp}{r^3}} = \frac{2kp}{r^3} \cdot \frac{r^3}{kp} = 2 \]
Thus, we find that:
\[ E_a = 2E_e \]

### Conclusion
The electric field strength at the axial line \( E_a \) is twice that at the equatorial line \( E_e \):
\[ E_a = 2E_e \]