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)=2E_(a)`
B. `E_(a)= 2E_(e)`
C. `E_(a)=E_(e )`
D. None of the above

Explanation

To find the relationship between the electric field strengths \( E_a \) (on the axial line) and \( E_e \) (on the equatorial line) of a short dipole at the same distance, we can follow these steps:

### Step 1: Understand the Electric Field Formulas
For a short dipole, the electric field strength at a point on the axial line is given by:
\[ E_a = \frac{2kp}{r^3} \]
where \( k \) is the Coulomb's constant, \( p \) is the dipole moment, and \( r \) is the distance from the dipole center to the point.

For the electric field strength at a point on the equatorial line, the formula is:
\[ E_e = \frac{kp}{r^3} \]

### Step 2: Set Up the Ratio of Electric Fields
Since we are given that both fields are calculated at the same distance \( r \), we can set up the ratio of \( E_e \) to \( E_a \):
\[ \frac{E_e}{E_a} = \frac{\frac{kp}{r^3}}{\frac{2kp}{r^3}} \]

### Step 3: Simplify the Ratio
When we simplify the ratio, we notice that \( k \) and \( r^3 \) cancel out:
\[ \frac{E_e}{E_a} = \frac{1}{2} \]

### Step 4: Rearranging the Equation
From the ratio, we can rearrange it to express \( E_a \) in terms of \( E_e \):
\[ E_a = 2E_e \]

### Conclusion
Thus, the relationship between the electric field strengths at the axial and equatorial lines of a short dipole is:
\[ E_a = 2E_e \]