If the electric potential on the axis of an electric dipole at a distance 'r' from it is V, then the potential at a point on its equatorial line at the same distance away from it will be

Options

A. `2V`
B. `(V)/(2)`
C. `-V`
D. zero

Answer

To solve the problem, we need to understand the electric potential due to an electric dipole at different positions relative to the dipole.

### Step-by-Step Solution:

1. **Understanding Electric Dipole**:
An electric dipole consists of two equal and opposite charges (+Q and -Q) separated by a small distance (2L). The dipole moment (p) is defined as \( p = Q \cdot 2L \).

2. **Electric Potential on the Axis of the Dipole**:
The electric potential (V) at a point on the axis of the dipole at a distance \( r \) from the center of the dipole is given by the formula:
\[ V = \frac{k \cdot p \cdot \cos \theta}{r^2} \]
where \( k \) is the Coulomb's constant, \( p \) is the dipole moment, and \( \theta \) is the angle between the dipole moment and the line connecting the dipole to the point where the potential is being calculated.

3. **Calculating Potential on the Axis**:
For a point on the axis of the dipole, \( \theta = 0^\circ \). Therefore, \( \cos 0 = 1 \):
\[ V = \frac{k \cdot p}{r^2} \]

4. **Electric Potential on the Equatorial Line**:
Now, we need to find the electric potential at a point on the equatorial line of the dipole at the same distance \( r \). For a point on the equatorial line, \( \theta = 90^\circ \). Therefore, \( \cos 90 = 0 \):
\[ V = \frac{k \cdot p \cdot \cos 90}{r^2} = \frac{k \cdot p \cdot 0}{r^2} = 0 \]

5. **Conclusion**:
The potential at a point on the equatorial line at the same distance \( r \) from the dipole is \( 0 \).

### Final Answer:
The potential at a point on the equatorial line at a distance \( r \) from the dipole is \( 0 \).

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