The electric potential at a point on the equatorial line of a electric dipole is

Options

A. directly proportional to the square of the distance
B. indirectly proportional to the square of the distance
C. directly proportional to the charge
D. none of the above

Explanation

To find the electric potential at a point on the equatorial line of an electric dipole, we can follow these steps:

### Step 1: Understand the formula for electric potential due to a dipole
The electric potential \( V \) at a distance \( r \) from the dipole is given by the equation:
\[ V = \frac{1}{4\pi \epsilon_0} \frac{P \cos \theta}{r^2} \]
where:
- \( P \) is the dipole moment,
- \( \theta \) is the angle between the dipole moment and the line joining the dipole to the point where the potential is being calculated,
- \( r \) is the distance from the dipole.

### Step 2: Analyze the situation on the equatorial line
On the equatorial line of the dipole, the angle \( \theta \) is \( 90^\circ \) (or \( \frac{\pi}{2} \) radians). This means that the dipole moment is perpendicular to the line connecting the dipole to the point of interest.

### Step 3: Calculate \( \cos \theta \)
Since \( \theta = 90^\circ \):
\[ \cos(90^\circ) = 0 \]

### Step 4: Substitute \( \cos \theta \) into the potential formula
Substituting \( \cos(90^\circ) = 0 \) into the potential formula:
\[ V = \frac{1}{4\pi \epsilon_0} \frac{P \cdot 0}{r^2} = 0 \]

### Step 5: Conclusion
The electric potential at a point on the equatorial line of an electric dipole is zero. Therefore, the correct answer is:
- **None of the above.**