Which of the following is not true?

Options

A. for a point charge, electrostatic potential varies as `(1)/(r )`
B. for a dipole the potential depends on the magnitude of position vector and dipole moment vector
C. the electric dipole potential varies as `(1)/(r )` at large distance
D. for a point charge, the electro static field varies as `(1)/(r^(2))`

Answer

To determine which of the statements is not true regarding electrostatic potential and capacitance, we will analyze each statement step by step.

### Step-by-Step Solution:

1. **Understanding Electrostatic Potential for a Point Charge**:
- The electrostatic potential \( V \) due to a point charge \( Q \) at a distance \( r \) from the charge is given by the formula:
\[ V = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r} \]
- This shows that the potential \( V \) is inversely proportional to the distance \( r \).

2. **Electrostatic Potential for an Electric Dipole**:
- For an electric dipole consisting of charges \( +q \) and \( -q \) separated by a distance \( 2l \), the potential \( V \) at a point making an angle \( \theta \) with the dipole axis is given by:
\[ V = \frac{k \cdot p \cdot \cos \theta}{r^2} \]
- Here, \( p \) is the dipole moment defined as \( p = q \cdot 2l \).

3. **Behavior of Potential at Large Distances**:
- At distances much greater than the separation of the dipole (i.e., \( r \gg l \)), the potential behaves as:
\[ V \propto \frac{1}{r^2} \]
- This indicates that the potential decreases with the square of the distance from the dipole.

4. **Electric Field of a Point Charge**:
- The electric field \( E \) due to a point charge \( Q \) is given by:
\[ E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r^2} \]
- This shows that the electric field also varies inversely with the square of the distance \( r \).

5. **Conclusion**:
- After analyzing the statements, we find that:
- The potential due to a point charge is inversely proportional to \( r \).
- The potential due to a dipole at large distances varies as \( \frac{1}{r^2} \).
- The electric field due to a point charge varies as \( \frac{1}{r^2} \).
- Therefore, if any statement contradicts these relationships, it would be the incorrect one.

Based on the analysis, the statement that is not true is the one that claims the electric dipole potential varies as \( \frac{1}{r^2} \) at large distances, which is incorrect because it actually varies as \( \frac{1}{r^2} \) with respect to the dipole moment and the angle \( \theta \).

### Final Answer:
The incorrect statement is related to the electric dipole potential, which does not vary as \( \frac{1}{r^2} \) at large distances.