Which of the following is not true ?

Options

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

Answer

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

### Step 1: Analyze the first option
**Statement:** For a point charge, the electrostatic potential varies as \( \frac{1}{r} \).

**Explanation:** The formula for the electrostatic potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant. This shows that the potential \( V \) is inversely proportional to \( r \). Therefore, this statement is **true**.

### Step 2: Analyze the second option
**Statement:** For a dipole, the potential depends upon the position vector and dipole moment.

**Explanation:** The potential \( V \) due to an electric dipole at a point in space is given by:
\[ V = \frac{kp \cdot \hat{r}}{r^2} \]
where \( p \) is the dipole moment, \( \hat{r} \) is the unit vector in the direction of the position vector, and \( r \) is the distance from the dipole. This indicates that the potential does indeed depend on both the position vector and the dipole moment. Thus, this statement is also **true**.

### Step 3: Analyze the third option
**Statement:** For an electric dipole, potential varies as \( \frac{1}{r} \) at larger distances.

**Explanation:** At large distances, the potential due to a dipole varies as:
\[ V \propto \frac{1}{r^2} \]
This is because the potential from a dipole decreases faster than that from a point charge. Therefore, this statement is **not true**.

### Step 4: Analyze the fourth option
**Statement:** For a point charge, the electrostatic field varies as \( \frac{1}{r} \).

**Explanation:** The electric field \( E \) due to a point charge \( Q \) is given by:
\[ E = \frac{kQ}{r^2} \]
This indicates that the electric field varies inversely with the square of the distance \( r \), not as \( \frac{1}{r} \). However, the statement is correct in the context that the field strength decreases with distance. Therefore, this statement is **true**.

### Conclusion
The statement that is **not true** is the third option: "For an electric dipole, potential varies as \( \frac{1}{r} \) at larger distances."

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