A solid sphere of radius `R` is charged uniformly. The electrostatic potential `V` is plotted as a function of distance `r` from the centre of th sphere. Which of the following best represents the resulting curve?

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the shape of the electrostatic potential \( V \) as a function of distance \( r \) from the center of a uniformly charged solid sphere, we will analyze the potential in three distinct regions: outside the sphere, at the surface, and inside the sphere.

### Step-by-Step Solution:

1. **Understanding the Regions**:
- We have a solid sphere of radius \( R \) that is uniformly charged with total charge \( Q \).
- We need to analyze the potential \( V \) for three cases:
- Case 1: \( r > R \) (outside the sphere)
- Case 2: \( r = R \) (at the surface of the sphere)
- Case 3: \( r < R \) (inside the sphere)

2. **Case 1: Outside the Sphere (\( r > R \))**:
- The potential \( V \) at a distance \( r \) from the center of the sphere is given by the formula:
\[ V(r) = \frac{KQ}{r} \]
- Here, \( K \) is the electrostatic constant.
- This equation represents a hyperbolic relationship, where the potential decreases as \( r \) increases.

3. **Case 2: At the Surface of the Sphere (\( r = R \))**:
- At the surface, we can substitute \( r = R \) into the equation:
\[ V(R) = \frac{KQ}{R} \]
- This gives us the potential at the surface of the sphere.

4. **Case 3: Inside the Sphere (\( r < R \))**:
- The potential inside a uniformly charged solid sphere is given by:
\[ V(r) = \frac{KQ}{2R} \left( 3 - \frac{r^2}{R^2} \right) \]
- This equation shows that the potential is a quadratic function of \( r \) and opens downward, indicating that as \( r \) approaches \( R \), the potential approaches \( \frac{KQ}{R} \).

5. **Combining the Results**:
- For \( r < R \), the potential is a downward-opening parabola.
- At \( r = R \), the potential reaches a maximum value \( \frac{KQ}{R} \).
- For \( r > R \), the potential decreases hyperbolically as \( \frac{KQ}{r} \).

6. **Graph Representation**:
- The graph of \( V \) as a function of \( r \) will show:
- A downward parabola for \( r < R \).
- A constant value at \( r = R \).
- A hyperbolic decrease for \( r > R \).

### Conclusion:
The best representation of the resulting curve for the electrostatic potential \( V \) as a function of distance \( r \) from the center of the sphere is a combination of a downward parabola for \( r < R \) and a hyperbola for \( r > R \). The correct option is the one that reflects this behavior.