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.

Answer

To solve the problem of how the electrostatic potential \( V \) varies as a function of distance \( r \) from the center of a uniformly charged solid sphere, we can break down the analysis into several steps:

### Step-by-Step Solution

1. **Understanding the Charge Distribution**:
- A solid sphere of radius \( R \) is uniformly charged. This means that the charge \( Q \) is distributed evenly throughout the volume of the sphere.

2. **Electrostatic Potential Inside the Sphere**:
- For a point inside the sphere (where \( r < R \)), the electrostatic potential \( V \) can be expressed as:
\[ V = \frac{kQ}{2R} \left( 3 - \frac{r^2}{R^2} \right) \]
- Here, \( k \) is Coulomb's constant, and \( r \) is the distance from the center of the sphere.

3. **Electrostatic Potential at the Surface of the Sphere**:
- At the surface of the sphere (where \( r = R \)), the potential can be calculated as:
\[ V(R) = \frac{kQ}{R} \]

4. **Electrostatic Potential Outside the Sphere**:
- For points outside the sphere (where \( r > R \)), the potential behaves like that of a point charge:
\[ V = \frac{kQ}{r} \]

5. **Behavior of the Potential**:
- Inside the sphere, as \( r \) increases from \( 0 \) to \( R \), the potential decreases from a maximum value at the center to the value at the surface.
- Outside the sphere, as \( r \) increases beyond \( R \), the potential decreases inversely with distance.

6. **Graphical Representation**:
- The graph of \( V \) as a function of \( r \) will show a decrease in potential as you move from the center of the sphere to the surface and then continue to decrease as you move further away from the sphere.
- The curve will start at a maximum value at the center, decrease to a certain value at the surface, and then follow a hyperbolic decline for \( r > R \).

7. **Choosing the Correct Option**:
- Based on the analysis, the correct representation of the potential \( V \) as a function of distance \( r \) from the center of the sphere is best represented by a curve that initially decreases and then follows an inverse relationship for \( r > R \). This corresponds to option C.

### Final Answer:
The best representation of the electrostatic potential \( V \) as a function of distance \( r \) from the center of the uniformly charged solid sphere is option C.