Which graph best represents the variation of electric potential as a function of distance from the centre of a uniformly charged solid sphere of charge of radius R ?

Options

A.

B.
C.
D.

Answer

To determine the variation of electric potential as a function of distance from the center of a uniformly charged solid sphere, we can analyze the behavior of electric potential both inside and outside the sphere.

### Step-by-Step Solution:

1. **Understanding Electric Potential Inside the Sphere**:
- For a uniformly charged solid sphere of radius \( R \) and total charge \( Q \), the electric potential \( V \) at any point inside the sphere (at a distance \( r \) from the center, where \( r < R \)) is constant and equal to the potential at the surface of the sphere.
- The formula for the electric potential at the surface is given by:
\[ V = \frac{kQ}{R} \]
- Therefore, for \( r < R \):
\[ V = \frac{kQ}{R} \quad \text{(constant)} \]

2. **Understanding Electric Potential Outside the Sphere**:
- For points outside the sphere (where \( r \geq R \)), the electric potential \( V \) behaves as if all the charge \( Q \) were concentrated at the center of the sphere.
- The formula for the electric potential at a distance \( r \) from the center (where \( r \geq R \)) is:
\[ V = \frac{kQ}{r} \]
- This indicates that the potential decreases inversely with distance \( r \).

3. **Graphical Representation**:
- From the above analysis, we can summarize the behavior of the electric potential:
- Inside the sphere (\( r < R \)): The potential is constant at \( \frac{kQ}{R} \).
- At the surface (\( r = R \)): The potential is \( \frac{kQ}{R} \).
- Outside the sphere (\( r > R \)): The potential decreases as \( \frac{1}{r} \).
- Therefore, the graph of electric potential \( V \) versus distance \( r \) will show:
- A horizontal line at \( V = \frac{kQ}{R} \) for \( r < R \).
- A hyperbolic decline for \( r \geq R \), approaching zero as \( r \) increases.

4. **Choosing the Correct Graph**:
- Based on the described behavior, the correct graph will show:
- A constant potential for \( r < R \).
- A decreasing potential for \( r \geq R \) that approaches zero.
- Thus, the graph that best represents this variation is option 2.

### Summary:
- The electric potential is constant inside the sphere and decreases inversely with distance outside the sphere.