The electric field due to uniformly charged sphere of radius `R` as a function of the distance from its centre is represented graphically by

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the electric field due to a uniformly charged sphere of radius \( R \) as a function of the distance from its center, we can break it down into two regions: inside the sphere (when the distance \( x < R \)) and outside the sphere (when the distance \( x \geq R \)).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a uniformly charged sphere with total charge \( Q \) and radius \( R \).
- We need to find the electric field \( E \) at a distance \( x \) from the center of the sphere.

2. **Electric Field Inside the Sphere (\( x < R \))**:
- According to Gauss's law, the electric field inside a uniformly charged sphere is given by:
\[ E = \frac{Q \cdot x}{4 \pi \epsilon_0 R^3} \]
- Here, \( \epsilon_0 \) is the permittivity of free space.
- This can be rewritten as:
\[ E \propto x \]
- This indicates that the electric field inside the sphere increases linearly with distance from the center.

3. **Electric Field Outside the Sphere (\( x \geq R \))**:
- For points outside the sphere, the electric field behaves as if all the charge were concentrated at the center of the sphere. Thus, the electric field is given by:
\[ E = \frac{Q}{4 \pi \epsilon_0 x^2} \]
- This can be rewritten as:
\[ E \propto \frac{1}{x^2} \]
- This indicates that the electric field decreases with the square of the distance from the center.

4. **Graphical Representation**:
- For \( x < R \), the electric field \( E \) increases linearly, which can be represented as a straight line starting from the origin (0,0).
- For \( x \geq R \), the electric field \( E \) decreases as \( \frac{1}{x^2} \), which creates a curve that approaches the x-axis but never touches it as \( x \) tends to infinity.

5. **Conclusion**:
- The graph of the electric field \( E \) as a function of distance \( x \) from the center of the sphere will show a straight line for \( 0 < x < R \) and a curve for \( x \geq R \) that approaches zero as \( x \) increases.

### Final Answer:
The correct graphical representation of the electric field due to a uniformly charged sphere of radius \( R \) as a function of the distance from its center is option B.