In a uniformly charged sphere of total charge Q and radius R, the electric field E is plotted as function of distance from the centre, The graph which would correspond to the above will be:

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Answer

To solve the problem of finding the electric field \( E \) as a function of distance from the center of a uniformly charged sphere with total charge \( Q \) and radius \( R \), we will analyze the electric field both inside and outside the sphere using Gauss's law.

### Step-by-Step Solution:

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

2. **Electric Field Inside the Sphere**:
For points inside the sphere (where \( r < R \)):
- According to Gauss's law, the electric field \( E \) at a distance \( r \) from the center can be calculated using a Gaussian surface of radius \( r \).
- The charge enclosed by this Gaussian surface is given by:
\[ Q_{\text{enc}} = \rho \cdot V = \rho \cdot \frac{4}{3} \pi r^3 \]
where \( \rho \) is the charge density and \( V \) is the volume of the sphere of radius \( r \).
- The charge density \( \rho \) can be expressed as:
\[ \rho = \frac{Q}{\frac{4}{3} \pi R^3} \]
- Substituting this into the expression for \( Q_{\text{enc}} \):
\[ Q_{\text{enc}} = \frac{Q}{\frac{4}{3} \pi R^3} \cdot \frac{4}{3} \pi r^3 = Q \cdot \frac{r^3}{R^3} \]
- Applying Gauss's law:
\[ E \cdot 4 \pi r^2 = \frac{Q_{\text{enc}}}{\epsilon_0} \]
- Therefore:
\[ E \cdot 4 \pi r^2 = \frac{Q \cdot \frac{r^3}{R^3}}{\epsilon_0} \]
- Solving for \( E \):
\[ E = \frac{Q}{4 \pi \epsilon_0 R^3} r \]
- This shows that the electric field inside the sphere increases linearly with \( r \).

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

4. **Graphical Representation**:
- For \( r < R \), the electric field \( E \) increases linearly from \( 0 \) to a maximum value at \( r = R \).
- For \( r \geq R \), the electric field \( E \) decreases with \( \frac{1}{r^2} \).
- Therefore, the graph of \( E \) as a function of \( r \) will show a linear increase for \( r < R \) and a hyperbolic decrease for \( r \geq R \).

5. **Conclusion**:
The correct graph corresponding to the electric field \( E \) as a function of distance \( r \) from the center of the uniformly charged sphere will show a linear increase for \( r < R \) and a \( \frac{1}{r^2} \) decrease for \( r \geq R \).