A sphere of radius `R` has a uniform distribution of electric charge in its volume. At a distance `x` from its centre, for `x lt R`, the electric field is directly proportional to

Options

A. `(1)/(x^(2))`
B. `(1)/(x)`
C. `x`
D. `x^(2)`

Explanation

To solve the problem, we need to determine how the electric field behaves inside a uniformly charged sphere at a distance \( x \) from its center, where \( x < R \) (the radius of the sphere).

### Step-by-Step Solution:

1. **Understanding the Problem**: We have a sphere of radius \( R \) with a uniform distribution of electric charge throughout its volume. We are interested in finding the electric field at a distance \( x \) from the center of the sphere, where \( x < R \).

2. **Using Gauss's Law**: To find the electric field inside the sphere, we can use Gauss's Law, which states:
\[ \Phi_E = \frac{Q_{\text{enc}}}{\epsilon_0} \]
where \( \Phi_E \) is the electric flux through a closed surface, \( Q_{\text{enc}} \) is the charge enclosed by that surface, and \( \epsilon_0 \) is the permittivity of free space.

3. **Choosing a Gaussian Surface**: We choose a Gaussian surface that is a sphere of radius \( x \) (where \( x < R \)). The electric field \( E \) at this distance will be uniform over the surface of the Gaussian sphere.

4. **Calculating the Charge Enclosed**: The charge enclosed within the Gaussian surface can be calculated using the volumetric charge density \( \rho \):
\[ Q_{\text{enc}} = \rho \cdot V = \rho \cdot \left(\frac{4}{3} \pi x^3\right) \]
where \( V \) is the volume of the sphere of radius \( x \).

5. **Calculating the Electric Flux**: The electric flux through the Gaussian surface is given by:
\[ \Phi_E = E \cdot A = E \cdot (4 \pi x^2) \]
where \( A \) is the surface area of the Gaussian sphere.

6. **Applying Gauss's Law**: Setting the electric flux equal to the charge enclosed divided by \( \epsilon_0 \):
\[ E \cdot (4 \pi x^2) = \frac{\rho \cdot \left(\frac{4}{3} \pi x^3\right)}{\epsilon_0} \]

7. **Solving for \( E \)**: Rearranging the equation to solve for the electric field \( E \):
\[ E = \frac{\rho \cdot \left(\frac{4}{3} \pi x^3\right)}{4 \pi x^2 \epsilon_0} \]
Simplifying this gives:
\[ E = \frac{\rho x}{3 \epsilon_0} \]

8. **Conclusion**: The electric field \( E \) at a distance \( x \) from the center of the sphere is directly proportional to \( x \):
\[ E \propto x \]

### Final Answer:
The electric field at a distance \( x \) from the center of a uniformly charged sphere is directly proportional to \( x \).