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)`

Answer

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 \( x < R \). We will use Gauss's Law to derive the relationship.

### 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 need to find the electric field \( E \) at a distance \( x \) from the center of the sphere, where \( x < R \).

2. **Applying Gauss's Law**: According to Gauss's Law, the electric flux through a closed surface is proportional to the charge enclosed within that surface. Mathematically, it is given by:
\[ \Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \]
where \( \Phi_E \) is the electric flux, \( \vec{E} \) is the electric field, \( d\vec{A} \) is the differential area vector, \( Q_{\text{enc}} \) is the charge enclosed, and \( \epsilon_0 \) is the permittivity of free space.

3. **Choosing a Gaussian Surface**: For a point inside the sphere at a distance \( x \) from the center, we choose a spherical Gaussian surface of radius \( x \). The area \( A \) of this surface is:
\[ A = 4\pi x^2 \]

4. **Finding the Enclosed Charge**: The charge density \( \rho \) is given by:
\[ \rho = \frac{Q}{\frac{4}{3}\pi R^3} \]
where \( Q \) is the total charge of the sphere. The charge enclosed \( Q_{\text{enc}} \) within the Gaussian surface of radius \( x \) is:
\[ Q_{\text{enc}} = \rho \cdot \text{Volume of sphere of radius } x = \rho \cdot \frac{4}{3}\pi x^3 \]

5. **Substituting for Charge Density**: Substitute \( \rho \) into the expression for \( Q_{\text{enc}} \):
\[ Q_{\text{enc}} = \left(\frac{Q}{\frac{4}{3}\pi R^3}\right) \cdot \frac{4}{3}\pi x^3 = Q \cdot \frac{x^3}{R^3} \]

6. **Using Gauss's Law**: Now substituting \( Q_{\text{enc}} \) into Gauss's Law:
\[ E \cdot 4\pi x^2 = \frac{Q \cdot \frac{x^3}{R^3}}{\epsilon_0} \]

7. **Solving for Electric Field \( E \)**: Rearranging gives:
\[ E = \frac{Q}{4\pi \epsilon_0 R^3} \cdot x \]
This shows that the electric field \( E \) is directly proportional to \( x \).

8. **Conclusion**: Therefore, the electric field inside the uniformly charged sphere at a distance \( x \) from its center is directly proportional to \( x \).

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