There is a uniformly charged non-conducting solid sphere. A spherical cavity is made whose centre does not coincide with the centre of solid sphere. Electric field intensity inside the cavity is

Options

A. directly proportional to distance of the point from centre of cavity
B. directly proportional to the distance of the point from centre of sphere
C. uniform
D. not predictable due to insufficient information

Answer

To find the electric field intensity inside a spherical cavity within a uniformly charged non-conducting solid sphere, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have a uniformly charged non-conducting solid sphere with a certain radius \( R \).
- A spherical cavity is created within this sphere, and the center of the cavity does not coincide with the center of the solid sphere.

2. **Identifying the Electric Field Inside the Cavity**:
- According to Gauss's law, the electric field inside a uniformly charged solid sphere is not uniform; however, the presence of the cavity alters the situation.
- The electric field inside the cavity can be treated as the superposition of the electric field due to the entire sphere and the electric field due to the absence of charge in the cavity.

3. **Using Superposition Principle**:
- The electric field inside the cavity is uniform and directed along the line connecting the center of the solid sphere to the center of the cavity.
- The electric field intensity \( E \) inside the cavity can be expressed as:
\[ E = \frac{\rho L}{3 \epsilon_0} \]
- Here, \( \rho \) is the volume charge density of the solid sphere, \( L \) is the distance between the center of the solid sphere and the center of the cavity, and \( \epsilon_0 \) is the permittivity of free space.

4. **Direction of the Electric Field**:
- The direction of the electric field inside the cavity is along the vector that connects the center of the solid sphere to the center of the cavity. This vector can be denoted as \( \mathbf{L} \).

5. **Final Expression for Electric Field**:
- The electric field intensity inside the cavity can be written in vector form as:
\[ \mathbf{E} = \frac{\rho \mathbf{L}}{3 \epsilon_0} \]
- This indicates that the electric field is uniform throughout the cavity.

### Conclusion:
The electric field intensity inside the cavity of a uniformly charged non-conducting solid sphere is uniform and given by the expression \( \mathbf{E} = \frac{\rho \mathbf{L}}{3 \epsilon_0} \), where \( \mathbf{L} \) is the position vector from the center of the solid sphere to the center of the cavity.

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