A capacitor is made of two concentric spherical shells A and B of radii a and b respectively. Where `a lt b`. The external shell B is grounded `(V_(B) = 0)` Inner shel i.e `(r le a )` is filled with a uniform positive charge of density `rho`. Variation of electric field versus distance from the centre.

Options

A.
B.
C.
D.

Explanation

To solve the problem of finding the variation of the electric field versus distance from the center for the given capacitor made of two concentric spherical shells, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have two concentric spherical shells: inner shell A with radius \( a \) and outer shell B with radius \( b \).
- The outer shell B is grounded, meaning its potential \( V_B = 0 \).
- The inner shell A is filled with a uniform positive charge density \( \rho \).

2. **Applying Gauss's Law**:
- To find the electric field \( E \) at a distance \( r \) from the center, we can use Gauss's law, which states:
\[ \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \]
- Here, \( Q_{\text{enc}} \) is the charge enclosed by a Gaussian surface of radius \( r \).

3. **Case 1: Inside the Inner Shell ( \( r < a \) )**:
- For \( r < a \), there is no charge enclosed within this Gaussian surface, so:
\[ Q_{\text{enc}} = 0 \implies E = 0 \]

4. **Case 2: Within the Inner Shell ( \( a \leq r < b \) )**:
- For \( a \leq r < b \), we need to calculate the charge enclosed within a sphere of radius \( r \):
\[ Q_{\text{enc}} = \rho \cdot \text{Volume of sphere with radius } r = \rho \cdot \frac{4}{3} \pi r^3 \]
- Applying Gauss's law:
\[ E \cdot 4 \pi r^2 = \frac{\rho \cdot \frac{4}{3} \pi r^3}{\epsilon_0} \]
- Simplifying this gives:
\[ E = \frac{\rho r}{3 \epsilon_0} \]

5. **Case 3: Outside the Outer Shell ( \( r \geq b \) )**:
- For \( r \geq b \), since the outer shell B is grounded, the electric field outside the shell will be zero:
\[ E = 0 \]

6. **Summary of Electric Field Variation**:
- For \( r < a \): \( E = 0 \)
- For \( a \leq r < b \): \( E = \frac{\rho r}{3 \epsilon_0} \) (linear increase with \( r \))
- For \( r \geq b \): \( E = 0 \)

### Final Result:
The electric field \( E \) varies as follows:
- \( E = 0 \) for \( r < a \)
- \( E = \frac{\rho r}{3 \epsilon_0} \) for \( a \leq r < b \)
- \( E = 0 \) for \( r \geq b \)