A positive charge `+q` is placed at the centre of a hollow metallic sphere of inner radius a and outer radius b. The electric field at a distance r from the centre is denoted by E. In this regard, which one of the following statements is correct?

Options

A. `E=0" for "altrltb`
B. `E=0" for "rlta`
C. `E=q//4piepsilon_(0)r" for "altrltb`
D. `E=q//4piepsilon_(0)a" for "rlta`

Explanation

To solve the problem, we need to analyze the situation of a positive charge \( +q \) placed at the center of a hollow metallic sphere. The sphere has an inner radius \( a \) and an outer radius \( b \). We will determine the electric field \( E \) at various distances \( r \) from the center of the sphere.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a positive charge \( +q \) at the center of a hollow metallic sphere.
- The sphere has an inner radius \( a \) and an outer radius \( b \).

2. **Applying Gauss's Law**:
- According to Gauss's Law, the electric field \( E \) at a distance \( r \) from the center of a charge distribution can be found using the formula:
\[ \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \]
- Here, \( Q_{\text{enc}} \) is the charge enclosed by the Gaussian surface.

3. **Analyzing Different Regions**:
- **Region 1**: Inside the sphere (i.e., \( r < a \)):
- The electric field \( E \) is determined by the charge \( +q \) at the center. Since we are inside the hollow sphere, the electric field is given by:
\[ E = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{r^2} \]
- **Region 2**: Within the metallic shell (i.e., \( a < r < b \)):
- In this region, the electric field \( E \) is zero. This is because the charges in the metallic shell will redistribute themselves in response to the electric field created by the charge \( +q \), resulting in no net electric field within the conductor.
- **Region 3**: Outside the sphere (i.e., \( r > b \)):
- The electric field behaves as if all the charge were concentrated at the center. Thus, the electric field is:
\[ E = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{r^2} \]

4. **Conclusion**:
- The correct statement regarding the electric field \( E \) at a distance \( r \) from the center of the hollow metallic sphere is that:
- For \( a < r < b \), the electric field \( E = 0 \).

### Final Answer:
The electric field \( E \) is zero for \( a < r < b \).

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