A current I ampere flows along an infinitely long straight thin walled tube, then the magnetic induction at any point inside the tube is .

Options

A. Infinite
B. Zero
C. `mu_(0)/4pi .2I/R`
D. `2I/R`

Answer

To find the magnetic induction at any point inside an infinitely long straight thin-walled tube carrying a current \( I \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- Visualize the thin-walled tube as a cylindrical shell with a circular cross-section. The current \( I \) flows uniformly along the circumference of the tube.

2. **Identify the Magnetic Field Inside the Tube**:
- We need to determine the magnetic field at a point inside the tube. Since the tube is hollow, there is no current flowing through the area inside the tube.

3. **Apply Ampère's Circuital Law**:
- According to Ampère's Circuital Law, the line integral of the magnetic field \( \mathbf{B} \) around a closed loop is equal to the permeability of free space \( \mu_0 \) times the total current \( I_{\text{enc}} \) enclosed by that loop:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} \]

4. **Choose an Amperian Loop**:
- Consider a circular Amperian loop of radius \( r \) inside the tube. Since the tube is hollow, the current enclosed by this loop is zero because all the current flows along the circumference of the tube.

5. **Evaluate the Integral**:
- Since there is no current enclosed by the Amperian loop, we have:
\[ I_{\text{enc}} = 0 \]
- Therefore, applying Ampère's law gives:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \cdot 0 = 0 \]

6. **Conclude the Magnetic Field Inside the Tube**:
- Since the line integral of the magnetic field around the loop is zero, it implies that the magnetic field \( \mathbf{B} \) inside the tube must also be zero:
\[ \mathbf{B} = 0 \]

### Final Answer:
The magnetic induction at any point inside the tube is \( \mathbf{B} = 0 \).