A current of I ampere flows along an infinitely long straight thin walled hollow metallic cylinder of radius r. The magnetic field at any point inside the cylinder at a distance x from the axis of the cylinder is

Options

A. `oo`
B. `(mu_0I)/(2pir)`
C. `(mu_0I)/(2pix)`
D. zero

Explanation

To solve the problem of finding the magnetic field at any point inside a hollow metallic cylinder carrying a current \( I \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
We have an infinitely long, thin-walled hollow metallic cylinder of radius \( r \). The current \( I \) flows along the surface of this cylinder.

2. **Identify the Region of Interest**:
We need to find the magnetic field at a point inside the cylinder, specifically at a distance \( x \) from the axis of the cylinder, where \( x < r \) (since we are considering points inside the hollow cylinder).

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

4. **Determine the Enclosed Current**:
Since the cylinder is hollow and the current flows only on its surface, if we consider a circular path of radius \( x \) (where \( x < r \)), there is no current enclosed within this path. Therefore:
\[ I_{\text{enclosed}} = 0 \]

5. **Substitute into Ampere's Law**:
Substituting \( I_{\text{enclosed}} = 0 \) into Ampere's Circuital Law gives:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \cdot 0 = 0 \]
This implies that the line integral of the magnetic field around the path is zero.

6. **Conclude About the Magnetic Field**:
Since the line integral of the magnetic field is zero and there is no current enclosed, it follows that the magnetic field \( \mathbf{B} \) inside the hollow cylinder is also zero:
\[ \mathbf{B} = 0 \quad \text{(for } x < r\text{)} \]

### Final Answer:
The magnetic field at any point inside the hollow metallic cylinder at a distance \( x \) from the axis of the cylinder is:
\[ \mathbf{B} = 0 \]