For a cylindrical current carrying conductor, the magnetic field at any point on the axis is.........

Answer

To determine the magnetic field at any point on the axis of a cylindrical current-carrying conductor, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- Consider a cylindrical conductor with a hollow center. The current flows through the periphery of the cylinder.

2. **Identify the Axis**:
- Draw an axis along the length of the cylinder. This is the line where we want to find the magnetic field.

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

4. **Choose a Circular Loop**:
- For our analysis, consider a circular loop of radius \( r \) that lies inside the hollow part of the cylinder.

5. **Determine the Enclosed Current**:
- Since the current flows only through the periphery of the cylindrical conductor, the current enclosed by the circular loop (inside the hollow part) is zero:
\[ I_{\text{enclosed}} = 0 \]

6. **Substitute into Ampere's Law**:
- Substitute \( I_{\text{enclosed}} = 0 \) into Ampere's Law:
\[ \oint B \cdot dl = \mu_0 \cdot 0 = 0 \]

7. **Conclusion**:
- Since the integral of \( B \cdot dl \) is zero, it implies that the magnetic field \( B \) at any point on the axis of the hollow cylindrical conductor is also zero:
\[ B = 0 \]

### Final Answer:
The magnetic field at any point on the axis of a cylindrical current-carrying conductor is **zero**.

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