A steady electric current is flowing through a cylindrical conductor. Then,

Options

A. The electric field at the axis of the conductor is zero.
B. The magnetic field at the axis of the conductor is zero.
C. The electric field in the vicinity of the conductor is zero.
D. The magnetic field in the vicinity of the conductor is zero.

Answer

To solve the question regarding the behavior of electric and magnetic fields in a cylindrical conductor carrying a steady electric current, we can analyze each option systematically.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a cylindrical conductor with a steady electric current flowing through it. We need to analyze the electric and magnetic fields both inside and outside the conductor.

2. **Option A - Electric Field at the Axis**:
- The electric field inside a conductor carrying a steady current is not zero. The current implies that there is a potential difference causing the flow of charge. Therefore, the electric field at the axis of the conductor is **not zero**.
- **Conclusion**: Option A is incorrect.

3. **Option B - Magnetic Field at the Axis**:
- According to Ampere's law, the magnetic field inside a cylindrical conductor at the axis (where the radius tends to zero) is zero because there is no current enclosed by an Amperian loop at the center. Hence, the magnetic field at the axis of the conductor is **zero**.
- **Conclusion**: Option B is correct.

4. **Option C - Electric Field in the Vicinity**:
- In the vicinity of the conductor (just outside), the net charge is zero because the positive charges in the atomic lattice balance the negative charges (electrons) flowing. Thus, the electric field in the vicinity of the conductor is also **zero**.
- **Conclusion**: Option C is correct.

5. **Option D - Magnetic Field in the Vicinity**:
- Outside the cylindrical conductor, the magnetic field is not zero. According to Ampere's law, if we take an Amperian loop around the conductor, the current flowing through the conductor produces a magnetic field in the surrounding space. Therefore, the magnetic field in the vicinity of the conductor is **not zero**.
- **Conclusion**: Option D is incorrect.

### Final Answer:
- The correct options are **B and C**. Options A and D are incorrect.