Given statements:

I) The electric field at the axis of  conductor due  to the current is zero

II) The magnetic field at the axis of the conductor is zero

III) The Electric field in the vicinity of the conductor is zero

IV) The magnetic field in the vicinity of the conductor is zero

Detailed Solution:

If a current passes through any conductor, net charge through any section is zero. hence, electric field in the vicinity of the conductor is zero. We can observe using Biot and Savart law that the magnetic field at the axis of the cylindrical wire will be zero. Electric field at the axis of the wire cannot be zero otherwise the current through the wire cannot flow.

 Hence, the correct option is C.