Explanation:

To determine the electric field produced by a long wire carrying a time-varying current, we can apply Ampere's law. Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the product of the current enclosed by the loop and the permeability of free space.

Applying Ampere's Law:
Consider a circular loop of radius r and center at the point (x, y, z) near the middle of the wire. The current enclosed by this loop is given by the current passing through the wire within the area bounded by the loop.

As the wire carries a time-varying current, the magnetic field around the wire will vary with time. This variation in the magnetic field will induce an electric field according to Faraday's law of electromagnetic induction.

Electric Field due to Time-Varying Magnetic Field:
According to Faraday's law, the induced electric field is given by the negative rate of change of magnetic flux through the loop with respect to time. Since the magnetic field around the wire is changing with time, the induced electric field will not be zero.

Direction of the Electric Field:
The induced electric field will be perpendicular to the loop and will form closed loops around the wire.

Components of the Electric Field:
The electric field induced by the time-varying magnetic field will have components in all three directions (Ex, Ey, and Ez). The exact values of these components will depend on the specific geometry and the current variation of the wire.

Conclusion:
In conclusion, the electric field produced by a long wire carrying a time-varying current at a point near its middle will not be zero. It will have components in all three directions (Ex, Ey, and Ez). The exact values of these components will depend on the specific geometry and the current variation of the wire.