Q: suppose that I know that outside of some conductor circulates a constant electric current , I have $\nabla \times \vec{E} = 0$ and $\nabla \cdot \vec{E} = 0$. How do I prove that $\vec{E} = 0 $ outside of the conductor.
More details:
The conductor is: An ideal solenoid is an infinite cylindric conductor with a cylindric cavity along the same axis. On the surface of the conductor circulates a constant electric current in the direction perpendicular to its axis. Show that the electromagnetic field created by the current vanishes outside the conductor and the cavity
I proved that the magnetic field $\vec{B} = 0$ outside of the conductor and I want to prove that the electric field $\vec{E}$ is also zero outside. With Faraday's law, I have $\nabla \times \vec{E} = rot(\vec{E}) = curl(\vec{E}) = 0$. With Gauss's law I have $\nabla \cdot \vec{E} = div(\vec{E}) = 0$
How with that information, I can prove that $\vec{E} = 0$ outside of the ideal solenoid.
Thanks!!