Consider a charged particle (electron or proton) at rest. It is surrounded by its own electric field.

Now consider an electron moving with certain velocity. Is there still an electric field around it?

1. If it has an electric field around it, why is it that when electrons are moving in a conductor (i.e. current is flowing in a conductor) there is no electric field outside the conductor?

2. Now, when a current is flowing in a conductor (I'm not sure what happens if the motion is not inside conductor) it produces a magnetic field around it. I'm lost. What happened to the electric field? Is it still there? Are there both electric fields & magnetic fields? Why don't we talk about them?