Do moving charged particles have both magnetic and electric fields? 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?


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*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?

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

This is not the case in general. A conducting material that has reached a steady state will have zero electric field inside the conductor, but the field outside the conductor is determined by the net charge on the conductor as well as its shape. That being said, it is definitely possible to have a setup with flowing electrons and no electric field. If you have a loop of wire with a number of unmoving positive charges that matches the number of moving electrons, there will be no electric field outside the wire, because the net charge is zero. 
To be more specific, the electric field produced by an electron is superimposed on the electric field produced by a proton, and the resultant field is effectively zero when you are far enough away compared to the separation distance of the two charges.

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

Again, if there is a net charge, there will be a net electric field. If there is not, then there will be no electric field. There will also be a magnetic field as a result of the charges moving.
A: 
2.Now consider an electron moving with certain velocity (less than speed of light), Still is there electric field around it?

There is still an electric field associated with the moving charge, but there is now also a magnetic field. Only moving charges can experience this magnetic force.

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

For an ideal conductor of zero resistance and carrying a finite current, the electric force on the conduction electrons approaches zero. Therefore the electric field is zero inside. This is achieved by a static rearrangement of the electrons inside to create a static electric field that cancels any other internal electric fields.
However, there is still an external electric field from the excess charge in the wire that comes from the current source. Think of a capacitor slowly discharging where there is an electric field in the region between the plates, but no electric field inside the plates carrying the current.

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

A moving charge has a magnetic and electric field, which can be derived from the Lienard-Wiechert potentials This is all discussed in any book on electricity and magnetism at the undergraduate level.

3.Hypothetically, If the electron is moving with speed more than that of light. What happens now?

Special relativity teaches us that there is a universal speed limit that the speed of light happens to travel at and electrons therefore cannot travel faster than the speed of light, no matter how much energy you put in.
A: 
Still is there electric field around it?

Yes.  The electron is moving (in our reference frame), so now there is a magnetic field (in our reference frame), but nothing happens to the electric field.

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

The electrons in the conductor produce an electric field outside the conductor; however, realistically, there will be just as many protons in the conductor as electrons, and hence the net electric field outside the conductor is zero.

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

The electric field is still there (in some sense), but its zero, because the electrons and protons in the conductor cancel each other out, so we don't care about it.  (Actually, I believe that if you take into account relatavistic effects, which is probably silly not to do in the context of electrodynamics, there will be a nonzero electric field).  That being said, if for some reason there were stream of moving electrons with no protons, then we would observe both a (nonzero) magnetic and electric field.

  
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*Hypothetically, If the electron is moving with speed more than that of light. What happens now?
  

Special relativity says this can't happen :).  In any case, if we play dumb for a moment, the only thing that would change is the current, and hence the strength of the magnetic field.  
