Electric field of a moving charge?

[physics user1]

I have a moving charge in a generic motion, and I pick a point p, how do I get the electric field caused by the charge on point p, at any time?
Can I use the coulomb definition of electric field that relates it's strenght with the Distance?
Or does that law works only when the charge is static?
How can I Do?
Btw this is not a homework question oregarding something, I'm just wondering cause I never encountered such a problem and they told me that coulomb law work for electrostatics

I know all maxwell equations

 

[QuantumQuest]

First, I have to tell you that in order to calculate the fields (electric and magnetic) of a moving charge , the math are somewhat involved. You can derive the fields from the (retarded) potentials. For a such derivation you can take a look here. Also, for a derivation using first principles take a look here.

 

[physics user1]

First, I have to tell you that in order to calculate the fields (electric and magnetic) of a moving charge , the math are somewhat involved. You can derive the fields from the (retarded) potentials. For a such derivation you can take a look here. Also, for a derivation using first principles take a look here.

What are the retarded Potential? Why can't I use the coulomb Law?
The magnetic field is not conservative, how can it have a Potential?

 

[QuantumQuest]

What are the retarded Potential?

Retarded potentials are the electromagnetic potentials generated by time-varying electric current or charge distributions in the past. For more see Wikipedia.

Why can't I use the coulomb Law?

Coulomb's Law can be applied for static charge(s) or in integral form for charge distributions. Now, if we have a moving charge we have a magnetic force as well. So, we can't consider only the electrostatic force in this case.

The magnetic field is not conservative, how can it have a Potential?

The magnetic field itself is neither conservative nor non-conservative. In order to characterize a field in general as conservative or not, you have to take the force applied on a particle moving along any closed path and calculate the net work done. But in this case, the magnetic field (effectively force) on a moving particle is at right angles to the motion so the work is always zero. So, there is no proper application of the concept.