upon closing a electric circuit DC or AC , the potential difference generates an electrical field and electrons start moving through the circuit based on the electric field force.

The distribution of this electric field is setup at a very high speed. A light bulb lights up almost instantaneously.
Where does the electric field starts first? at the terminal where electrons are pushed into the circuit, where electrons are pulled out from the circuit , or at both terminals simultaneously ? There must be some initial time frame to get the electrical field throughout the entire circuit.
can someone explain this ?

 

The electric field travels at the speed of light and emanates from the source generating the voltage difference across the lines (or line and return).

 

The electric field travels at the speed of light and emanates from the source generating the voltage difference across the lines (or line and return).

does the electric field is distributed at both + and - terminal of the source at the same time ?
or only at the terminal where electrons enters the circuit , almost at speed of light, but there must be a delay, heading towards the other terminal ?
i am looking for a mathematical formulation of this phenomena.

 

I'm not sure if you have read this article: https://en.wikipedia.org/wiki/Speed_of_electricity
"The speed at which energy or signals travel down a cable is actually the speed of the electromagnetic wave, not the movement of electrons. Electromagnetic wave propagation is fast and depends on the dielectric constant of the material."

So if you assume you have a voltage source that is terminated to ground or some other constant potential, and the voltage source turns on at some time t, then the electric field will propagate along the wire at a speed determined by the dielectric constant of the material. For a PCB trace, this will mostly depend on the dielectric constant of the PCB material, I believe. It's still quite fast, about 66% of the speed of light for most pcb materials.

Here is a blog post that goes a bit more in depth about the propagation speed for pcb traces: https://blogs.mentor.com/hyperblog/blog/tag/velocity-of-propagation/

 

does the electric field is distributed at both + and - terminal of the source at the same time ?
or only at the terminal where electrons enters the circuit , almost at speed of light, but there must be a delay, heading towards the other terminal ?
i am looking for a mathematical formulation of this phenomena.

It's not Either/Or, the electrons in the wire and the electric field act as a system together to guide electrical energy from both terminals at the source to the load.

https://en.wikipedia.org/wiki/Energy_current
https://en.wikipedia.org/wiki/Poynting_vector