# Do we have an electrostatic field in a closed circuit with current?

I was wondering why the voltage drop were the same over resistors in parallel and I got the explanation that $$\nabla\times E=0$$ for the field in the circuit. This means that the field is conservative, so a closed line integral over it is zero.

I looked for a proof of this and I found a compendium that proved it for an electrostatic field. It proved it by looking at the field as point charges, and using Coloumbs law at each point charge.

What I am wondering is if we have a circuit with a battery and multiple resistors will there then be an electrostatic field over the circuit? The reason I am wondering is:

1. We have introduced a battery with chemical energy, does this change anything?
2. The electrons are moving, so I am wondering if we still have an electrostatic field. Static means something that is not moving?

And a static magnetic field, according to the Maxwell-Faraday law implies that $$\nabla \times \mathbf E = 0$$