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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?
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Electrostatic field means that the fields are static. But that is compatible with a static flow of current.

According to the Ampére-Maxwell law, that implies a static magnetic field.

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

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Yes after a short while, the battery, the resistors and the wires will establish stationary state in the circuit where electric field is constant and electrostatic everywhere. In the battery there are other forces than macroscopic EM forces (microscopic forces due to chemical reactions) that push the mobile charges against the electrostatic field inside the battery.

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