To begin with, none. Drop two dissimilar terminals into an electrolyte solution (making a battery) and they are initially at zero volts. Then one of them accumulates electrons and one loses electrons, at which point an electric field is present and any additional electrons that are moved do so in opposition to this field.
In short, it’s own
If you mean in a circuit, when we connect the terminals in the battery, the flow of electrons itself produces not only an overall current but a multitude of tiny eddy currents.
No surface is perfectly uniform so inevitably those charges create dissimilar and competing fields because of how the electrons are distributed. The net result is the real field that we care about and can measure
A battery is a thing that is capable of holding an internal electric field. If you flow electricity through a wire it just heats up, it can’t store a meaningful amount of charge. A battery and a capacitor can store charge, partly because they are internally insulated, meaning electricity cannot flow around freely inside them to even things out.
In fluid flow a bucket is a bit analogous, you flow water into a bucket and it can store it. If you dump water on the grass it just dissipates. If you flow water through a pipe there is some storage, but not much. A bucket has walls and prevents the water from flowing out, so you can make use of it later.
This might be pedantic, but there is only one electric field and one magnetic field, and they are linked. Everything operates against the same fields. By definition, a field takes on a value everywhere.
Electrons need a low resistance path relative to their voltage to move freely, and that’s what a circuit provides. The movement of electrons induces a magnetic field, which in turn powers the system.
The electric and magnetic energies are parts of the same force, and we take advantage of the batteries voltage to push electrons to create the magnetic field. The push of electrons is backwards from what we consider “classical” electricity flow (electrons go from negative to positive).
Putting a resistance on the circuit draws energy from the field induced from the moving electrons. This converts the electromagnet energy into heat, movement, vibration, etc. in other words: work.
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