Question

In electrostatic systems, a potential difference (i.e, a voltage) always corresponds to an electric field. Is this also true about circuits? Specifically, does the potential difference between the two terminals of the battery correspond to an electric field anywhere? Or do circuits allow us to have "voltages without fields?"

Answer 1

A battery is made of two electrodes, inside an electrolyte. The two electrodes are at different potentials. Positive charges gather at the anode and negative charges at the cathode. If the battery is in open circuit, the charges will simply accumulate at the surfaces of the electrodes. The battery will have an open-circuit voltage and an electric field exists inside the battery, between the anode and cathode. The shape and nature of this electric field, will depend on the type, structure and composition of the battery.

Once these charges are accumulated, further conduction is opposed, by the formation of a Helmholtz Double layer (interfacial).

When a load is connected between the battery terminals, current flows through the conductor. The electric field will be shaped around the conductor, depending on the potential change along that conductor. So E is defined by the load as,

Hence we can not say that its the "voltage without foelds". Its just that the field is governed by the nature of load.