The battery does work on the electrons. Work is a transfer of energy. So the electrons store the energy, not the field. No field, except the combined EM-field, contains energy! So even the E- and M-fields of EM-radiation don't carry energy. The Poynting vector tells you that energy only goes in the direction of propagation!!! And it is all kinetic. Any potential energy mentioned in this video is the potential energy of electrons, not fields. Don't forget, a battery is like the slope of a mountain and the electrons are like balls on that slope. Mountains don't contain energy, but balls do!
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