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well you should note however that it would only do no work for a homogeneous electric field, you don't need high powered math since the displacement integral is zero the work is zero very simple... these days I have been working on a system that gets energy out of the E field now the hard stuff is how to get that energy out say you accelerate electrons without energy cost and the total kinetic energy is increased how do you get energy out wihout thermal loses to make overunity?