For the same total energy and power output would a battery powered vehicle be lighter than one which uses a fuel burning engine (Currently)?
This question is intended for helicopters.
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Poorly. That is a reason why we don't see battery powered aircraft beyond some experiments. The specific power is still much worse for batteries.
That is more than an order of magnitude difference in favour of hydrocarbon fuel (almost 2 orders by weight, one by volume). Even taking into account the 2-3 times higher efficiency of electric engines (which do not suffer the theoretical efficiency limits associated with heat engines) hydrocarbon fuel is an order of magnitude better by weight.
Electric engine won't give any advantage to electricity.
Well, the difference in energy density is so huge that even large difference in engine weight wouldn't help much anyway. For models, electricity is OK as due to the square-cube law they have a lot of power and lift and small-scale heat engines are difficult. But for aircraft carrying humans, batteries are very far from usable.