To my knowledge, neither Volta nor Ampere nor Oersted, the founding fathers of electric current, knew that current is the transport of charge. Who, then, and by which experiment, proved the nature of electric current?

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    $\begingroup$ There was no such experiment. It was a view that gradually emerged after Volta's invention from measurements and theoretical musings done by multiple people. Volta already suggested a simple version of it (with electric fluid pushed by "electromotive force"). Ampere and Ohm probably contributed the most to its acceptance. $\endgroup$ – Conifold Feb 21 at 22:06