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We need an electric solution for the electrons to move from the anode to the cathode but why ?

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You need the solution to move the cations around and maintain local equilibrium for a sustained current in a closed current loop, otherwise you just get an initial current pulse to equilibrate the charge. But two metals create a potential difference on their own, due to different fermi levels / work function.

https://en.wikipedia.org/wiki/Volta_potential

orion
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