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Can I build a battery with 2 cathodes and 1 electrolyte 

Dear student,
We cannot create a battery having 2 cathode as in a complete circuit electron should follow a closed path moves in a particular  direction. So we should have one positive negative terminal and electrolyte.
Regards

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no, you cant build a circuit with 2 cathode and electrolyte. The anode when reacts with electrolyte produces more electrons and has extra.on the other hand when the cathode reacts with the electrolyte in reduces the amount of electrolyte thus has a vacancy.In this circuit no electrons are being produced to go and fill the vacancy .Thus it is compulsory to have a cathode ,anode and electrolyte.

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