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In class we were learning about voltaic cells and my chemistry teacher joked that after modern society falls, we'll be the only ones who can recharge batteries with simple voltaic cells. And this got me thinking, is this possible? Could I create a salt bridge cell, attach wires to a store bought rechargeable battery, and charge it? I'm really curious and I can't ask my chem teacher as he's in the hospital recovering from an emergency appendix removal

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Any source of electricity of the required voltage and current could be used to charge a rechargeable ("secondary") cell. A home-made "primary" cell would work (though you might need a few in series to provide the needed voltage).

For that matter, you can generate DC to charge the cell many other ways:

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