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Since the salt bridge helps balance the ions on both sides of the solution, the solution of the anode will probably become overly postive whereas the cathode's solution will become overly negative. I think with the solution unbalanced, the transfer of electrons on the wire will most likely stop and electricty would not be produced.
Over time, the electrons flow from one side of the cell to the other side, and the cell eventually reaches an equilibrium where the potential difference of the cell does not have a positive value. If there isn't a positive value to this difference, then the cell no longer really has the potential to be a battery. The salt bridge is necessary to keep both of the two solutions to stay neutral, so that chemical energy can be transformed into electrical energy.
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