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With the movement of electrons from the anode to the cathode, the charges of the solutions in a galvanic cell can buildup to be increasingly positive and negative. When this buildup of charge occurs, it can stop the movement of the electrons since they wouldn't want to continue moving towards an increasingly negative area. The salt bridge works to keep the solutions neutrals (and thus keep the movement of electrons going) by releasing anions into the more positive solution and cations into the more negative solution.
Since e- move from left to right, eventually, the e- would build up on the right side and that would cause the battery to stop working/ make the e- stop moving. To avoid this, the salt bridge allows for the movement of ions so that e- don't build up on the right side.
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