Silver ions "(Ag^+)" react with the Chloride ions "(Cl^-)" to form a precipitate. The end product consume the free ions of Silver ions"(Ag^+)" , impairing current flow. Accordingly, the activities within the cell stops.
The net ionic equation for the reaction is;
"Ag^+_{(aq)}+Cl^-_{(aq)}\\to AgCl_{(s)}"
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