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(A) Let the charges on the four plates be $Q_1 = Q$, $Q_2 = 2Q$, $Q_3 = -3Q$, and $Q_4 = -4Q$. When the plates are isolated, the charge on the outer s

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$5Q$ from $A$ to $B$

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(A) Let the charges on the four plates be $Q_1 = Q$, $Q_2 = 2Q$, $Q_3 = -3Q$, and $Q_4 = -4Q$. When the plates are isolated, the charge on the outer surfaces is $(Q_1 + Q_2 + Q_3 + Q_4)/2 = (Q + 2Q - 3Q - 4Q)/2 = -4Q/2 = -2Q$. Thus, the outer surface of the first plate has $-2Q$ and the outer surface of the fourth plate has $-2Q$. Consequently, the inner surface of the first plate has $Q - (-2Q) = 3Q$. Since the plates are metal, the inner surface of the second plate must have $-3Q$. The total charge on the second plate is $2Q$, so its outer surface has $2Q - (-3Q) = 5Q$. Similarly, the inner surface of the third plate has $-5Q$. The total charge on the third plate is $-3Q$, so its outer surface has $-3Q - (-5Q) = 2Q$. When the middle two plates are connected by a wire, they reach the same potential. Let the total charge on the combined system be $Q_{total} = Q_2 + Q_3 = 2Q - 3Q = -Q$. After connection, the charge redistributes such that the outer surfaces remain $-2Q$ each. The total charge on the two inner plates is $-Q$. Let the charge on the inner surfaces be $q$ and $-q$. Then, the charge on the outer surface of the second plate is $Q_{total}/2 = -Q/2$ and the third plate is $-Q/2$. Initial charge on the inner surface of the second plate was $5Q$. Final charge is $-Q/2$. Charge flowed through the wire $= 	ext{Initial} - 	ext{Final} = 5Q - (-Q/2) = 5.5Q$. Wait, re-evaluating: The charge on the inner surfaces of the middle plates before connection is $5Q$ (on the right side of plate $2$) and $-5Q$ (on the left side of plate $3$). When connected, the potential difference becomes zero. The charge $5Q$ on the right side of plate $2$ and $-5Q$ on the left side of plate $3$ neutralize. Thus, $5Q$ charge flows from $A$ to $B$.

Four very large metal plates are given the charges as shown in the figure. The middle two plates are then connected through a wire. Find the charge that will flow through the wire.

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