Question

Three identical metallic plates are kept parallel to one another at separation $a$ and $b$ as known in figure. The outer plates are connected by a thin conducting wire and a charge $Q$ is placed on the central plate. Find the charge on all the six surfaces.

Answer 1

Let the three plates be denote by $A,B$ and $C$ and the six surfaces as $A-1,A_2,B_1,B_2,C_1,C_2$ as shown in figure. Taking into account the capacitive effects, one can easily get the charges on $A_2,B_1,B_2$ & $C_1$.
If $C_1$ and $C_2$ be the capacitance of capacitors formed by plates $A$ & $B$ and $B$ & $C$.
respectively, then $C_1=\frac{{\in }_{0}A}{a}$ & $C_2=\frac{{\in }_{0}A}{b}$
Where $A=$ area of each plate as $C_1$ & $C_2$ are in parallel,
$Q_1=Q\frac{C_1}{C_1+C_2}$ and $Q_2=Q\frac{C_2}{C_1+C_2}$
Solving we get,
$Q_1=\frac{Qb}{a+b}$ and $Q_2=\frac{Qa}{a+b}$
Hence, change on $B_1{Q}_1=\frac{Qb}{a+b}$
charge on $B_2,Q_2=\frac{Qb}{a+b}$
charge on $A_2,Q_1=\frac{Qb}{a+b}$
charge on $C_2,Q_2=\frac{Qb}{a+b}$
As net charge on plates $A$& $C$ must be zero, charges of $A-1$ & $C_2$ together should be negative of charges on $A_2$ & $C_1$ $Q$. Using the fact the electric field must we get,
charge on $A_1=Q/2$
charge on $C_2=Q/2$