Question

Four identical parallel conductors, carrying charges $Q,-2Q,3Q$ and $4Q$ are placed as shown in the figure. The plates carrying charges $Q$ and $4Q$, are connected through a switch, as shown. The amount of charge flowing through the switch, after closing it is found to be $nQ$. Find $n$.

Answer 1

Ref. image I

$\frac{Q-2Q+3Q+4Q}{2=3Q}$

Ref. image II

$V_{AD}=0$

$\Rightarrow E_{1}d+E_{2}2d+E_{3}4d=0$

$\Rightarrow \frac{Q_1-3Q}{A{\in }_0}d+\frac{Q_1-5Q}{A{\in }_0}2d+\frac{Q_1-2Q}{A{\in }_0}4d=0$

$\Rightarrow (Q_1-3Q)+(Q_1-5Q)2+(Q_1-2Q)4=0$

$\Rightarrow Q_1-3Q+2Q_1-10Q+4Q_1-8Q=0$

$\Rightarrow 7Q_1=21Q$

$\Rightarrow Q_1=\frac{21}{7}Q$

$Q_1>Q$ $\Rightarrow Q_1=3Q$

$\therefore$ Charge flow from D to A

Charge flow through the switch $=Q_1-Q=3Q-Q=2Q$

$n=2$