Question

Three identical large metal plates of area $A$ are small at distances $d$ and $2d$ from each other. Top metal plate is uncharged, while the other metal plates have charges $+Q$ and $-Q$. Top and bottom metal plates are connected by switch $S$ through a resistor of unknown resistance. What energy (in mJ) is dissipated in the resistor when switch is closed?
(given: $\frac{{ϵ}_{0}A}{d}=6\mu F,Q=60\mu C$)

Answer 1

$C_1=\frac{{\epsilon }_{0}A}{2d}=3\mu F{C}_2=6\mu F$

$Q=60\mu F$

when switch is closed

$\frac{q_1}{C_1}=\frac{q_2}{C_2}=\frac{q_1}{3}=\frac{q_2}{6}$

$q_2=2q_1$

$q_1+q_2=60$

$q_1+2q_1=60$

$q_1=20$

$q_2=40$

Initial energy $=\frac{1}{2}\frac{Q^2}{C^2}=\frac{1}{2}\times \frac{60\times 60}{6\times 6}=50\times {10}^{-12}\times 6J=300\times {10}^{-12}$

Finial $=\frac{1}{2}[\frac{20\times 20}{3}+\frac{40\times 40}{6}]\times {10}^{-12}J$

$=\frac{1}{2}\left[\frac{2\times 400+1600}{6}\right]\times {10}^{-12}J$

$=\frac{1}{2}\left[\frac{800+1600}{6}\right]=\frac{2400}{12}\times {10}^{-12}$

Change in energy $=$ heat lost

$=2800\times {10}^{-12}J$