Figure shows four plates each of plate area A and separated between plates is d. Two switches `S_(w_1) and S_(w_2)` can activate two batteries in the circuit.

Switch `S_(w_1) and S_(w_2)` are closed, What is the charge on plate 2?

Options

A. `3/2 (epsilon_0AV)/(D)`
B. `3/2 (epsilon_0AV)/(d)`
C. `1/3 (epsilon_0AV)/(d)`
D. `4/3(epsilon_0AV)/(d)`

Answer

D

Explanation

When both the switches are closed, circuit is
Charge on plate 2
`C_(eq)=(2C)/(3)=(2epsilon_(0)A)/(3d)`
`q_(0)=((2C)/(3))(2V)=(4CV)/(3)`
Charge on plate 1
Where switch `Sw_(2)` is opened, circuit is

`[((x-V))/(-2V)]C+(x-2V)C+xC=0`
or `x=(5)/(3)V`
Charge on plate 4 is `-(5)/(3)CV=(-5epsilon_(0)AV)/(2d)`
Alternative method
`(-q_(1))/(C)+V+(q_(2))/(C)=0` or `-q_(1)+q_(2)=-CV`
`(-q_(1))/(C)-((q_(1)+q_(2)))/(C)+2V=0`
or `q_(1)+2q_(2)=2CV`
`q_(1)=(4CV)/(3),q_(2)=(CV)/(3)`