Question

Consider an arrangement of six large parallel plates as shown in the figure, each of the plate’s area $A$. Initially when battery was not connected all plates were uncharged. The emf of the battery is $V$. Choose the correct option(s) after connecting the battery (in steady state):

• A. Total work done by battery is $\frac{{\epsilon }_{0}A}{d}{V}^2$
• B. Electric field between plates 3 and 4 is $\frac{V}{2d}$
• C. Potential difference between plate 2 and 5 is $\frac{V}{2}$
• D. Potential difference between plate 1 and 5 is $\frac{V}{2}$

Answer 1

Answer: (A), (B), (C), (D)

Potential difference between plate 1 and 5 is $\frac{V}{2}$

Given circuit can be reduced as,


$C_1=C_2=C_3=C_4=C_5=\frac{{\epsilon }_{0}A}{d}$
Circuit can be rearranged as
this is a balanced wheatstone bridge, hence charge on third capacitor is $Q_3=\frac{{\epsilon }_{0}AV}{2d}$
Electric field is
$E_3=\frac{Q_3}{A{\epsilon }_0}=\frac{\frac{{\epsilon }_{0}AV}{2d}}{A{\epsilon }_0}=\frac{V}{2d}$
Potential difference between 2 and 5
$=V_Q-V_S=\frac{V}{2}$
Potential difference between 1 and 4
$=V_P-V_S=\frac{V}{2}$