Question

Four identical metal plates are places in air parallel to each other with distance $d$ from one another. the area of each plate is equal to $A$. The arrangement is shown in the figure. Find the capacitance of the system between plates $P$ and $R$.

Answer 1

In order to find the equivalent capacitance between two given plates, we can connect a battery across them and then find the charge distribution on the surfaces of each plate using Gauss theorem or the concept:
"Net electric field in a conductor is always zero."
Let the plates $P$ and $R$ have the potential $V$ and $0$ respectively and the plates $1$ and $4$ are the potential $V_1$.
Charge distribution is shown in the figure.
$\therefore q_1=C(V-V_1)$ from $(4,3)$ ...(1)
$q_2=CV$ from $(3,2)$ ...(2)
$q_1=C{V}_1$ from $(2,1)$ ...(3)
From equation (1) and (2)
$q_1=\frac{CV}{2}$ ...(4)
Charge supplied by the battery is
$Q=q_1+q_2=\frac{3}{2}V$ using (2) and (4)
$⟹CV=\frac{2}{3}Q$ ...(5)
If equivalent capacitance be $C^{\prime }$ the $C^{\prime }V=Q$ ...(6)
Dividing (5) by (6)
$\frac{C}{C^{\prime }}=\frac{2}{3}⟹C^{\prime }=\frac{3}{2}C=\frac{3}{2}\frac{{\in }_{0}A}{d}$