Four identical metal plates are placed in air parallel to each other with distances 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.

Explanation

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 at 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_(2)=CV" from (3,2) "...(2)`
` q_(1)=CV_(1)" from (2,1) "...(3)`
From equation (1) and (2)
`q_(1)=(CV)/(2)" "...(4)`
Charge supplied by the battery is
`Q=q_(1)+q_(2)=(3)/(2)V` using (2) ad (4)
`implies CV=(2)/(3)Q" "...(5)`
If equivalent capacitance be C' then C'V = Q `" "...(6)`
Dividing (5) by (6)
`(C)/(C)=(2)/(3)impliesC'=(3)/(2)C=(3)/(2)(epsilon_(0)A)/(d)`