Find the equivalent capacitance of the combination shown in figure (31.11a) between the point P and N.

Explanation

Let us connect a battery between the points P and N. The charge and the potentials are shown in figure The positive terminal of the battery supplies a charge+Q which appears on the plate `A_1`. The facing plate `A_2` gets a charge_Q. The plates `A_2`. `A_3` and `A_5` taken together from an isolated system. The total charge on these three plates should be zero. Let a charge `Q_1` appear on `A_3`, then a charge `Q-Q_1` will appeat on `A_5` to make the total charge zero on the three plates. The plate `A_4` will get a charge`-Q_1 (facing plate of `A_3) and `A_6` will get a charge `-(Q-Q_1) ( facing plate of `A_5`). The total charge `_Q on A_4 and A_6` is supplied by the negative terminal of the battery. This completes the charge distribution.
Next, suppose the potential at the point N is zero and at P it is V. The potential of the plates `A_4 and A_6` is also zero. The potential of the plate `A_1` is V. The plates `A_2, A_3 and A_5` are at the same potential. Let this common potential be `V_1`. This completes the potential distributino.
Applying the capacitor equation `Q=CV` to the three capacitors,
`Q=C_1(V-V_1)................(1)`
`Q_1= C_2V_1...............(2)`
and `Q-Q_1=C_3V_1..................(3)
Adding (2) and (3),
`Q =(C_2+C_3)V_1`
or `Q/C_2+C_3= V_1`...............(4)
From(1), `Q/C_1=V-V_1`...................(5)
Adding (4) and (5)
`Q/C_2+C_3+ Q/C_1= V`
or`)C_1+C_2+C_3)Q/C_1(C_2+C_3)=V`
or `C=Q/V=C_1(C_2+C_3)/C_1+C+N64_2+C_3`.
It may be noted that the above example could be solved bby using the equations for series-parallel combinations. However, the general method wax used to demonstrate its application.