(a) Capacitor in series
(i) Consider three capacitors of capacitance `C_(1),C_(2)` and `C_(3)` connected in series with a batter of voltage V as shown in the Figure (a)
(ii) As soon as the battery is connected to the capacitors in series, the electrons of charge -Q are transferred from negative terminal to the right plate of C3 which pushes the electrons of same amount -Q from left plate of `C_(3)` to the right plate of `C_(2)` due to electrostatic induction.
(a) Capacitors connected in series
(b) Equivalence capacitors `C_(s)`
(iii) Similarly, the left plate `C_(2)` pushes the charges of -Q to the right plate of `C_(1)` which induces the positive charge +Q on the left plate of `C_(1)`.
(iv) At the same time, electrons of charge -Q are transferred from left plate of `C_(1)` to positive terminal of the battery.
(v) By these processes, each capacitor stores the same amount of charge Q.
(vi) The capacitances of the capacitors are in general different, so that the voltage acorss each capacitor is also different and are denoted as `V_(1),V_(2)` and `V_(3)` respectively.
The total voltage across each capacitor must be equal to the voltage of the battery.
`V=V_(1)+V_(2)+V_(3)" ".......(1)`
Since, `Q=CV,`
we have `V=(Q)/(C_(1))+(Q)/(C_(2))+(Q)/(C_(3))`
`=Q[(1)/(C_(1))+(1)/(C_(2))+(1)/(C_(3))]" "......(2)`
(viii) If three capacitors in series are considered to form an equivalent single capacitor `C_(s)` shown in Figure (b), then we have `V=(Q)/(C_(s))`. Substituting this expression into equation (2), we get
`(Q)/(C_(s))=Q((1)/(C_(1))+(1)/(C_(2))+(1)/(C_(3)))`
`(1)/(C_(s))=(1)/(C_(1))+(1)/(C_(2))+(1)/(C_(3))" "......(3)`
(ix) Thus, the inverse of the equivalent capacited in series is equal to the sum of the smallest individual capacitance in the series.
(b) Capacitance in parallel
(i) Consider three capacitors of capacitance `C_(1),C_(2)` and `C_(3)` connected in parallel with a battery of voltage V as shown in Figure (a).
(ii) Since corresponding sides of the capacitors are connected in positive and negative terminals of the battery, the voltage across each capacitor is equal to the battery.s voltage.
(a) capacitors in parallel
(b) equivalent capacitance with the same total charge
(iii) Since capacitance of the capacitors is different, the charge stored in each capacitor is not the same. Let the charge stored in the three capacitors be `Q_(1),Q_(2)`, and `Q_(3)` respectively.
(iv) According to the law of conservation of total charge, the sum of these three charges is equal to the charge Q transferred by the battery,
`Q=Q_(1)+Q_(2)+Q_(3)" "......(1)`
Now, since Q=CV, we have
`Q=C_(1)V+C_(2)V+C_(3)V" "......(2)`
(v) If these three capacitors are considered to form a single capacitance `C_(p)` which stores the total charge Q as shown in the Figure (b) then we can write `Q=C_(p)V`. Substituting this in equation (2) we get
`C_(P)V=C_(1)V+C_(2)V+C_(3)V`
`C_(P)=C_(1)+C_(2)+C_(3)" ".......(3)`
(vi) Thus, the equivalent capacitance of capacitors connected in parallel is equal to the sum of the individual capacitances.
(vii) The equivalent capacitance `C_(p)` in a parallel connection is always greater than the largest individual capacitance. In a parallel connection, it is equivalent as area or each capacitance adds to give more effective area such that total capacitance increases.
