Capacitors
10.0 Combination of capacitors
10.0 Combination of capacitors
There are majorly two types of combination are found in capacitors:
1. Series Combination:
In series connection, charge on the all plates of capacitors will remain same and total voltage applied by battery will divide on different plates in the ratio of their capacitance. And then total voltage will be equal to the addition of the individual voltage drops. So from figure, $$\begin{equation} \begin{aligned} \frac{{{V_1}}}{{{V_2}}} = \frac{{{C_2}}}{{{C_1}}} \\ {V_1} = \left( {\frac{{{C_2}}}{{{C_1} + {C_2}}}} \right)V \\ {V_2} = \left( {\frac{{{C_1}}}{{{C_1} + {C_2}}}} \right)V \\ V = {V_1} + {V_2} \\ \frac{q}{C} = \frac{q}{{{C_1}}} + \frac{q}{{{C_2}}} \\ \frac{1}{C} = \frac{1}{{{C_2}}} + \frac{1}{{{C_2}}} \\\end{aligned} \end{equation} $$
Here $C$ is the equivalent capacitance. Further we can extend this result for any number of capacitors connected in series,$$\frac{1}{C} = \frac{1}{{{C_2}}} + \frac{1}{{{C_2}}} + \frac{1}{{{C_3}}} + ...$$
Following points are important in case of series combination of capacitors.
- In case of series connection the equivalent capacitance is always less than the individual capacitance.
- If $n$ capacitors, of equal capacity $C$ are connected in series, then their equivalent capacitance is $\frac{C}{n}$.
2. Parallel Combination:
In parallel combination of capacitor the voltage difference between the plates of capacitor remain same and the total
charge is equal to the sum of the charges on each capacitor. So,
$$\begin{equation} \begin{aligned} \frac{{{q_1}}}{{{q_2}}} = \frac{{{C_1}}}{{{C_2}}} \\ {q_1} = \left( {\frac{{{C_1}}}{{{C_1} + {C_2}}}} \right)q \\ {q_2} = \left( {\frac{{{C_2}}}{{{C_1} + {C_2}}}} \right)q \\ q = {q_1} + {q_2} \\ CV = {C_1}V + {C_2}V \\ C = {C_1} + {C_2} \\\end{aligned} \end{equation} $$In this way we can extend this result for any number of capacitors. $$C = {C_1} + {C_2} + {C_3} + ....$$
