Combination of Capacitors
We can combine two or more capacitors to get some effective capacitance C. There are two ways to combine capacitors, series combination, and parallel combination.
Series Combination
In series combination capacitors are connected one after the other. The charge that flows in the positive terminal goes through all the capacitors by induction. So the charge is the same in all the capacitors if initially all are uncharged.
Consider three capacitors of capacitance `C_1,\ C_2`, and `C_3` are connected in series and the conductors of the series combination are connected to a source of EMF which gives a potential difference of V volt.
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| Series Combination of Capacitors |
Equivalent Capacitance in Series Combination
The potential of the battery is divided among the capacitors.
`V\ =\ V_1+\ V_2+\ V_3` ………………… eq. (1)
The potential difference across each capacitors are given by
`V_1\ =\ \frac{Q}{C_1} , V_2\ =\ \frac{Q}{C_2} and V_3\ =\ \frac{Q}{C_3}`
Let C_s is the equivalent capacitance of the series arrangement so the same charge Q will be on C_s with potential difference V across the plate
`V\ =\ \frac{Q}{C_s}` ………………… eq. (2)
Combining eq (1) and (2)
`V\ =\ V_1+\ V_2+\ V_3`
`\frac{Q}{C_s}\ =\ \frac{Q}{C_1}\ +\ \frac{Q}{C_2}\ +\ \frac{Q}{C_3} `
`\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2}\ +\ \frac{1}{C_3}`
Case I
For two capacitors in a series combination
`\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2}`
`\frac{1}{C_s} = \frac{C_1+C_2}{C_1C_2}`
`C_s\ =\ \frac{C_1 C_2}{C_1+ C_2}`
If `C_1=C_2\ =\ C`
`C_s\ =\ \frac{C\ \ C}{C+C}`
`C_s\ =\ \frac{C}{2\ }\ `
Case II
For three capacitors in a series combination
`\frac{1}{C_s}\ =\ \frac{1}{C_1}\ +\ \frac{1}{C_2}\ +\ \frac{1}{C_3}`
`\frac{1}{C_s}\ =\ \frac{C_1C_2+C_2C_3+C_3C_1}{C_1C_2C_3}`
`C_s=\ \frac{C_1C_2C_3}{C_1C_2+C_2C_3+C_3C_1}`
Case III
For n capacitors in a series combination
`\frac{1}{C_s}\ =\ \frac{1}{C_1}\ +\ \frac{1}{C_2}\ +\ \frac{1}{C_3}\ +\ ............\ +\ \frac{1}{C_n}\ `
Parallel Combination of Capacitors
In a parallel combination, capacitors are connected such that one plate of each capacitor is connected to one terminal (positive terminal) of the battery and the other plate of each capacitor is connected to the negative terminal of the battery.
In a parallel combination, the voltage across each capacitor remains the same, while the total charge stored in the combination is the sum of the charges on each capacitor. The distribution of charge among the capacitors is proportional to their capacitances.
Equivalent Capacitance in Parallel
Consider three capacitors of capacitance `C_1,\ C_2` and `C_3` are connected in parallel. If this combination is connected to a source with a potential difference of V, then each capacitor has the same voltage V.
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| Parallel combination of Capacitors |
To fine the equivalent capacitance of capacitors in parallel, consider that the total charge Q stored by the combination is the sum of charges on each individual capacitor.
`Q\ =\ Q_1\ +\ Q_2+\ Q_3`
Where,
`Q_1\ =\ C_1\ V`
`Q_2\ =\ C_2\ V`
`Q_3\ =\ C_3\ V`
Thus, the total charge can be written as
`Q\ =\ Q_1\ +\ Q_2+\ Q_3`
`Q\ =\ C_1\ V\ +\ C_2\ V+\ C_3\ V`
`Q\ =\ (C_1\ +\ C_2\ +\ C_3\ )\ V`
If C_p is the equivalent capacitance for the parallel combination, then `Q\ =\ C_p\ V`. Thus, we have
`C_p\ V\ =(C_1\ +\ C_2\ +\ C_3\ )\ V`
`C_p = C_1 + C_2 + C_3`
Thus, the equivalent capacitance for a parallel combination of capacitors is the sum of the capacitances of the individual capacitors.
Case I
For two capacitors in a parallel combination
`C_p\ =\ C_1\ +\ C_2`
If `C_1\ =\ C_2\ =\ C`
`C_p\ =\ 2\ C`
Case II
For three capacitors in a parallel combination
`C_p\ =\ C_1\ +\ C_2\ +\ C_3`
Case III
For n capacitors in a parallel combination
`C_p\ =\ C_1\ +\ C_2\ +\ C_3+\ .............+\ C_n`
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