7.2 Capacitors in series and parallel  (Page 2/2)

Capacitors in parallel

[link] (a) shows a parallel connection of three capacitors with a voltage applied. Here the total capacitance is easier to find than in the series case. To find the equivalent total capacitance $C_{\text{p}}$ , we first note that the voltage across each capacitor is $V$ , the same as that of the source, since they are connected directly to it through a conductor. (Conductors are equipotentials, and so the voltage across the capacitors is the same as that across the voltage source.) Thus the capacitors have the same charges on them as they would have if connected individually to the voltage source. The total charge $Q$ is the sum of the individual charges:

Using the relationship $Q=\text{CV}$ , we see that the total charge is $Q=C_{\text{p}}V$ , and the individual charges are $Q_1=C_{1}V$ , $Q_2=C_{2}V$ , and $Q_3=C_{3}V$ . Entering these into the previous equation gives

Canceling $V$ from the equation, we obtain the equation for the total capacitance in parallel $C_{\text{p}}$ :

Total capacitance in parallel is simply the sum of the individual capacitances. (Again the “ ... ” indicates the expression is valid for any number of capacitors connected in parallel.) So, for example, if the capacitors in the example above were connected in parallel, their capacitance would be

The equivalent capacitor for a parallel connection has an effectively larger plate area and, thus, a larger capacitance, as illustrated in [link] (b).

More complicated connections of capacitors can sometimes be combinations of series and parallel. (See [link] .) To find the total capacitance of such combinations, we identify series and parallel parts, compute their capacitances, and then find the total.

Section summary

• Total capacitance in series $\frac{1}{C_{\text{S}}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\text{.}\text{.}\text{.}$
• Total capacitance in parallel $C_{\text{p}}=C_1+C_2+C_3+\text{.}\text{.}\text{.}$
• If a circuit contains a combination of capacitors in series and parallel, identify series and parallel parts, compute their capacitances, and then find the total.

Conceptual questions

Problems&Exercises

Find the total capacitance of the combination of capacitors in [link] .

$\text{0.293 μF}$

Find the total capacitance of the combination of capacitors shown in [link] .

$2\text{.}\text{79 µF}$

Find the total capacitance of the combination of capacitors shown in [link] .