Physics · Ch 2 — Electrostatic Potential and Capacitance

Combination of Capacitors

2.15

Combination of Capacitors

Exactly as resistors can be wired together in series or in parallel to obtain a resultant resistance different from any single available resistor, capacitors are routinely combined in a circuit to obtain a resultant (equivalent) capacitance different from any single capacitor on hand. The two combination rules for capacitance, worked out in detail in the following two subsections, come out the OPPOSITE way round from the corresponding rules for resistors: capacitors in SERIES combine like resistors in PARALLEL (reciprocals add), and capacitors in PARALLEL combine like resistors in SERIES (values add directly) -- a contrast worth keeping firmly in mind, since it is a common source of error. …

Table 1Series versus parallel combination of capacitors: formula comparison

	Series combination	Parallel combination
Equivalent capacitance	$1/C_s = 1/C_1 + 1/C_2 + \cdots$ (always smaller than the smallest $C_i$ )	$C_p = C_1 + C_2 + \cdots$ (always larger than the largest $C_i$ )
Charge on each capacitor	Same on every capacitor, $Q$	Different on each; splits in proportion to $C_i$
Potential difference across each	Different across each; splits in inverse proportion to $C_i$	Same across every capacitor, $V$

Figure 2Series and parallel capacitor circuits

What this figure shows. Two small circuit diagrams are drawn side by side. The LEFT diagram shows three capacitors $C_1$ , $C_2$ , $C_3$ drawn as standard parallel-line capacitor symbols connected one after another in a single chain (the right plate of $C_1$ wired directly to the left plate of $C_2$ , and the right plate of $C_2$ wired directly to the left plate of $C_3$ ), with the free left terminal of $C_1$ and the free right terminal of $C_3$ each connected by wire to opposite terminals of a battery labelled $V$ , illustrating a SERIES combination. The RIGHT diagram shows the same three capacitors $C_1$ , $C_2$ , $C_3$ instead drawn as three separate vertical branches between a shared top horizontal wire and a shared bottom horizontal wire, so that all three capacitors' left plates connect to the same top node and all three right plates connect to the same bottom …

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