Capacitance in Series

Figure 16.1(a) shows a series connection of three capacitors with a voltage applied. As for any capacitor, the capacitance of the combination is related to charge and voltage by [latex]C=\frac{Q}{V}[/latex]. In a series connection, the capacitors are connected “end-to-end,” so there is only one path for charge to move. That single-path feature is what forces all series capacitors to share the same magnitude of charge.

Note in Figure 16.1 that opposite charges of magnitude [latex]Q[/latex] flow to either side of the originally uncharged combination of capacitors when the voltage [latex]V[/latex] is applied. Conservation of charge requires that equal-magnitude charges be created on the plates of the individual capacitors, since charge is only being separated in these originally neutral devices. To see why, imagine pushing a small amount of negative charge onto the leftmost outer plate. That charge cannot cross the insulating gap inside the first capacitor, so it forces an equal amount of negative charge to appear on the adjacent inner plate of that same capacitor. But that inner plate is physically connected by a wire to the next capacitor’s plate, so the “extra” negative charge cannot simply sit there without affecting the next capacitor. The only way the conductor can remain overall neutral (no net charge piling up in the wire segments between capacitors) is for charge separation to occur on each capacitor, producing the same magnitude [latex]Q[/latex] on every capacitor in the series chain. In other words, series wiring does not allow different capacitors to “take different charges”; the chain behaves like a single conduit for charge separation.

This charge constraint has an important geometric interpretation. Each capacitor stores charge by separating it across a gap. If you place capacitors in series, you are effectively placing multiple gaps one after another. The combination resembles a single capacitor with an effective plate separation greater than that of any individual capacitor alone. (See Figure 16.1(b).) Larger plate separation means smaller capacitance, because it is harder (for a given voltage) to pull large amounts of opposite charge onto plates that are effectively farther apart. That is why a series combination always has a total capacitance smaller than any single capacitor in the series.

We can find an expression for the total capacitance by considering the voltage across the individual capacitors shown in Figure 16.1. Start from the definition [latex]C=\frac{Q}{V}[/latex]. Solving for the voltage across a single capacitor gives [latex]V=\frac{Q}{C}[/latex]. In series, the magnitude of the charge on each capacitor is the same, so the voltages across the individual capacitors are
[latex]{V}_{1}=\frac{Q}{{C}_{1}}[/latex], [latex]{V}_{2}=\frac{Q}{{C}_{2}}[/latex], and [latex]{V}_{3}=\frac{Q}{{C}_{3}}[/latex]. The total voltage supplied by the source is the sum of the voltage drops across each capacitor (just as voltage drops add in series resistor circuits):

Now define the equivalent series capacitance [latex]{C}_{\text{S}}[/latex] as the capacitance of a single capacitor that would store the same charge [latex]Q[/latex] when connected to the same total voltage [latex]V[/latex]. By definition,

Because the total voltage is also the sum of the individual voltages, we can write

Substitute [latex]{V}_{1}=\frac{Q}{{C}_{1}}[/latex], [latex]{V}_{2}=\frac{Q}{{C}_{2}}[/latex], and [latex]{V}_{3}=\frac{Q}{{C}_{3}}[/latex]:

Canceling the common factor [latex]Q[/latex] (which is nonzero whenever the capacitors are charged) yields the key series formula:

The “...” indicates that this expression works for any number of capacitors in series. This form immediately explains why the series capacitance is always smaller than any individual capacitance. Adding positive terms on the right-hand side makes [latex]\frac{1}{C_{\text{S}}}[/latex] larger than [latex]\frac{1}{C_1}[/latex], [latex]\frac{1}{C_2}[/latex], etc., which forces [latex]C_{\text{S}}[/latex] to be smaller than each [latex]C_i[/latex]. Physically, placing capacitors in series increases the effective separation and therefore reduces the ability of the combination to store charge for a given applied voltage.

Capacitors in Parallel

Figure 16.2(a) shows a parallel connection of three capacitors with a voltage applied. In a parallel connection, each capacitor is connected directly across the same two nodes of the circuit (the same two “rails” of potential). Because conductors are equipotentials, the potential difference between those two nodes is the same for every capacitor in the parallel group. Therefore, the voltage across each capacitor is the same, and it is equal to the source voltage [latex]V[/latex]. This “same voltage” constraint is the defining feature of a parallel connection.

If each capacitor has the same voltage, then each stores whatever charge it would store if it were connected alone to that same voltage. Using [latex]Q=\text{CV}[/latex], a capacitor with a larger capacitance stores more charge at the same [latex]V[/latex]. The total charge drawn from the source is simply the sum of the charges placed on the individual capacitors, because the source supplies charge into multiple branches:

To derive the parallel formula, write the total charge and individual charges using [latex]Q=\text{CV}[/latex]. The equivalent capacitor has [latex]Q={C}_{\text{p}}V[/latex]. Each individual capacitor has [latex]{Q}_{1}={C}_{1}V[/latex], [latex]{Q}_{2}={C}_{2}V[/latex], and [latex]{Q}_{3}={C}_{3}V[/latex]. Substitute these into [latex]Q={Q}_{1}+{Q}_{2}+{Q}_{3}[/latex]:

Cancel the common factor [latex]V[/latex] (which is the same across each branch in a parallel connection):

So total capacitance in parallel is simply the sum of the individual capacitances. This makes physical sense if you think in terms of geometry: connecting capacitors in parallel is like increasing the effective plate area, because the source can store charge on multiple sets of plates at the same voltage. Increasing plate area increases capacitance, which is why the equivalent capacitor in Figure 16.2(b) is drawn as having larger effective area.

More complicated connections of capacitors can sometimes be combinations of series and parallel. (See Figure 16.3.) To find the total capacitance of such combinations, we identify series and parallel parts, compute their equivalent capacitances step by step, and then combine those equivalents until the entire network reduces to one capacitance.