Two Capacitors in Series

Capacitors in series combine the way resistors in parallel do, and the reason is worth following rather than memorising. Wire two capacitors end to end and the middle section — the bottom plate of the first joined to the top plate of the second — is isolated from everything else. Whatever charge leaves that plate must arrive at this one, so every capacitor in a series string carries the identical charge $Q$. Each then holds its own voltage, $V_1 = Q/C_1$ and $V_2 = Q/C_2$, and those add to the applied voltage. Divide through by the common $Q$ and you get $1/C_t = 1/C_1 + 1/C_2$, which for two parts is the product-over-sum form on this page.

Two equal 10 µF capacitors in series make 5 µF, not 20. A 10 µF with a 1 µF gives $(10 \times 1)/(10 + 1) = 0.91\ \mu\text{F}$ — barely less than the small one, because the small capacitor needs the most volts to accept the shared charge and therefore dominates the total. That is the general behaviour: the result is always smaller than the smallest member, and a large capacitor in series with a small one is very nearly the small one alone. Voltage divides in inverse proportion to capacitance, so in that pair, 100 V applied puts about 91 V across the 1 µF part and only 9 V across the 10 µF.

The geometric picture explains why it must be so. Capacitance for parallel plates is $C = \varepsilon A/d$, and stacking capacitors in series is effectively increasing $d$ — putting more insulating distance between the outermost plates — which lowers capacitance. Stacking them in parallel increases $A$, which raises it. That is also the practical use of a series string: the working voltage adds even as the capacitance falls, so two 400 V parts in series will stand 800 V, and high-voltage DC links are routinely built this way.

The famous error is the one this page exists to prevent: series capacitors are not added. The mirror-image mistake — treating parallel capacitors with product-over-sum — is just as common. If you can only remember one anchor, remember that a series connection always makes capacitance worse and parallel always makes it better, then check your answer against that. The second error is far more hazardous and almost never taught. The voltage division above holds only at DC in an ideal world; real capacitors, and electrolytics in particular, have leakage currents that differ from part to part, and over minutes the string will redistribute itself until one capacitor carries most of the applied voltage and fails. Series electrolytic banks therefore need balancing resistors across each part, sized so their current is many times the worst-case leakage. Finally, do not assume two parts marked 400 V in series give a comfortable 800 V rating — they give it only while they share, and sharing is what the balancing resistors are for.