Capacitors in Series

Capacitors are said to be connected in SERIES when they are wired end to end -- the second plate of the first capacitor joined directly to the first plate of the second, and so on down the chain -- such that the SUM of the individual potential differences across every capacitor in the chain equals the single total applied potential difference: $V=V_1+V_2+V_3$ for three capacitors in series.

The key physical fact about a series chain is that every capacitor in it carries the SAME MAGNITUDE of charge $Q$ -- charge is induced in strict sequence down the chain by simple electrostatic induction from one plate to the next, so no capacitor in a series chain can end up with a different charge magnitude from any other. Since each individual capacitor's own potential difference is $V_i=Q/C_i$, the total applied PD becomes $V=\dfrac{Q}{C_1}+\dfrac{Q}{C_2}+\dfrac{Q}{C_3}$.

Defining the single equivalent series capacitance as $C_S=Q/V$ (so the SAME charge $Q$ and the SAME total voltage $V$ characterise the equivalent capacitor as the real chain), substituting gives $\dfrac{Q}{C_S}=\dfrac{Q}{C_1}+\dfrac{Q}{C_2}+\dfrac{Q}{C_3}$, and the common factor $Q$ cancels from every term, leaving $\dfrac{1}{C_S}=\dfrac{1}{C_1}+\dfrac{1}{C_2}+\dfrac{1}{C_3}$ for three capacitors -- generalising directly, by the same argument extended to any number of capacitors, to $\dfrac{1}{C_{eq}}=\dfrac{1}{C_1}+\dfrac{1}{C_2}+\cdots+\dfrac{1}{C_n}$ for $n$ capacitors in series: the RECIPROCAL of the equivalent series capacitance equals the SUM of the individual reciprocals. A direct mathematical consequence of this reciprocal-addition rule is that $C_S$ is always SMALLER than the smallest individual capacitance in the whole chain -- adding another capacitor in series can only ever shrink the overall equivalent capacitance further, never grow it. If all $n$ capacitors happen to be identical, each of value $C$, this simplifies neatly to $C_{eq}=C/n$. …