Capacitors in series have an equivalent capacitance equal to what?

• A. The reciprocal of the sum of reciprocals
• B. The sum of their capacitances
• C. The product of the capacitances
• D. The difference of the capacitances

Answer: (A)

When capacitors are in series, the same amount of charge flows through each one, so the charge on every capacitor is the same. The total voltage across the chain is the sum of the voltages across each capacitor. For each capacitor, V_i = Q / C_i, so the total voltage is V_total = Q(1/C1 + 1/C2 + ...). The equivalent capacitor carries the same charge Q with the total voltage V_total, so Q = Ceq · V_total. Substituting gives Q = Ceq · Q · (1/C1 + 1/C2 + ...). Canceling Q (assuming nonzero charge) yields 1 = Ceq · (1/C1 + 1/C2 + ...), hence Ceq = 1 / (1/C1 + 1/C2 + ...). For two capacitors, this becomes Ceq = 1 / (1/C1 + 1/C2) = (C1 C2)/(C1 + C2). This is the reciprocal of the sum of reciprocals. The other forms don’t describe series behavior: summing capacitances applies to parallel connections, and the product or difference isn’t the general rule for series.