Combination of Capacitors 🤝

Putting capacitors together lets us “tune” the total capacitance of a circuit. Two super-useful ways to hook them up are series and parallel. Let’s see how each one works. 🧐

1 · Capacitors in Series 🔗

  • The same charge \(Q\) flows through every capacitor because any surplus would push charges until balance returns. :contentReference[oaicite:0]{index=0}
  • Total potential difference is the sum of the drops:
    \(V = V_1 + V_2 + \dots + V_n\). :contentReference[oaicite:1]{index=1}
  • Using \(V_i = Q/C_i\) and \(V = Q/C\), we get the famous reciprocal rule:
    \[ \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}. \] :contentReference[oaicite:2]{index=2}
  • 📝 Tip: Series always reduces capacitance—the smallest capacitor dominates!

2 · Capacitors in Parallel 🛤️

  • Each capacitor feels the same potential difference \(V\). :contentReference[oaicite:3]{index=3}
  • Plate charges can differ: \(Q_i = C_i V\). :contentReference[oaicite:4]{index=4}
  • Total charge adds up, giving a direct-sum rule for capacitance:
    \[ C = C_1 + C_2 + \dots + C_n. \] :contentReference[oaicite:5]{index=5}
  • 💡 Tip: Parallel always increases capacitance—just add them like buckets placed side-by-side.

3 · Worked Example 💪

Four \(10\ \text{μF}\) capacitors form the network below (three in series, that combo in parallel with the fourth) and are connected to \(500\ \text{V}\).

  1. Series part: \[ \frac{1}{C’} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} \;\;\Longrightarrow\;\; C’ = \frac{10}{3}\ \text{μF}. \] :contentReference[oaicite:6]{index=6}
  2. Parallel step: \[ C_{\text{eq}} = C’ + 10 = 13.3\ \text{μF}. \] :contentReference[oaicite:7]{index=7}
  3. Charge on each series capacitor: \[ Q = C_1 V_1 = 1.7 \times 10^{-3}\ \text{C}. \] :contentReference[oaicite:8]{index=8}
  4. Charge on the lone parallel capacitor: \[ Q’ = C_4 V = 5.0 \times 10^{-3}\ \text{C}. \] :contentReference[oaicite:9]{index=9}

🚀 Strategy: Reduce complex combos step-by-step—series first, then parallel (or vice-versa) until just one capacitor remains.

NEET High-Yield Nuggets 🎯

  1. Series ⇒ charge same, potential adds, use reciprocals: \(\tfrac{1}{C} = \sum \tfrac{1}{C_i}\). :contentReference[oaicite:10]{index=10}
  2. Parallel ⇒ potential same, charge adds, just add capacitances directly. :contentReference[oaicite:11]{index=11}
  3. Series lowers total \(C\); parallel raises it—easy trick for quick checks. :contentReference[oaicite:12]{index=12}
  4. Break any network into simple series/parallel blocks; simplify ruthlessly. :contentReference[oaicite:13]{index=13}