## 19.4.1 Series Combination of Capacitors In a series combination, capacitors are connected sequentially, plate to plate. The right plate of one capacitor connects to the left plate of the next capacitor in the chain. ### Charge Distribution In a series circuit, due to electrostatic induction, the charge on each capacitor is the same and is equal to the total charge supplied by the battery. $$Q = Q_1 = Q_2 = Q_3 = \ldots$$ ### Voltage Distribution The total potential difference ($V$) supplied by the battery is divided among the individual capacitors. It is the sum of the potential differences across each capacitor. $$V = V_1 + V_2 + V_3 + \ldots$$ ### Equivalent Capacitance ($C_e$) This is the single capacitance that would have the same effect as the entire series combination. The formula is derived as follows: 1. Start with the voltage equation: $V = V_1 + V_2 + V_3$ 2. Substitute $V = Q/C$ for each term: $V = \frac{Q_1}{C_1} + \frac{Q_2}{C_2} + \frac{Q_3}{C_3}$ 3. Since charge is the same for all capacitors ($Q_1 = Q_2 = Q_3 = Q$): $V = Q \left( \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \right)$ 4. Rearrange the equation: $\frac{V}{Q} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}$ 5. The equivalent capacitance is defined as $C_e = \frac{Q}{V}$, so $\frac{1}{C_e} = \frac{V}{Q}$. 6. This gives the final formula for three capacitors: $$\frac{1}{C_e} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}$$ ### General Formula for n Capacitors For any number $n$ of capacitors connected in series, the formula is: $$\frac{1}{C_e} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \ldots + \frac{1}{C_n}$$ ### Key Property The equivalent capacitance of a series combination is always less than the smallest individual capacitance in the combination. Connecting capacitors in series effectively increases the distance between the plates of the equivalent capacitor. | Formula/Law | Equation | |---|---| | Total Voltage (Series) | $V = V_1 + V_2 + V_3 + \ldots$ | | Total Charge (Series) | $Q = Q_1 = Q_2 = Q_3 = \ldots$ | | Equivalent Capacitance | $\frac{1}{C_e} = \sum_{i=1}^{n} \frac{1}{C_i}$ | The primary effect of connecting capacitors in series is to increase the total voltage they can withstand, but at the cost of reducing the overall capacitance of the circuit.