## 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.

19.4.1 Series Combination Of Capacitors | Electric Potential And Capacitor | Physics 12