8.2 Capacitors in Series and in Parallel

Capacitors in series and parallel are key to understanding electrical circuits. Series connections reduce overall capacitance, while parallel connections increase it. This impacts how charge and voltage distribute across the network.

Knowing how to calculate equivalent capacitance helps predict circuit behavior. Series capacitors share charge but have different voltages, while parallel capacitors share voltage but have different charges. This affects energy storage and distribution in the network.

Capacitors in Series and Parallel

Equivalent capacitance in circuits

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• Capacitors connected in series have an equivalent capacitance ($C_{eq}$) that is always less than the smallest individual capacitance in the series
  • Calculated using the formula: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$
  • Example: Two capacitors with capacitances of 2 µF and 4 µF connected in series have an equivalent capacitance of 1.33 µF
• Capacitors connected in parallel have an equivalent capacitance that is always greater than the largest individual capacitance in the parallel network
  • Calculated using the formula: $C_{eq} = C_1 + C_2 + ... + C_n$
  • Example: Two capacitors with capacitances of 2 µF and 4 µF connected in parallel have an equivalent capacitance of 6 µF

Charge and voltage across capacitors

• Capacitors in series have the same charge ($Q$) across all capacitors
  • Voltage ($V$) across each capacitor calculated using the formula: $V_i = \frac{Q}{C_i}$, where $C_i$ is the capacitance of the $i$-th capacitor
  • Total voltage across the series network is the sum of voltages across individual capacitors
  • Example: Two capacitors with capacitances of 2 µF and 4 µF connected in series with a total charge of 10 µC will have voltages of 5 V and 2.5 V, respectively
• Capacitors in parallel have the same voltage across all capacitors
  • Charge on each capacitor calculated using the formula: $Q_i = C_i V$, where $C_i$ is the capacitance of the $i$-th capacitor
  • Total charge in the parallel network is the sum of charges on individual capacitors
  • Example: Two capacitors with capacitances of 2 µF and 4 µF connected in parallel with a voltage of 10 V will have charges of 20 µC and 40 µC, respectively
• The potential difference across capacitors in parallel is the same, while in series, it varies based on capacitance

Effects of capacitor networks

• Total stored energy ($U$) in a capacitor network calculated using the formula: $U = \frac{1}{2} C_{eq} V^2$, where $C_{eq}$ is the equivalent capacitance and $V$ is the voltage across the network
  • For capacitors in series, the total stored energy is less than the sum of the energies that would be stored in each capacitor individually
  • For capacitors in parallel, the total stored energy is equal to the sum of the energies stored in each capacitor
  • Example: Two capacitors with capacitances of 2 µF and 4 µF connected in series with a voltage of 10 V will have a total stored energy of 33.3 µJ, while the same capacitors connected in parallel will have a total stored energy of 100 µJ
• Charge distribution in capacitor networks
  • In series, the charge is the same across all capacitors, but the voltage divides according to the inverse of the capacitance values
  • In parallel, the voltage is the same across all capacitors, but the charge distributes according to the capacitance values
  • Total charge in a parallel network is the sum of the charges on each capacitor, while in a series network, the total charge is equal to the charge on any individual capacitor
  • Example: Three capacitors with capacitances of 1 µF, 2 µF, and 3 µF connected in parallel with a voltage of 12 V will have charges of 12 µC, 24 µC, and 36 µC, respectively, while the same capacitors connected in series with a total charge of 6 µC will have equal charges of 6 µC on each capacitor
• The electric potential energy stored in a capacitor network depends on the arrangement of capacitors and their individual capacitances

Circuit Analysis with Capacitors

• Kirchhoff's laws are essential for analyzing complex circuits containing capacitors
• Capacitance affects the behavior of circuits, influencing current flow and voltage distribution
• Circuit analysis techniques can be applied to determine equivalent capacitance, charge distribution, and potential differences in capacitor networks