Capacitors in Series and Parallel
Reconnect two capacitors to compare shared charge, shared voltage and stored energy.
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Capacitors in Series and Parallel
For initially uncharged ideal capacitors after a DC source has established equilibrium, each capacitor obeys Q=CV. Values in μF multiplied by volts give μC. Energy is U=(1/2)CV2; using μF gives μJ.
In series, the neutral intermediate conductor enforces equal charge magnitudes Q1=Q2=Q, while V=V1+V2. Thus V=Q(1/C1+1/C2) and Ceq=C1C2/(C1+C2). In parallel, both capacitors share V and their charges add, so Ceq=C1+C2.
Worked example: same selectable capacitors, two connections
Set Capacitance C1=2 μF, Capacitance C2=4 μF and Source voltage V=3 V.
- Select Series. Ceq=2×4/(2+4)=4/3 μF≈1.33333333333 μF.
- Q=CeqV=(4/3)×3=4 μC, so each capacitor has charge magnitude4 μC.
- V1=4/2=2 V and V2=4/4=1 V. Their sum is3 V; the smaller capacitor carries the greater voltage.
- Stored energy is U=(1/2)(4/3)×32=6 μJ. Summing the two individual energies also gives (1/2)×2×22+(1/2)×4×12=4+2=6 μJ.
- Now select Parallel. Ceq=2+4=6 μF and both voltages are3 V.
- Q1=2×3=6 μC; Q2=4×3=12 μC. Battery charge is18 μC and stored energy is (1/2)×6×9=27 μJ.
Experiment: constraints and boundaries
Use the Capacitors in Series and Parallel module with the named controls.
- Reproduce the worked example, changing only Connection. Series shares charge; parallel shares voltage.
- Set Capacitance C1=1, Capacitance C2=8, Connection=Series and Source voltage V=0. All voltages, charges and stored energy are0, but equivalent capacitance remains8/9 μF.
- Increase Source voltage V to12. Equal series charge does not mean equal voltage: V1=32/3 V and V2=4/3 V.
- Inspect the circuit and use Reset view after panning. The exact fraction shown next to Ceq distinguishes mathematical exactness from a finite decimal display.
Limits of the equilibrium model
Capacitors are ideal and initially uncharged. This model omits switching transients, series resistance, leakage, arbitrary trapped charge and dielectric breakdown. Battery charge is the charge delivered to the equivalent capacitance, and stored energy is the sum in the two capacitors; no claim about transient charging losses is made.
Primary reference
OpenStax University Physics Volume2, section8.2: Capacitors in Series and in Parallel explains equal series charge, equal parallel voltage and equivalent capacitance.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Capacitors in Series and Parallel module.
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