Series & Parallel Combinations
Same charge in series, same voltage in parallel.
Lesson at a glanceShow the key points
- Equivalent capacitance C = Q/V for the network.
- Series: same charge; 1/C = 1/C₁ + 1/C₂ + … (31.5).
- Parallel: same voltage; C = C₁ + C₂ + … (31.6).
- General method: assign charges and potentials to plates, apply Q = CV, eliminate.
Real circuits join capacitors together. Any network with two terminals behaves like one capacitor, whose capacitance we can work out.
Equivalent capacitance
Connect a battery across the two terminals of a network. If it supplies charge Q (+Q from one terminal, −Q from the other) at potential difference V, the network’s equivalent capacitance is C = Q/V. A single capacitor of that value would store the same charge at the same voltage.
Series
Capacitors joined end to end, one after another. The battery puts +Q on the first plate and −Q on the last. Each inner pair of connected plates is isolated, so if one plate gets −Q, its partner must be left with +Q. Result: every capacitor in series carries the same charge Q. The voltages add:
A 6.0 µF and a 3.0 µF capacitor are joined in series across an 18 V battery. Find the charge and the voltage on each.
Parallel
Capacitors connected across the same two points. All their positive plates are at one potential and all their negative plates at another, so each capacitor in parallel has the same voltage V. The charges add: Q = C₁V + C₂V + C₃V.
Same charge on each. Voltages add. 1/C adds, so C is smaller than the smallest one.
Same voltage on each. Charges add. C adds, so C is bigger than the biggest one.
A 4 µF and an 8 µF capacitor are in parallel, and this pair is in series with a 12 µF capacitor. Find the equivalent capacitance.
A 6 µF and a 3 µF capacitor in series are connected in parallel with a 4 µF capacitor, across a 12 V battery. Find the charge on each and the voltage across the 3 µF.
A general method
When a network is not simply series and parallel, go back to first principles:
- Name the two terminals P and N.
- Imagine a battery with + at P and − at N, supplying +Q and −Q.
- Write the charge on every plate. Facing plates carry equal and opposite charge; isolated groups of connected plates keep zero total charge. Use unknowns Q₁, Q₂ … where needed.
- Set VN = 0 and VP = V; connected plates share a potential. Use unknowns V₁, V₂ … where needed.
- Write Q = CV for each capacitor and eliminate the unknowns to get C = Q/V.
Example of the method
C₁ = 6 µF runs from P to a junction. From that junction, C₂ = 1 µF and C₃ = 2 µF each run to N. Find the equivalent capacitance using the general method.
- 1Equivalent capacitance: C = Q/V for the whole network.
- 2Series: same Q; 1/C = Σ1/Cᵢ (31.5).
- 3Parallel: same V; C = ΣCᵢ (31.6).
- 4General method: assign plate charges and potentials, write Q = CV for each, eliminate.
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.