Capacitors In Series And Parallel

1. Definitions (Must Know)

A. Capacitance, C

Capacitance, C, is charge stored per unit potential difference:

C = Q/V

Unit: farad (F) = C V⁻¹.

B. Equivalent (combined) capacitance, C_eq

C_eq is the capacitance of a single capacitor that can replace a capacitor network between two terminals.

C. Parallel capacitors

Capacitors are in parallel if they are connected across the same two nodes. They have the same potential difference.

D. Series capacitors

Capacitors are in series if they are connected end-to-end and the junction between them has no other connections.

In series, the magnitude of charge on each capacitor is the same.

2. Key Ideas (What Earns Marks)

• Parallel:
  • same V across each capacitor
  • charges add: Qₜₒₜₐₗ = Q₁ + Q₂ + …
  • C_eq = C₁ + C₂ + …
• Series:
  • same charge magnitude on each capacitor: Q₁ = Q₂ = … = Q
  • voltages add: Vₜₒₜₐₗ = V₁ + V₂ + …
  • 1/C_eq = 1/C₁ + 1/C₂ + …
• Quick checks:
  • series: C_eq is smaller than the smallest capacitor
  • parallel: C_eq is larger than the largest capacitor

Series capacitors: how voltage splits (scaled)

Two curves showing the fraction of total voltage across each capacitor in a two-capacitor series pair as the capacitance ratio changes.

Scroll across the graph to read all labels.

Open full-size graph

View figure data

Values for Series capacitors: how voltage splits (scaled)

Capacitance ratio (C1/C2)	Across C1: V1/V = C2/(C1+C2)	Across C2: V2/V = C1/(C1+C2)
0.25	0.8	0.2
0.5	0.667	0.333
1	0.5	0.5
2	0.333	0.667
4	0.2	0.8

Values for Series capacitors: how voltage splits (scaled)

Capacitance ratio (C1/C2)	Across C1: V1/V = C2/(C1+C2)	Across C2: V2/V = C1/(C1+C2)
0.25	0.8	0.2
0.5	0.667	0.333
1	0.5	0.5
2	0.333	0.667
4	0.2	0.8

3. Detailed Explanations

A. Derivation (parallel)

In parallel, the potential difference is the same: V₁ = V₂ = … = V

Using Q = CV: Qₜₒₜₐₗ = Q₁ + Q₂ + …

Since Qᵢ = CᵢV,

Qₜₒₜₐₗ = (C₁ + C₂ + …)V

So: Qₜₒₜₐₗ = (C₁ + C₂ + …)V

Hence C_eq = Qₜₒₜₐₗ/V = C₁ + C₂ + ….

B. Derivation (series)

In series, charge cannot accumulate at the isolated junction. The same charge magnitude appears on each capacitor: Q₁ = Q₂ = … = Q

Voltages add: Vₜₒₜₐₗ = V₁ + V₂ + …

Using V = Q/C: Vₜₒₜₐₗ = Q/C₁ + Q/C₂ + …

Vₜₒₜₐₗ = Q(1/C₁ + 1/C₂ + …)

But Vₜₒₜₐₗ = Q/C_eq, so: 1/C_eq = 1/C₁ + 1/C₂ + …

C. Workflow for mixed networks

1. Identify any capacitors that share the same two nodes (parallel).
2. Identify any capacitors that are end-to-end with an isolated junction (series).
3. Reduce step-by-step until one C_eq remains.
4. If the network is connected to a supply, use Q = C_eqV to find the total charge, then work back to find charges/voltages on individual capacitors.

4. Common Mistakes

• Treating capacitors like resistors (series/parallel rules are reversed).
• Forgetting to convert μF or nF into F before calculations.
• Saying “series” when the junction has a third connection (then it is not series).
• For series capacitors: assuming the voltage is the same (it is not).
• For parallel capacitors: assuming the charge is the same (it is not).

5. Exam Tips

• For series, start with “Q is the same”, then use V = Q/C.
• For parallel, start with “V is the same”, then use Q = CV.
• If one capacitor in series has a much smaller C, it gets a much larger share of the total voltage.

6. Worked Examples

7. Mind Stretchers