10 min readConcept sheet2-mark definitions, 3-mark derivations for series and parallel combinations, and 2 or 3-mark network numericals

Capacitors, capacitance and their combinations

A capacitor is two conductors separated by an insulator. Capacitance C = Q/V. In parallel capacitances add; in series their reciprocals add.

CapacitanceC = Q/V; SI unit farad (F); 1 F = 1 C V⁻¹; 1 μF = 10⁻⁶ F, 1 pF = 10⁻¹² F

Series1/C = 1/C₁ + 1/C₂ + 1/C₃; the same charge Q on each

ParallelC = C₁ + C₂ + C₃; the same potential difference V across each

What C depends onSize, shape and separation of the conductors, and the medium between them; not Q or V

What is a capacitor?

A capacitor is two conductors separated by an insulator (a dielectric or air). When it is charged, one conductor holds +Q and the other −Q. The potential difference V between them is proportional to Q. The ratio C = Q/V is the capacitance. It tells how much charge the capacitor stores per volt. The farad is a very large unit, so we use μF and pF. Too high a voltage breaks down the insulator. For air this happens at about 3 × 10⁶ V m⁻¹, its dielectric strength. The circuit symbol is two short parallel lines of equal length.

Derivation: capacitors in series

C₁, C₂ and C₃ are joined end to end across a supply of potential difference V.

1
Each capacitor gets the same charge Q: the joined inner plates of two neighbouring capacitors are isolated, so their charges (−Q and +Q) add up to zero.
2
The potential differences are V₁ = Q/C₁, V₂ = Q/C₂ and V₃ = Q/C₃.
3
They add up to the supply: V = V₁ + V₂ + V₃ = Q(1/C₁ + 1/C₂ + 1/C₃).
4
For the equivalent capacitor C, V = Q/C.
5★
So, 1/C = 1/C₁ + 1/C₂ + 1/C₃. The equivalent is smaller than the smallest capacitor.

★ key step

Derivation: capacitors in parallel

C₁, C₂ and C₃ are joined between the same two points, across a supply V.

1
Each capacitor has the same potential difference V.
2
The charges are Q₁ = C₁V, Q₂ = C₂V and Q₃ = C₃V.
3
Total charge: Q = Q₁ + Q₂ + Q₃ = (C₁ + C₂ + C₃)V.
4
For the equivalent capacitor, Q = CV.
5★
So, C = C₁ + C₂ + C₃. The equivalent is larger than the largest capacitor.

★ key step

How to draw series and parallel combinations

Capacitor symbol: two short parallel lines of equal length, with a gap between them.
Series: draw C₁, C₂ and C₃ one after another on a single wire, with the battery across the ends. Mark +Q and −Q on each, and V₁, V₂, V₃ below.
Parallel: draw two horizontal wires joined to the battery, and put C₁, C₂ and C₃ as vertical rungs between them. Mark Q₁, Q₂, Q₃ and the common V.

Worked example: three capacitors

Problem

Capacitors of 2 μF, 3 μF and 6 μF are connected to a 12 V battery. Find the equivalent capacitance (a) in series and (b) in parallel. For the series case, find the charge and the voltage on each.

Working

(a) 1/C = 1/2 + 1/3 + 1/6 = (3 + 2 + 1)/6 = 1, so C = 1 μF.
Charge on each: Q = CV = 1 μF × 12 V = 12 μC.
Voltages: 12/2 = 6 V, 12/3 = 4 V and 12/6 = 2 V. Check: 6 + 4 + 2 = 12 V.
(b) C = 2 + 3 + 6 = 11 μF; total charge = 11 μF × 12 V = 132 μC.

Answer

(a) 1 μF; 12 μC on each; 6 V, 4 V and 2 V; (b) 11 μF, 132 μC

Worked example: a mixed network

Problem

C₁ = 3 μF and C₂ = 6 μF are in series. This pair is in parallel with C₃ = 4 μF, across a 10 V supply. Find the equivalent capacitance and the charge on each capacitor.

Working

Series pair: C₁₂ = (3 × 6)/(3 + 6) = 18/9 = 2 μF.
With C₃ in parallel: C = 2 + 4 = 6 μF.
Charge on C₃ = 4 μF × 10 V = 40 μC.
Charge on the series pair = 2 μF × 10 V = 20 μC, so C₁ and C₂ each carry 20 μC.
Check: total = 40 + 20 = 60 μC = 6 μF × 10 V.

Answer

6 μF; C₃ carries 40 μC; C₁ and C₂ carry 20 μC each

In real life

A ceiling fan has a small capacitor (a few microfarads) that helps it start and run smoothly. When a fan hums but turns slowly, an electrician often replaces this capacitor. A camera flash also stores its charge in a capacitor.

Exam angle

3 marks: “Derive the expression for the equivalent capacitance of three capacitors connected in series.” Same Q on each; V = V₁ + V₂ + V₃ = Q(1/C₁ + 1/C₂ + 1/C₃) = Q/C; so 1/C = 1/C₁ + 1/C₂ + 1/C₃.

2 marks: “Define capacitance and give its SI unit. On what factors does it depend?” C = Q/V; unit farad. It depends on the size, shape and separation of the conductors and on the medium between them, not on Q or V.

3 marks: “You have three 3 μF capacitors. How will you connect them to get (a) 1 μF, (b) 9 μF and (c) 4.5 μF?” (a) All three in series. (b) All three in parallel. (c) Two in series (1.5 μF), then this pair in parallel with the third: 1.5 + 3 = 4.5 μF.

Remember it

Capacitors are the opposite of resistors: in series, reciprocals add; in parallel, values simply add.