recall and use C = Q / V

Capacitors and Capacitance (Cambridge IGCSE/A‑Level 9702 – 19 Capacitance)

Learning Objective

Recall and use the fundamental relationship

\$C=\frac{Q}{V}\$

where C is the capacitance, Q the charge stored on the plates and V the potential difference between them.

1. What Is a Capacitor?

• A passive component that stores energy in an electric field.
• Made of two conductors (plates) separated by an insulating material (dielectric).
• When a voltage is applied, equal and opposite charges accumulate on the plates.

2. Capacitance – Definition, Units & Geometry

2.1 Definition & SI Unit

• Definition: ability of a system to store charge per unit voltage.
  \$C=\frac{Q}{V}\$
• Unit: farad (F) 1 F = 1 C V⁻¹.
• Common practical ranges: µF (10⁻⁶ F), nF (10⁻⁹ F), pF (10⁻¹² F).

2.2 Geometry‑Based Formulas (syllabus requirement)

3. Energy Stored in a Capacitor

Work to move a small charge dq onto a plate when the voltage is V is V dq. Since V = q/C, the total work (energy) is

\$U=\int_{0}^{Q}\frac{q}{C}\,dq=\frac{1}{2}\frac{Q^{2}}{C}=\frac{1}{2}CV^{2}=\frac{1}{2}QV\$

• All three forms are interchangeable – choose the one that best matches the data given.
• Useful for AO2 (explain) questions about energy transfer.

4. Series and Parallel Combinations

4.1 Why the Formulas Work

• Parallel connection: each capacitor has the same voltage V. Total charge is the sum of individual charges.

  \$Q{\text{tot}}=C1V+C2V+\dots =V(C1+C_2+\dots)\$

  Hence,

  \$C{\text{tot}}=C1+C_2+\dots\$

• Series connection: the same charge Q passes through each capacitor, but voltages add.

  \$V{\text{tot}}= \frac{Q}{C1}+ \frac{Q}{C2}+ \dots = Q\!\left(\frac{1}{C1}+ \frac{1}{C_2}+ \dots\right)\$

  Rearranging,

  \$\frac{1}{C{\text{tot}}}= \frac{1}{C1}+ \frac{1}{C_2}+ \dots\$

4.2 Summary Table

5. Discharging a Capacitor (19.3)

5.1 RC Time‑Constant

• When a charged capacitor is connected across a resistor R, the voltage, charge and current decay exponentially.
• Time‑constant: τ = RC (units of seconds).
• After a time τ the voltage (and charge) has fallen to \(e^{-1}\) ≈ 37 % of its initial value.

5.2 Exponential Decay Equations

\$V(t)=V0 e^{-t/RC}=V0 e^{-t/\tau}\$

\$Q(t)=Q0 e^{-t/RC}=Q0 e^{-t/\tau}\$

\$I(t)=\frac{V0}{R}e^{-t/RC}=I0 e^{-t/\tau}\$

where \(V0, Q0, I_0\) are the initial values at t = 0.

5.3 Sketch of V‑t and I‑t Curves

5.4 Numerical Example

Find the voltage after 3 s for a 10 µF capacitor discharged through a 2 kΩ resistor, initially charged to 12 V.

• Time‑constant: τ = RC = (2 × 10³ Ω)(10 × 10⁻⁶ F) = 0.020 s.
• Using \(V(t)=V_0e^{-t/τ}\):

  \$V(3\text{ s}) = 12\,\text{V}\;e^{-3/0.020}=12\,\text{V}\;e^{-150}\approx 0\;\text{V}\$

• After just a few τ (≈0.06 s) the capacitor is essentially discharged – a useful point for AO2 questions.

6. Example Calculations (Illustrating Core Formulas)

6.1 Parallel‑Plate Capacitor

Find C for a capacitor with:

• Plate area \(A = 0.020\;\text{m}^2\)
• Separation \(d = 1.0\;\text{mm}=1.0\times10^{-3}\;\text{m}\)
• Dielectric: air (\(\varepsilon_r\approx1\))

\$\$C = \varepsilon0\varepsilonr\frac{A}{d}

= (8.85\times10^{-12})(1)\frac{0.020}{1.0\times10^{-3}}

= 1.77\times10^{-10}\;\text{F}=177\;\text{pF}\$\$

6.2 Energy in a Charged Capacitor

A 5 µF capacitor is charged to 200 V.

\$Q = CV = (5\times10^{-6})(200)=1.0\times10^{-3}\;\text{C}\$

\$U = \tfrac12 CV^{2}= \tfrac12 (5\times10^{-6})(200)^{2}=0.10\;\text{J}\$

6.3 Series Combination

Two capacitors, 3 µF and 6 µF, in series across a 12 V battery.

\$\frac{1}{C{\text{eq}}}= \frac{1}{3}+ \frac{1}{6}= \frac{1}{2}\;\Rightarrow\;C{\text{eq}}=2\;\mu\text{F}\$

Charge on each:

\$Q = C_{\text{eq}}V = (2\;\mu\text{F})(12\;\text{V}) = 24\;\mu\text{C}\$

Voltages:

\$\$V1=\frac{Q}{C1}= \frac{24}{3}=8\;\text{V},\qquad

V2=\frac{Q}{C2}= \frac{24}{6}=4\;\text{V}\$\$

6.4 Mixed Combination

Three identical 2 µF capacitors in parallel, then in series with a 4 µF capacitor.

Parallel block: \(C_P = 2+2+2 = 6\;\mu\text{F}\)

Series with 4 µF:

\$\frac{1}{C{\text{tot}}}= \frac{1}{6}+ \frac{1}{4}= \frac{5}{12}\;\Rightarrow\;C{\text{tot}}= \frac{12}{5}=2.4\;\mu\text{F}\$

7. Practical Considerations

• Dielectric breakdown: Exceeding the material’s electric‑field limit makes the dielectric conductive, causing failure.
• Leakage current: Real dielectrics are not perfect insulators; a small current may flow over time, slowly discharging the capacitor.
• Temperature coefficient: Capacitance can vary with temperature, especially for electrolytic types.
• Polarity: Polarised capacitors (e.g., electrolytic) must be connected with the correct orientation; reverse bias can damage them.
• Measuring C: Use an LCR meter or a known RC time‑constant method (measure τ, then \(C = τ/R\)).

8. Summary Checklist (AO1)

1. Definition: \(C = Q/V\).
2. Unit: farad (F) = coulomb per volt.
3. Parallel‑plate formula: \(C = \varepsilon0\varepsilonr A/d\).
4. Isolated spherical capacitor: \(C = 4\pi\varepsilon_0 r\) (multiply by εᵣ if a dielectric fills the space).
5. Energy stored: \(U = \tfrac12 CV^{2} = \tfrac12 QV = Q^{2}/(2C)\).
6. Series combination: \(1/C{\text{tot}} = \sum 1/Ci\); reduces total capacitance.
7. Parallel combination: \(C{\text{tot}} = \sum Ci\); adds capacitances.
8. Discharging: \(V(t)=V_0e^{-t/RC}\), τ = RC, exponential decay of V, Q and I.

9. Practice Questions (AO2 & AO3)

1. A 5 µF capacitor is charged to 200 V. (a) Find the charge stored. (b) Find the energy stored.
2. Two capacitors, 3 µF and 6 µF, are connected in series across a 12 V battery. Determine the voltage across each capacitor.
3. Three identical 2 µF capacitors are connected in parallel. What is the total capacitance? If this combination is then connected in series with a 4 µF capacitor, what is the final total capacitance?
4. Derive the expression for the capacitance of an isolated spherical conductor of radius 5 cm in vacuum.
5. A parallel‑plate capacitor has plates of area 0.015 m² separated by 0.5 mm of mica (\(\varepsilon_r = 5.4\)). Find its capacitance and the energy stored when charged to 150 V.
6. A 10 µF capacitor, initially charged to 12 V, is discharged through a 2 kΩ resistor. (a) Calculate the time‑constant τ. (b) What voltage remains after 0.1 s? (c) Sketch the V‑t curve and indicate τ on the diagram.