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)

Configuration	Capacitance Formula	Key Variables
Parallel‑plate (dielectric of constant εᵣ)	$$C=\varepsilon_0\varepsilon_r\frac{A}{d}$$	A – plate area (larger → larger C) d – separation (smaller → larger C) ε₀ – permittivity of free space, 8.85 × 10⁻¹² F m⁻¹ εᵣ – relative permittivity (dielectric constant)
Isolated spherical conductor (radius r)	$$C=4\pi\varepsilon_0 r$$	r – radius of the sphere If a dielectric of constant εᵣ fills the space, multiply by εᵣ.

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. $$Q_{\text{tot}}=C_1V+C_2V+\dots =V(C_1+C_2+\dots)$$ Hence, $$C_{\text{tot}}=C_1+C_2+\dots$$
• Series connection: the same charge Q passes through each capacitor, but voltages add. $$V_{\text{tot}}= \frac{Q}{C_1}+ \frac{Q}{C_2}+ \dots = Q\!\left(\frac{1}{C_1}+ \frac{1}{C_2}+ \dots\right)$$ Rearranging, $$\frac{1}{C_{\text{tot}}}= \frac{1}{C_1}+ \frac{1}{C_2}+ \dots$$

4.2 Summary Table

Configuration	Formula for Total Capacitance	Reasoning
Parallel	$$C_{\text{tot}} = C_1 + C_2 + C_3 + \dots$$	Same voltage on each capacitor; charges add.
Series	$$\frac{1}{C_{\text{tot}}}= \frac{1}{C_1}+ \frac{1}{C_2}+ \frac{1}{C_3}+ \dots$$	Same charge on each capacitor; voltages add.

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

$$V(t)=V_0 e^{-t/RC}=V_0 e^{-t/\tau}$$ $$Q(t)=Q_0 e^{-t/RC}=Q_0 e^{-t/\tau}$$ $$I(t)=\frac{V_0}{R}e^{-t/RC}=I_0 e^{-t/\tau}$$

where \(V_0, Q_0, I_0\) 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 (\(\varepsilon_r\approx1\))

$$C = \varepsilon_0\varepsilon_r\frac{A}{d} = (8.85\times10^{-12})(1)\frac{0.020}{1.0\times10^{-3}} = 1.77\times10^{-10}\;\text{F}=177\;\text{pF}$$

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: $$V_1=\frac{Q}{C_1}= \frac{24}{3}=8\;\text{V},\qquad V_2=\frac{Q}{C_2}= \frac{24}{6}=4\;\text{V}$$

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: \(C = \varepsilon_0\varepsilon_r A/d\).
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: \(1/C_{\text{tot}} = \sum 1/C_i\); reduces total capacitance.
7. Parallel combination: \(C_{\text{tot}} = \sum C_i\); 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.