Capacitance
1. Definition and Fundamental Relation
- Capacitance is the ability of a system to store electric charge per unit potential difference.
- For any capacitor the capacitance C is defined by
$$C=\frac{Q}{V}$$
where Q is the magnitude of charge on each plate (or conductor) and V is the potential difference between the conductors.
2. Symbol, SI Unit and Common Multiples
- Symbol: C
- SI unit: farad (F) \(1\;\text{F}=1\;\text{C V}^{-1}\)
- Common sub‑multiples (used in examinations):
- microfarad (µF) \(1\;\mu\text{F}=10^{-6}\;\text{F}\)
- nanofarad (nF) \(1\;\text{nF}=10^{-9}\;\text{F}\)
- picofarad (pF) \(1\;\text{pF}=10^{-12}\;\text{F}\)
3. Capacitance of Simple Conductors
3.1 Parallel‑plate capacitor
For an ideal parallel‑plate arrangement the capacitance is
$$C=\frac{\varepsilon A}{d}$$
- \(A\) – plate area (m²)
- \(d\) – separation between the plates (m)
- \(\varepsilon\) – absolute permittivity of the material between the plates.
Effect of a dielectric
If a dielectric of relative permittivity \(\varepsilon_{r}\) fills the space,
$$\varepsilon=\varepsilon_{0}\varepsilon_{r},\qquad
\varepsilon_{0}=8.85\times10^{-12}\;\text{F m}^{-1}$$
The capacitance is increased by the factor \(\varepsilon_{r}\).
Quick derivation (AO2)
The electric field between the plates is \(E=V/d\).
Surface charge density \(\sigma = Q/A\).
From Gauss’s law \(E=\sigma/\varepsilon\) ⇒ \(\sigma=\varepsilon V/d\).
Substituting \(\sigma=Q/A\) gives \(Q=(\varepsilon A/d)V\) ⇒ \(C=Q/V=\varepsilon A/d\).
3.2 Isolated spherical conductor
An isolated conducting sphere of radius \(r\) behaves as a single‑plate capacitor whose other “plate’’ is at infinity. Its capacitance is
$$C_{\text{sphere}}=4\pi\varepsilon_{0}r$$
- Only the radius matters – the larger the sphere, the larger the capacitance.
- Useful for AO1 questions that ask for the capacitance of a single conductor (e.g. a metal ball of radius 5 cm has \(C\approx5.6\;\text{pF}\)).
4. Combination of Capacitors
4.1 Series connection
- Same charge \(Q\) flows through each capacitor.
- Total voltage is the sum of individual voltages:
$$V_{\text{tot}}=V_{1}+V_{2}+\dots+V_{n}$$
- Using \(V_i=Q/C_i\) leads to
$$\frac{1}{C_{\text{eq}}}= \frac{1}{C_{1}}+\frac{1}{C_{2}}+\dots+\frac{1}{C_{n}}$$
4.2 Parallel connection
- All capacitors experience the same voltage \(V\).
- Total charge is the sum of the individual charges:
$$Q_{\text{tot}}=Q_{1}+Q_{2}+\dots+Q_{n}$$
- Since \(Q_i=C_iV\), the equivalent capacitance is
$$C_{\text{eq}}=C_{1}+C_{2}+\dots+C_{n}$$
4.3 Example – Series‑parallel network
- Two capacitors, \(C_{1}=4.0\;\mu\text{F}\) and \(C_{2}=6.0\;\mu\text{F}\), are in series; the combination is placed in parallel with \(C_{3}=3.0\;\mu\text{F}\). The network is connected to a 12 V battery.
- Find the equivalent capacitance, the charge on each capacitor, and the total energy stored.
Solution
- Series part:
$$\frac{1}{C_{s}}=\frac{1}{4.0\;\mu\text{F}}+\frac{1}{6.0\;\mu\text{F}}=\frac{5}{12\;\mu\text{F}}\;\Rightarrow\;C_{s}=2.4\;\mu\text{F}$$
- Parallel addition:
$$C_{\text{eq}}=C_{s}+C_{3}=2.4\;\mu\text{F}+3.0\;\mu\text{F}=5.4\;\mu\text{F}$$
- Charge (same for the series pair):
$$Q=C_{\text{eq}}V=5.4\;\mu\text{F}\times12\;\text{V}=64.8\;\mu\text{C}$$
- Energy stored:
$$U=\tfrac12 C_{\text{eq}}V^{2}= \tfrac12(5.4\;\mu\text{F})(12\;\text{V})^{2}=3.89\times10^{-4}\;\text{J}=388.8\;\mu\text{J}$$
4.4 Capacitors as a potential divider (Paper 2/4 topic)
For two capacitors in series, the voltage across each is proportional to the *other* capacitance:
$$V_{1}=V_{\text{tot}}\frac{C_{2}}{C_{1}+C_{2}},\qquad
V_{2}=V_{\text{tot}}\frac{C_{1}}{C_{1}+C_{2}}$$
This is frequently used to obtain a required fraction of a supply voltage without resistors.
5. Energy Stored in a Capacitor
Starting from the definition \(U=\displaystyle\int_{0}^{Q}V\,\mathrm{d}Q\) and using \(V=Q/C\):
$$U=\frac12\frac{Q^{2}}{C}$$
Because \(Q=CV\), the same expression can be written in two equivalent forms:
$$U=\frac12CV^{2}= \frac12QV$$
- Energy varies with the square of the voltage – doubling \(V\) quadruples the stored energy.
- The three forms are useful in different exam questions (e.g. when \(Q\) is known, use \(U=\tfrac12Q^{2}/C\)).
6. Charging and Discharging – RC Time Constant
6.1 Definition
When a capacitor of capacitance \(C\) is connected to a resistor \(R\), the circuit has a time constant
$$\tau = RC$$
After a time \(\tau\) the voltage (and charge) has fallen to \(\displaystyle\frac{1}{e}\approx36.8\%\) of its initial value.
6.2 Exponential decay (AO2)
Applying Kirchhoff’s loop rule to a discharging circuit (\(V_{C}+V_{R}=0\)) gives
$$\frac{Q}{C}+R\frac{\mathrm{d}Q}{\mathrm{d}t}=0$$
Integrating,
$$Q(t)=Q_{0}\,e^{-t/RC},\qquad V(t)=V_{0}\,e^{-t/RC}$$
where \(Q_{0}=CV_{0}\) is the initial charge.
6.3 Example – Discharge calculation
A 10 µF capacitor is charged to 15 V and then discharged through a 200 kΩ resistor. Find the voltage after 0.5 s.
- \(\tau = RC = (2.0\times10^{5}\;\Omega)(1.0\times10^{-5}\;\text{F}) = 2\;\text{s}\)
- \(V(t)=V_{0}e^{-t/\tau}=15\,e^{-0.5/2}=15\,e^{-0.25}\approx15\times0.779=11.7\;\text{V}\)
7. Common Types of Capacitors
| Type |
Dielectric Material |
Typical Applications |
Capacitance Range |
| Ceramic |
Metal‑oxide ceramic |
High‑frequency circuits, decoupling |
pF – µF |
| Electrolytic |
Aluminium oxide (wet) or tantalum |
Power‑supply filtering, bulk storage |
µF – mF |
| Film |
Polypropylene, polyester |
Audio, precision timing, low‑loss filters |
nF – µF |
| Mica |
Natural mica |
RF circuits, temperature‑stable capacitance |
pF – nF |
8. Practical Considerations
- Voltage rating: Never exceed the specified maximum; breakdown destroys the component.
- Leakage current: Real capacitors slowly discharge even when isolated – important for timing circuits.
- Polarity: Electrolytic and tantalum capacitors are polarized; reversing them causes failure.
- Temperature coefficient: Dielectric constant may vary with temperature; choose a type with suitable stability for precision work.
- Physical size vs. capacitance: Higher capacitance generally requires larger plate area or higher‑\(\varepsilon_{r}\) dielectrics.
9. Summary of Key Points (AO1)
- Capacitance quantifies how much charge a system can store per volt: \(C=Q/V\).
- For a parallel‑plate capacitor \(C=\varepsilon A/d\); inserting a dielectric multiplies \(C\) by \(\varepsilon_{r}\).
- An isolated spherical conductor has \(C=4\pi\varepsilon_{0}r\).
- Series combination: \(\displaystyle\frac{1}{C_{\text{eq}}}= \sum\frac{1}{C_i}\) – the smallest capacitor dominates the total.
- Parallel combination: \(C_{\text{eq}}=\sum C_i\) – capacitances add directly.
- Energy stored: \(U=\tfrac12CV^{2}= \tfrac12Q^{2}/C= \tfrac12QV\); it grows with the square of the voltage.
- RC time constant \(\tau = RC\) governs charging and discharging; voltage decays exponentially as \(V=V_{0}e^{-t/\tau}\).
- Series capacitors act as a potential divider: \(V_{1}=V_{\text{tot}}C_{2}/(C_{1}+C_{2})\).
- Always respect voltage rating, polarity, leakage, and temperature specifications when selecting a capacitor for a circuit.