Capacitance

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 \(Vi=Q/Ci\) 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 \(Qi=CiV\), the equivalent capacitance is

  \$C{\text{eq}}=C{1}+C{2}+\dots+C{n}\$

4.3 Example – Series‑parallel network

1. 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.
2. 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

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: \(\displaystyle\frac{1}{C{\text{eq}}}= \sum\frac{1}{Ci}\) – the smallest capacitor dominates the total.
• Parallel combination: \(C{\text{eq}}=\sum Ci\) – 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.