Capacitors and Capacitance – Cambridge International AS & A Level Physics 9702
Learning Objective
Recall and use the expressions for the energy stored in a capacitor and related concepts:
\[
W=\frac12QV=\frac12CV^{2}
\]
1. Fundamental Concepts
- Capacitance – ability of a device to store charge per unit potential difference.
- Definition \(C=\dfrac{Q}{V}\) (where \(Q\) is the charge on one plate and \(V\) the potential difference).
- Parallel‑plate capacitor \(C=\displaystyle\frac{\varepsilon{0}A}{d}\) (\(A\) = plate area, \(d\) = separation, \(\varepsilon{0}=8.85\times10^{-12}\,\text{F m}^{-1}\)).
- Dielectric material – inserting a dielectric of relative permittivity \(\kappa\) multiplies the capacitance by \(\kappa\):
\[
C{\text{new}}=\kappa\,C{\text{vacuum}} .
\]
- Series and parallel combinations
| Configuration | Resulting capacitance |
|---|
| Parallel: \(C{\text{eq}} = C{1}+C_{2}+ \dots\) | Capacitances add directly. |
| Series: \(\displaystyle\frac{1}{C{\text{eq}}}= \frac{1}{C{1}}+\frac{1}{C_{2}}+\dots\) | Reciprocals add. |
- Capacitor in a potential‑divider – two capacitors in series share the total voltage in proportion to the inverse of their capacitances:
\[
V{1}=V{\text{total}}\frac{C{2}}{C{1}+C_{2}},\qquad
V{2}=V{\text{total}}\frac{C{1}}{C{1}+C_{2}} .
\]
This is the analogue of a resistive voltage divider (Syllabus 10.3).
- AC reactance – for a sinusoidal source of angular frequency \(\omega\),
\[
X_{C}= \frac{1}{\omega C}\qquad(\text{units }\Omega) .
\]
\(X_{C}\) appears in Paper 4 questions on alternating currents.
2. Energy Stored in a Capacitor
Derivation
- When a small charge \(dq\) is moved onto a plate that already carries charge \(q\), the instantaneous voltage is \(V=\dfrac{q}{C}\).
- The incremental work is
\[
dW = V\,dq = \frac{q}{C}\,dq .
\]
- Integrating from an uncharged state (\(q=0\)) to the final charge \(Q\):
\[
W = \int_{0}^{Q}\frac{q}{C}\,dq
= \frac{1}{2}\frac{Q^{2}}{C}
= \frac12 QV
= \frac12 CV^{2}.
\]
The factor \(\tfrac12\) arises because the voltage rises linearly from 0 to its final value.
Energy density
For a uniform electric field between the plates,
\[
u = \frac{W}{\text{volume}} = \frac12\varepsilon_{0}E^{2},
\qquad\text{with }E=\frac{V}{d}.
\]
This form is useful when comparing capacitors with other energy‑storage media.
3. Discharging a Capacitor – RC Time Constant
Example – Discharge
A 47 µF capacitor charged to 12 V is connected across a 10 kΩ resistor. Find the voltage after 0.5 s.
- \(\tau = RC = (10\,000\;\Omega)(47\times10^{-6}\,\text{F}) = 0.47\;\text{s}\).
- \(V(0.5) = 12\,e^{-0.5/0.47} \approx 12\,e^{-1.06} \approx 4.2\;\text{V}.\)
4. Units and Typical Ranges (A‑Level)
| Quantity | Symbol | SI unit | Typical range |
|---|
| Capacitance | C | farad (F) | pF – µF (parallel‑plate), mF – F (electrolytic) |
| Charge | Q | coulomb (C) | 10⁻⁹ – 10⁻³ C |
| Voltage | V | volt (V) | 1 – 500 V |
| Energy | W | joule (J) | 10⁻⁹ – 10⁻¹ J |
| Resistance | R | ohm (Ω) | 10 Ω – 10 MΩ (typical for RC circuits) |
| Angular frequency | \(\omega\) | rad s\(^{-1}\) | 10 – 10⁴ (AC circuits) |
5. Worked Examples
5.1 Energy Stored
Problem: A 47 µF capacitor is charged to 12 V. Calculate the energy stored.
- Use \(W = \dfrac12 CV^{2}\).
- \[
W = \frac12 (47\times10^{-6}\,\text{F})(12\;\text{V})^{2}
= \frac12 (47\times10^{-6})(144)
\approx 3.4\times10^{-3}\,\text{J}.
\]
- The capacitor stores about 3.4 mJ of energy.
5.2 Series–Parallel Combination
Problem: Two capacitors, \(C{1}=2.0\;\mu\text{F}\) and \(C{2}=3.0\;\mu\text{F}\), are connected in series, then this combination is placed in parallel with a third capacitor \(C_{3}=5.0\;\mu\text{F}\). Find the equivalent capacitance.
- Series part: \(\displaystyle\frac{1}{C{s}}=\frac{1}{C{1}}+\frac{1}{C_{2}}
=\frac{1}{2.0}+\frac{1}{3.0}=0.833\;\mu\text{F}^{-1}\)
\(\Rightarrow C_{s}=1.20\;\mu\text{F}\).
- Parallel addition: \(C{\text{eq}} = C{s}+C_{3}=1.20+5.0=6.20\;\mu\text{F}\).
5.3 AC Reactance
Problem: A 10 µF capacitor is connected to a 50 Hz AC supply. Find its reactance.
- \(\omega = 2\pi f = 2\pi(50)=314\;\text{rad s}^{-1}\).
- \(X_{C}= \dfrac{1}{\omega C}= \dfrac{1}{314\times10^{-5}} \approx 318\;\Omega.\)
6. Common Mistakes & How to Avoid Them
- Leaving out the factor \(\tfrac12\) – it originates from the integration of a linearly increasing voltage.
- Using the source voltage instead of the actual voltage across the capacitor when it is only partially charged.
- Failing to convert units (µF → F, kΩ → Ω, µH → H) before substitution.
- Mixing up symbols: \(C\) for capacitance, \(R\) for resistance, \(X_{C}\) for reactance.
- Treating series and parallel formulas for resistors as if they were the same for capacitors; remember the reciprocal rule for series.
7. Practical Investigation (AO3 Skill)
Objective: Measure the capacitance of an unknown parallel‑plate capacitor using the RC‑discharge method.
- Charge the unknown capacitor to a known voltage \(V_{0}\) (e.g., 10 V) with a DC supply.
- Connect a known resistor \(R\) (e.g., 10 kΩ) in series and start a timer the instant the circuit is closed.
- Record the voltage across the capacitor at regular intervals (e.g., every 0.1 s) using a digital voltmeter.
- Plot \(\ln\!\bigl(V/V_{0}\bigr)\) against time. The plot should be a straight line with slope \(-1/RC\).
- Calculate the capacitance:
\[
C = -\frac{1}{R\,\text{slope}} .
\]
- Compare the result with a direct reading from a capacitance meter and discuss sources of error (instrument resolution, lead resistance, leakage, stray capacitance).
8. Links to Other Syllabus Areas
- Energy stored in a capacitor → 19.2 (energy concepts).
- RC time constant → 20.5 (electromagnetic induction) and 21 (alternating currents).
- Energy density → 22 (energy of photons & comparison of storage media).
- Capacitor potential divider → 10.3 (potential dividers).
- AC reactance → 21.2 (reactive components in AC circuits).
9. Quick Revision Checklist
- Write the definition \(C = Q/V\) and the parallel‑plate formula \(C=\varepsilon_{0}A/d\).
- State the effect of a dielectric: \(C_{\text{new}}=\kappa C\).
- Recall series and parallel combination rules for capacitors.
- Remember the three equivalent energy expressions and the \(\tfrac12\) factor.
- State the energy‑density formula \(u=\tfrac12\varepsilon_{0}E^{2}\).
- Give the RC time constant \(\tau = RC\) and the discharge equation \(V(t)=V_{0}e^{-t/RC}\).
- Write the AC reactance \(X_{C}=1/(\omega C)\).
- Check units: \(C\) in farads, \(W\) in joules, \(\tau\) in seconds, \(X_{C}\) in ohms.
- Complete a numerical problem for (a) energy storage, (b) RC discharge, and (c) AC reactance.
- Outline the practical method for measuring an unknown capacitance and list at least three possible error sources.