Capacitors: Complete GCSE & A-Level Guide
A capacitor stores electrical energy in an electric field between two conducting plates separated by an insulator (dielectric). Understanding capacitors is essential for A-Level physics, electronics, and engineering — from timing circuits to power supplies and medical devices.
Fundamental Formula: Q = CV
Q = C × V
Where Q is charge in Coulombs (C), C is capacitance in Farads (F), and V is potential difference in Volts (V). Rearrangements: C = Q/V and V = Q/C.
Capacitance Units
| Unit | Symbol | Value in Farads | Typical Use |
|---|
| Farad | F | 1 F | Supercapacitors |
| Millifarad | mF | 10⁻³ F | Large electrolytics |
| Microfarad | μF | 10⁻⁶ F | Power supply smoothing |
| Nanofarad | nF | 10⁻⁹ F | RF circuits, filters |
| Picofarad | pF | 10⁻¹² F | High-frequency RF |
Energy Stored: E = ½CV²
E = ½CV² = Q²/(2C) = ½QV
All three forms are equivalent. A defibrillator uses this principle: charge a 32 μF capacitor to ~5,000 V, storing E = ½ × 32×10⁻⁶ × 5000² = 400 J, then discharge through the patient's chest in milliseconds.
RC Time Constant: τ = RC
τ = RC (seconds)
Charging: V(t) = V₀(1 − e−t/τ). At t = τ: 63.2% of V₀. At t = 5τ: 99.3% (fully charged).
Discharging: V(t) = V₀ × e−t/τ. At t = τ: 36.8% of V₀. At t = 5τ: 0.67% (fully discharged).
Capacitors in Parallel
C_total = C1 + C2 + C3 + …
In parallel, each capacitor sees the same voltage. Total charge = sum of individual charges. Total capacitance increases. This is opposite to resistors in parallel.
Capacitors in Series
1/C_total = 1/C1 + 1/C2 + 1/C3
Shortcut for two: C = (C1 × C2)/(C1 + C2). The charge on each capacitor is the same. Total voltage = sum of voltages across each. Series capacitance is always less than the smallest value.
Real-World Applications
Power supply smoothing: Large electrolytic capacitors (1,000–10,000 μF) smooth the ripple in DC power supplies by charging and discharging at the mains frequency (50 Hz). Timing circuits: The RC time constant determines the frequency of 555-timer oscillators and the delay in alarm circuits. Camera flash: A large capacitor (typically 1,000 μF, 330 V) stores ~54 J for the xenon flash tube. Touchscreens: Capacitive touch works by detecting changes in capacitance when a finger (conductor) approaches the screen surface.
Worked A-Level Example: RC Circuit
Q: A 470 μF capacitor is connected in series with a 10 kΩ resistor to a 9 V supply. Calculate τ, and the voltage at t = 2s.
- τ = RC = 10,000 × 470×10⁻⁶ = 4.7 s
- V(2) = 9 × (1 − e−2/4.7) = 9 × (1 − e−0.4255)
- e−0.4255 ≈ 0.6532
- V(2) = 9 × (1 − 0.6532) = 9 × 0.3468 = 3.12 V
Frequently Asked Questions
What is capacitance and how is charge calculated?
Capacitance C (Farads) measures a component's ability to store charge per unit voltage. The fundamental equation is Q = CV: charge (Coulombs) = capacitance (Farads) × voltage (Volts). A 100 μF capacitor charged to 12 V stores Q = 100×10⁻⁶ × 12 = 1.2 mC. Larger capacitance means more charge stored at the same voltage.
What is the RC time constant and why does it matter?
τ = RC (in seconds) defines how quickly a capacitor charges or discharges through a resistor. After one time constant: 63.2% charged (or 36.8% remaining on discharge). After 5τ: effectively complete (99.3%). This is used to design timer circuits, filters, and signal processing. Reducing R or C speeds up the circuit; increasing either slows it down.
How do capacitors combine in parallel?
Parallel capacitors add directly: C_total = C1 + C2 + C3. Each plate area effectively increases, giving more total capacitance. All capacitors see the same voltage. Total charge = C1V + C2V + C3V = V(C1+C2+C3). This is the opposite rule to resistors in parallel. Parallel capacitors are used to increase stored charge and provide better decoupling in power supplies.
How do capacitors combine in series?
1/C_total = 1/C1 + 1/C2 + 1/C3. For two: C = (C1×C2)/(C1+C2). Each capacitor stores the same charge Q. Total voltage = V1 + V2 + V3. Series capacitors are used to handle higher voltages (each shares the total voltage) and to reduce capacitance to precise values not available in single components.
How does a defibrillator use a capacitor?
A defibrillator charges a large capacitor (typically 32–150 μF) to a high voltage (1,000–5,000 V) using a battery-powered converter. The energy stored E = ½CV² can be 200–360 J. At the moment of shock, the capacitor discharges through the patient's chest in approximately 10 milliseconds, delivering current that briefly stops the chaotic electrical activity and allows the heart's natural pacemaker to re-establish rhythm.
What is the energy stored in a capacitor?
Energy E = ½CV² (Joules). Three equivalent forms: E = ½CV² = Q²/(2C) = ½QV. A 1,000 μF capacitor charged to 50 V stores E = ½ × 0.001 × 2500 = 1.25 J. This energy is stored in the electric field between the capacitor plates. Doubling voltage quadruples stored energy (because of the V² term), which is why high-voltage capacitors must be handled with extreme caution.
What is the charge/discharge equation for a capacitor?
Charging through resistor R from supply V₀: V(t) = V₀(1 − e−t/RC). Discharging: V(t) = V₀ × e−t/RC. At t=0 the voltage is 0 (charging) or V₀ (discharging). The exponential shape means the capacitor charges quickly at first, then more slowly. The minus sign in the exponent confirms that voltage approaches V₀ asymptotically during charging.