Capacitance (Copy)

A2 Level Physics – Section 19: Capacitance (Detailed Notes)

19.1 Capacitors and Capacitance

1. Definition of Capacitance

• Capacitance (C) is defined as the charge stored per unit potential difference across a capacitor:
  C = Q / V
• Unit: farad (F) = coulomb per volt (C/V)
• A capacitor stores electrical energy by accumulating opposite charges on two conductive plates.
• For an isolated spherical conductor:
  Capacitance depends on radius and permittivity of space.
• For a parallel-plate capacitor:
  C = ε₀εr·A / d
  • ε₀ = vacuum permittivity (8.85 × 10⁻¹² F/m)
  • εr = relative permittivity of dielectric
  • A = plate area (m²)
  • d = separation (m)

2. Capacitance Equation

• C = Q / V
  • Q = charge (C)
  • V = potential difference (V)

3. Combined Capacitance Formulae

a. Capacitors in Parallel:

• Same voltage across all
• Total capacitance:
  C_total = C₁ + C₂ + C₃ + …

b. Capacitors in Series:

• Same charge on all
• Total capacitance:
  1 / C_total = 1 / C₁ + 1 / C₂ + 1 / C₃ + …

Derivation from C = Q / V:

• In series, total voltage = sum of voltages
• Since Q is constant, use V = Q / C for each

19.2 Energy Stored in a Capacitor

1. Energy from Area Under Q–V Graph

• Q–V graph is a straight line (Q = CV)
• Area under line = energy stored
• Area of triangle:
  W = ½·Q·V

2. Energy Stored Formulae

• W = ½·Q·V
• Using C = Q / V:
  • W = ½·C·V²
  • W = ½·Q² / C

All give energy in joules (J)

• Energy is stored in the electric field between the plates

19.3 Discharging a Capacitor

1. Graphs of Discharge

• Exponential decay of:
  • Charge (Q)
  • Potential difference (V)
  • Current (I)
• All decrease to zero asymptotically

2. Time Constant (τ)

• τ = R·C
  • R = resistance (Ω)
  • C = capacitance (F)
  • Unit: seconds (s)
• After time = τ, values fall to ~37% of their initial value

3. Exponential Discharge Equation

For capacitor discharging through a resistor:

x = x₀·e^(–t / RC)

• x = Q, V, or I at time t
• x₀ = initial value
• RC = time constant
• e = Euler’s number ≈ 2.718

Charge: Q = Q₀·e^(–t / RC)
Voltage: V = V₀·e^(–t / RC)
Current: I = I₀·e^(–t / RC)