Key Equations

Capacitance & Charge/Discharge — AQA A-Level Physics

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Capacitance
$$C = \frac{Q}{V}$$
  • Where:
    • $C$ = capacitance (F)
    • $Q$ = charge stored (C)
    • $V$ = potential difference (V)
Q is the charge on one plate. 1 F is very large; typical values are uF, nF, or pF.
Relative permittivity
$$\varepsilon_r = \frac{\varepsilon}{\varepsilon_0}$$
  • Where:
    • $varepsilon_r$ = relative permittivity (dimensionless)
    • $varepsilon$ = permittivity of material (F m^-1)
    • $varepsilon_0$ = permittivity of free space (F m^-1)
Also called the dielectric constant. epsilon_r >= 1 for all materials.
Parallel plate capacitance
$$C = \frac{A \varepsilon_0 \varepsilon_r}{d}$$
  • Where:
    • $C$ = capacitance (F)
    • $A$ = area of one plate (\(m^{2}\))
    • $d$ = plate separation (m)
    • $varepsilon_r$ = relative permittivity
    • $varepsilon_0$ = permittivity of free space (F m^-1)
A is one plate only. Larger A and smaller d give larger C.
Half-life of capacitor discharge
$$t_{1/2} = 0.69 RC$$
  • Where:
    • $t_{1/2}$ = half-life (s)
    • $R$ = resistance (ohm)
    • $C$ = capacitance (F)
0.69 = ln 2. Derived by setting Q = Q0/2 in the discharge equation.
Energy stored (charge and voltage)
$$E = \frac{1}{2} QV$$
  • Where:
    • $E$ = energy stored (J)
    • $Q$ = charge (C)
    • $V$ = potential difference (V)
Area under the Q-V graph (triangle).
Energy stored (capacitance and voltage)
$$E = \frac{1}{2} CV^2$$
  • Where:
    • $E$ = energy stored (J)
    • $C$ = capacitance (F)
    • $V$ = potential difference (V)
Derived from E = 1/2 QV by substituting Q = CV.
Energy stored (charge and capacitance)
$$E = \frac{Q^2}{2C}$$
  • Where:
    • $E$ = energy stored (J)
    • $Q$ = charge (C)
    • $C$ = capacitance (F)
Derived from E = 1/2 QV by substituting V = Q/C.
Time constant
$$\tau = RC$$
  • Where:
    • $tau$ = time constant (s)
    • $R$ = resistance (ohm)
    • $C$ = capacitance (F)
Discharging: time to fall to 37% of initial value. Charging: time to rise to 63% of final value.
Charging equation (charge)
$$Q = Q_0 \left(1 - e^{-\frac{t}{RC}}\right)$$
  • Where:
    • $Q$ = charge at time t (C)
    • $Q_0$ = maximum charge (C)
    • $t$ = time (s)
    • $RC$ = time constant (s)
Same form for V. Current during charging still decays: I = I0 e^(-t/RC).
Discharge equation (charge)
$$Q = Q_0 e^{-\frac{t}{RC}}$$
  • Where:
    • $Q$ = charge at time t (C)
    • $Q_0$ = initial charge (C)
    • $t$ = time (s)
    • $RC$ = time constant (s)
Same form for I (I = I0 e^(-t/RC)) and V (V = V0 e^(-t/RC)).
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