Capacitance & Charge/Discharge

Capacitance definition, parallel plate capacitor, dielectrics, energy stored, exponential charge and discharge, and the time constant.

Spec Points Covered
  • Define capacitanceThe chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). stored per unit potential difference across a capacitor. Measured in farads (F). and apply $C = Q/V$.
  • Describe the structure and function of a parallel plate capacitor with a dielectricAn insulating material placed between the plates of a capacitor that increases its capacitance by reducing the electric field strength for a given chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C)...
  • Apply $C = Aepsilon0 epsilonr / d$ to calculate capacitanceThe chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). stored per unit potential difference across a capacitor. Measured in farads (F). from plate geometry.
  • Calculate energyThe capacity to do work. Measured in joules (J). stored using E = 1/2 QV, $E = 1/2 CV^2$, and $E = Q^2 / 2C$.
  • Sketch and interpret charge, p.d., and currentThe rate of flow of charge. Measured in amperes (A). graphs for charging and discharging capacitors.
  • Define the time constantThe product of resistanceThe opposition to currentThe rate of flow of charge. Measured in amperes (A). flow. The ratio of potential difference to currentThe rate of flow of charge. Measured in amperes (A).. Measured in ohms (Ω). and capacitanceThe charge stored per unit potential difference across a capacitor. Measured in farads (F). in an RC circuit. The time taken for the charge (or voltageThe energyThe capacity to do work. Measured in joules (J). transferred per unit charge between two points. Measured in volts (V). Informal term for potential difference.) to fall to 1/e (about 37%) of its initial value. $tau = RC$ and the half-lifeThe time taken for half the number of radioactive nuclei in a sample to decay, or for the activityThe number of nuclear decays per unit time. Measured in becquerels (Bq), where 1 Bq = 1 decay per second. to halve. t_1/2 = 0.69RC.
  • Apply the exponential decay equations for discharge: $Q = Q0 e^(-t/RC)$.
  • Apply the charging equations: $Q = Q0(1 - e^(-t/RC))$.
  • Describe the required practical for measuring capacitance from a discharge curve.
Notes
01 Capacitance is the charge stored per unit potential difference Capacitance 3.7.4.1 → 02 Dielectrics are polar molecules that increase capacitance Relative permittivity (dielectric constant) 3.7.4.2 → 03 Parallel plate capacitance depends on area, separation, and dielectric $C = \frac{A \varepsilon_0 \varepsilon_r}{d}$ 3.7.4.2 → 04 Energy stored is the area under the Q-V graph $E = \frac{1}{2} QV$ 3.7.4.3 → 05 Charging graphs: current decays, charge and p.d. rise 3.7.4.4 → 06 Discharging graphs: everything decays exponentially 3.7.4.4 → 07 The time constant tau = RC sets the rate of charge and discharge Time constant (discharging) 3.7.4.4 → 08 Discharge equations: Q, I, and V all decay as e^(-t/RC) $Q = Q_0 e^{-\frac{t}{RC}}$ 3.7.4.4 → 09 Charging equations use (1 - e^(-t/RC)) for Q and V $Q = Q_0 \left(1 - e^{-\frac{t}{RC}}\right)$ 3.7.4.4 → 10 Required practical: measuring capacitance from a discharge curve 3.7.4.4 →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
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).
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).
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.
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)).
Q Retrieval Practice All 14 Questions →
Q1. Define capacitance and state its unit.
  • The charge stored per unit potential differenceThe energyThe capacity to do work. Measured in joules (J). transferred per unit charge between two points. Measured in volts (V)..
  • Unit: farad (F).
  • C = Q/V.
Q2. What is a dielectricAn insulating material placed between the plates of a capacitor that increases its capacitance by reducing the electric field strength for a given charge. and how does it affect capacitance?
  • An insulating material between the plates.
  • Its polar molecules align with the applied field, creating an opposing field that reduces the p.d. and increases capacitance.
Q3. State the equation for the capacitance of a parallel plate capacitor.
C = A epsilon_0 epsilon_r / d, where A is the area of one plate, d is the separation, and epsilon_r is the relative permittivity.
Q4. State all three equations for the energy stored in a capacitor.
  • E = 1/2 QV, E = 1/2 CV^2, E = \(Q^{2}\) / (2C).
  • All three are on the data sheet.
Q5. Describe how the current varies with time when a capacitor charges through a resistor.
The current starts at a maximum value I0 and decreases exponentially to zero as the capacitor fully charges.