Capacitors

Storing charge and energy in electric fields, and the exponential behaviour of RC circuits.

Spec Points Covered
  • I can 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 use $C = Q/V$.
  • I can calculate the energyThe capacity to do work. Measured in joules (J). stored by a capacitor using all three forms.
  • I can derive and apply C = $\frac{\epsilon_0 \epsilon_r A}{d}$ for a parallel plate capacitor.
  • I can explain the role of dielectrics and use the relative permittivity.
  • I can calculate total 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). for capacitors in series and parallel.
  • I can describe the exponential charging and discharging of a capacitor through a resistor.
  • I can use $Q = Q_0 e^{-t/RC} and the equivalent expressions for$ I and V during discharge.
  • I can define the time constantThe product of resistanceThe opposition to currentThe rate of flow of chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C).. 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 explain its physical meaning.
  • I can sketch and interpret graphs of Q, V and I against time for charging and discharging.
  • I can determine 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 current. Measured in ohms (Ω). and capacitance 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. from a ln(Q) vs t graph.
  • I can describe a practical experiment to investigate charging and discharging curves.
Notes
01 Capacitance Capacitance → 02 The energy stored equals the area under a Q-V graph, which is a straight line through the $E = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$ → 03 Capacitance increases with larger plate area (more charge can be stored), smaller separation $C = \frac{\epsilon_0 \epsilon_r A}{d}$ → 04 Parallel $\text{Parallel: } C_{\text{total}} = C_1 + C_2 \qquad \text{Series: } \frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2}$ → 05 When a charged capacitor discharges through a resistor, the charge, current and p.d. all $Q = Q_0 e^{-t/RC}$ → 06 Time constant Time constant → 07 Capacitor discharge → 08 Capacitors in parallel → 09 Charge a capacitor to a known voltage using a d →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
Capacitance
$$C = \frac{Q}{V}$$
  • Where:
    • $C$ = F
    • $Q$ = C
    • $V$ = V
Definition of capacitance. Use when relating charge stored to p.d.
Energy stored (charge-voltage form)
$$E = \frac{1}{2}QV$$
  • Where:
    • $E$ = J
    • $Q$ = C
    • $V$ = V
Area under the Q-V graph. Primary energy formula.
Energy stored (capacitance-voltage form)
$$E = \frac{1}{2}CV^2$$
  • Where:
    • $E$ = J
    • $C$ = F
    • $V$ = V
Substitute Q = CV into E = \frac{1}{2}QV. Most commonly used form.
Energy stored (charge-capacitance form)
$$E = \frac{Q^2}{2C}$$
  • Where:
    • $E$ = J
    • $Q$ = C
    • $C$ = F
Substitute V = Q/C into E = \frac{1}{2}QV. Use when voltage is not given. Must be memorised.
Parallel plate capacitance
$$C = \frac{\epsilon_0 \epsilon_r A}{d}$$
  • Where:
    • $C$ = F
    • $\epsilon_0$ = F \(m^{-1}\)
    • $A$ = \(m^{2}\)
    • $d$ = m
For a parallel plate capacitor with dielectric. Without dielectric, \epsilon_r = 1.
Exponential discharge (charge)
$$Q = Q_0 e^{-t/RC}$$
  • Where:
    • $Q$ = C
    • $t$ = s
    • $R$ = \Omega
    • $C$ = F
Charge remaining on a discharging capacitor at time t.
Time constant
$$\tau = RC$$
  • Where:
    • $\tau$ = s
    • $R$ = \Omega
    • $C$ = F
Time for Q to fall to 37% of initial value. After 5\tau the capacitor is effectively discharged.
Capacitors in series
$$\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2}$$
  • Where:
    • $C$ = F
Total capacitance is always less than the smallest individual capacitance. Opposite rule to resistors. Must be memorised.