Formula Sheet: Capacitors & Dielectrics

Comprehensive formula sheet extracted from FAD1022 Tutorial 3 — Capacitors

1. Physical Constants

Symbol Value Description
$\varepsilon_0$ $8.85 \times 10^{-12} \text{ F m}^{-1}$ Permittivity of free space

2. Capacitance Definitions

2.1 Fundamental Definition

$$C = \frac{Q}{V}$$

Variable Description SI Unit
$C$ Capacitance $\text{F}$ (Farads)
$Q$ Charge stored on one plate $\text{C}$ (Coulombs)
$V$ Potential difference between plates $\text{V}$ (Volts)

2.2 Parallel-Plate Capacitor (Vacuum/Air)

$$C = \frac{\varepsilon_0 A}{d}$$

Variable Description SI Unit
$A$ Area of one plate $\text{m}^2$
$d$ Distance between plates $\text{m}$
$\varepsilon_0$ Permittivity of free space $\text{F m}^{-1}$

2.3 Parallel-Plate Capacitor (With Dielectric)

$$C = \frac{\kappa \varepsilon_0 A}{d} = \kappa C_0$$

Variable Description SI Unit
$\kappa$ (or $\varepsilon_r$) Dielectric constant / relative permittivity dimensionless
$C_0$ Capacitance without dielectric $\text{F}$

3. Electric Field in Capacitors

3.1 Uniform Field Between Parallel Plates (Vacuum)

$$E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}$$

Also:

$$E = \frac{V}{d}$$

Variable Description SI Unit
$E$ Electric field strength $\text{V m}^{-1}$ or $\text{N C}^{-1}$
$\sigma$ Surface charge density ($Q/A$) $\text{C m}^{-2}$

3.2 Electric Field With Dielectric

$$E = \frac{E_0}{\kappa}$$

Where $E_0$ is the field without dielectric.

4. Energy Stored in a Capacitor

4.1 In Terms of Capacitance and Voltage

$$U = \frac{1}{2} C V^2$$

4.2 In Terms of Charge and Voltage

$$U = \frac{1}{2} Q V$$

4.3 In Terms of Charge and Capacitance

$$U = \frac{Q^2}{2C}$$

Variable Description SI Unit
$U$ Energy stored $\text{J}$ (Joules)

4.4 Energy Density in Electric Field

$$u = \frac{1}{2} \varepsilon_0 E^2$$

Variable Description SI Unit
$u$ Energy per unit volume $\text{J m}^{-3}$

5. Capacitor Combinations

5.1 Capacitors in Series

$$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots$$

For two capacitors:

$$C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$$

Key property: Same charge $Q$ on each capacitor. Total voltage adds: $V_{total} = V_1 + V_2 + V_3 + \dots$

5.2 Capacitors in Parallel

$$C_{eq} = C_1 + C_2 + C_3 + \dots$$

Key property: Same voltage $V$ across each capacitor. Total charge adds: $Q_{total} = Q_1 + Q_2 + Q_3 + \dots$

6. RC Circuits

6.1 Time Constant

$$\tau = RC$$

Variable Description SI Unit
$\tau$ Time constant $\text{s}$ (seconds)
$R$ Resistance $\Omega$ (Ohms)
$C$ Capacitance $\text{F}$

6.2 Charging a Capacitor

Charge as function of time:

$$q(t) = Q_{max}\left(1 - e^{-t/\tau}\right)$$

Where $Q_{max} = C\varepsilon$ (maximum charge when fully charged by emf $\varepsilon$).

Current as function of time:

$$I(t) = I_{max} , e^{-t/\tau}$$

Where $I_{max} = \dfrac{\varepsilon}{R}$.

Voltage across capacitor:

$$V_C(t) = \varepsilon\left(1 - e^{-t/\tau}\right)$$

6.3 Discharging a Capacitor

Charge as function of time:

$$q(t) = Q_0 , e^{-t/\tau}$$

Where $Q_0$ is the initial charge at $t = 0$.

Current as function of time:

$$I(t) = I_0 , e^{-t/\tau}$$

Where $I_0 = \dfrac{V_0}{R} = \dfrac{Q_0}{RC}$.

Voltage across capacitor:

$$V_C(t) = V_0 , e^{-t/\tau}$$

Variable Description SI Unit
$q(t)$ Charge at time $t$ $\text{C}$
$Q_{max}, Q_0$ Maximum / initial charge $\text{C}$
$I(t)$ Current at time $t$ $\text{A}$ (Amperes)
$\varepsilon$ EMF of battery $\text{V}$
$t$ Time elapsed $\text{s}$

6.4 Special Time Values During Charging

Time Charge Fraction of $Q_{max}$
$t = \tau$ $q = Q_{max}(1 - e^{-1}) \approx 0.632 , Q_{max}$ 63.2%
$t = 2\tau$ $q = Q_{max}(1 - e^{-2}) \approx 0.865 , Q_{max}$ 86.5%
$t = 3\tau$ $q = Q_{max}(1 - e^{-3}) \approx 0.950 , Q_{max}$ 95.0%
$t = 5\tau$ $q = Q_{max}(1 - e^{-5}) \approx 0.993 , Q_{max}$ 99.3%

7. Summary of Key Relationships

Concept Formula
Capacitance (definition) $C = \dfrac{Q}{V}$
Parallel plate (air) $C = \dfrac{\varepsilon_0 A}{d}$
Parallel plate (dielectric) $C = \dfrac{\kappa \varepsilon_0 A}{d}$
Energy stored $U = \dfrac{1}{2}CV^2 = \dfrac{1}{2}QV = \dfrac{Q^2}{2C}$
Series combination $\dfrac{1}{C_{eq}} = \sum \dfrac{1}{C_i}$
Parallel combination $C_{eq} = \sum C_i$
Time constant $\tau = RC$
Charging charge $q(t) = Q_{max}(1 - e^{-t/\tau})$
Discharging charge $q(t) = Q_0 e^{-t/\tau}$

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