Capacitance

Overview

A capacitor is a device that stores electric charge and energy. It consists of two conductors separated by an insulator (dielectric).

Capacitance

Definition

Ratio of charge to potential difference:

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

Unit: Farad (F) = C/V

Common units: μF, nF, pF

Properties

• Depends only on geometry and materials
• Independent of $Q$ and $V$
• Always positive

Parallel Plate Capacitor

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

Where:

• $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m
• $A$ = plate area
• $d$ = plate separation

Electric Field Between Plates

$$E = \frac{V}{d} = \frac{\sigma}{\varepsilon_0}$$

Other Capacitor Geometries

Cylindrical Capacitor

$$C = \frac{2\pi\varepsilon_0 L}{\ln(b/a)}$$

Where $a$ = inner radius, $b$ = outer radius, $L$ = length

Spherical Capacitor

$$C = 4\pi\varepsilon_0 \frac{ab}{b-a}$$

Isolated Sphere

$$C = 4\pi\varepsilon_0 R$$

Capacitors in Circuits

Series Connection

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

For two capacitors:

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

• Same charge on each capacitor
• Voltage divides

Parallel Connection

$$C_{\text{eq}} = C_1 + C_2 + C_3 + \cdots$$

• Same voltage across each capacitor
• Charge divides

Energy Stored in Capacitor

$$U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{1}{2}\frac{Q^2}{C}$$

Energy Density

Energy per unit volume in electric field:

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

Dielectrics

Dielectric Constant ($\kappa$)

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

Where $C_0$ is capacitance without dielectric

Effect of Dielectric

• Increases capacitance by factor $\kappa$
• Decreases electric field by factor $\kappa$
• Increases maximum voltage before breakdown

Common Dielectric Constants

Material	$\kappa$
Vacuum	1.0
Air	1.0006
Paper	3.7
Glass	5-10
Mica	5.4
Water	80

With Dielectric

$$E = \frac{E_0}{\kappa} = \frac{\sigma}{\kappa\varepsilon_0}$$ $$V = \frac{V_0}{\kappa} \quad \text{(if } Q \text{ is fixed)}$$

Charging and Discharging

RC Circuit Charging

$$Q(t) = Q_{\max}(1 - e^{-t/\tau})$$ $$V(t) = V_{\max}(1 - e^{-t/\tau})$$

RC Circuit Discharging

$$Q(t) = Q_0 e^{-t/\tau}$$ $$V(t) = V_0 e^{-t/\tau}$$

Time Constant

$$\tau = RC$$

At $t = \tau$: charge reaches 63% (charging) or falls to 37% (discharging)

Examples

Example 1: Parallel Plate Capacitor

Plates of area 0.01 m² separated by 0.001 m.

$$C = \frac{\varepsilon_0 A}{d} = \frac{8.85 \times 10^{-12} \times 0.01}{0.001} = 8.85 \times 10^{-11} \text{ F} = 88.5 \text{ pF}$$

Example 2: Charge and Energy

A 10 μF capacitor is charged to 100 V.

$$Q = CV = 10 \times 10^{-6} \times 100 = 10^{-3} \text{ C} = 1 \text{ mC}$$ $$U = \frac{1}{2}CV^2 = \frac{1}{2} \times 10 \times 10^{-6} \times 10000 = 0.05 \text{ J} = 50 \text{ mJ}$$

Example 3: Capacitors in Series

Three capacitors (2 μF, 3 μF, 6 μF) in series.

$$\frac{1}{C_{\text{eq}}} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = 1$$ $$C_{\text{eq}} = 1 \text{ μF}$$

Example 4: Capacitors in Parallel

Same three capacitors in parallel.

$$C_{\text{eq}} = 2 + 3 + 6 = 11 \text{ μF}$$

Example 5: Adding Dielectric

A 100 pF capacitor has a dielectric ($\kappa = 4$) inserted while connected to 50 V battery.

New capacitance:

$$C = \kappa C_0 = 4 \times 100 = 400 \text{ pF}$$

New charge:

$$Q = CV = 400 \times 10^{-12} \times 50 = 2 \times 10^{-8} \text{ C}$$

Energy:

$$U = \frac{1}{2}CV^2 = \frac{1}{2} \times 400 \times 10^{-12} \times 2500 = 5 \times 10^{-7} \text{ J}$$

Example 6: RC Time Constant

A 5 μF capacitor with 100 kΩ resistor. How long to charge to 90%?

$$\tau = RC = 100 \times 10^3 \times 5 \times 10^{-6} = 0.5 \text{ s}$$ $$0.9 = 1 - e^{-t/0.5}$$ $$e^{-t/0.5} = 0.1$$ $$t = -0.5 \times \ln(0.1) = 1.15 \text{ s}$$

Applications

• Camera flash: Store and release energy quickly
• AC coupling: Block DC, pass AC
• Timing circuits: RC time constant
• Filters: Low-pass and high-pass
• Power supply smoothing: Reduce ripple