Capacitors

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Table of Contents

• Capacitors
  • Key Definitions
    • $C=\frac{Q}{V}[F]$
  • Properties of Conductors
  • Properties of Capacitance
  • Geometric Capacitances
    • Sphere
      • $C=\frac{Q}{V}=4\pi {ϵ}_{0}R=\frac{R}{k_e}[F]$
    • Parallel Plate Capacitor
      • $C=\frac{{ϵ}_{0}A}{d}[F]$
        • $V=Ed=\frac{\sigma d}{{ϵ}_0}=\frac{Qd}{{ϵ}_{0}A}$
  • Capacitors
    • Energy
      • $U=\frac{1}{2}QV=\frac{1}{2}\frac{Q^2}{C}=\frac{1}{2}C{V}^2$
      • $\frac{U}{\text{Vol}}=\frac{E^2}{2{ϵ}_0}$
    • Discharging
    • In Series
      • $\frac{1}{C_{eq}}=\sum _{i=1}^n\frac{1}{C_i}$
    • In Parallel
      • $C_{eq}=\sum _{i=1}^n{C}_i$

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Capacitors

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Key Definitions

Capacitor - 2 conductors of equal but opposite separated by a distance (used to store charge in an electric field)

• Anode - positive plate
• Cathode - negative plate

Capacitance - the ratio of electric charge stored to difference in :

$C=\frac{Q}{V}[F]$

Where $Q$ is the charge on the positive conductor and $V$ is the potential difference between conductors

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Properties of Conductors

1. $E=0$ inside a conductor
2. $\rho =0$ inside a conductor
3. Any net charge lies on the surface of a conductor
4. A conductor is an equipotential
5. $E$ is perpendicular to the surface outside the conductor

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Properties of Capacitance

1. Capacitance is purely a geometric quantity determined by size, shape, and separation of the 2 conductors
2. Units of capacitance are farads, $F$
3. Charge $Q$ is considered the charge on the positive conductor
4. Electric potential $V$ is of the positive conductor
5. Capacitance is always a positive quantity

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Geometric Capacitances

Sphere

Capacitance of a conducting sphere of radius $R$:

$C=\frac{Q}{V}=4\pi {ϵ}_{0}R=\frac{R}{k_e}[F]$

Where $k_e\approx 9\times {10}^9$

Parallel Plate Capacitor

Capacitance of two conducting plates of area $A$ and separation $d$:

$C=\frac{{ϵ}_{0}A}{d}[F]$

Given that

$V=Ed=\frac{\sigma d}{{ϵ}_0}=\frac{Qd}{{ϵ}_{0}A}$

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Capacitors

Energy

The electrical energy stored in a charged capacitor:

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

Energy stored in the electric field:

$\frac{U}{\text{Vol}}=\frac{E^2}{2{ϵ}_0}$

Discharging

Energy stored in capacitors can be released very quickly resulting in higher power

E.g. camera flash, lasers, spark-plugs, defibrillators

In Series

Series connections result in lower capacitance:

$\frac{1}{C_{eq}}=\sum _{i=1}^n\frac{1}{C_i}$

In Parallel

Parallel connections increase capacitance:

$C_{eq}=\sum _{i=1}^n{C}_i$