Capacitors

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Table of Contents

#UCLA #Y1Q3 #Physics1B

Capacitors

Key Definitions

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

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

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

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

Geometric Capacitances

Sphere

Capacitance of a conducting sphere of radius $R$:

\(C=\frac Q V=4\pi\epsilon_0R=\frac R {k_e} [F]\)

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

Parallel Plate Capacitor

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

\(C=\frac{\epsilon_0A}{d}[F]\)

Given that

\(V = Ed = \frac{\sigma d}{\epsilon_0} = \frac{Qd}{\epsilon_0 A}\)

Capacitors

Energy

The electrical energy stored in a charged capacitor:

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

Energy stored in the electric field:

\(\frac{U}{\text{Vol}}=\frac{E^2}{2\epsilon_0}\)

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:

\(C_{eq}=\sum_{i=1}^nC_i\)