Capacitance October 6, 2012

Maximum Charge on a Conductor A battery establishes a difference of potential that can pump electrons e- from a ground (earth) to a conductor Earth Battery Conductor - e- There is a limit to the amount of charge that a conductor can hold without leaking to the air. There is a certain capacity for holding charge.

Capacitance The capacitance C of a conductor is defined as the ratio of the charge Q on the conductor to the potential V produced. Earth Battery Conductor - e- Q, V Capacitance:

Capacitance in Farads One farad (F) is the capacitance C of a conductor that holds one coulomb of charge for each volt of potential. Example: When 40 μC of charge are placed on a conductor, the potential is 8 V. What is the capacitance?

Capacitance of Spherical Conductor At surface of sphere: +Q r E and V at surface. Capacitance, C

Example 1: What is the capacitance of a metal sphere of radius 8 cm? r = 0.08 m Capacitance, C +Q r Note: The capacitance depends only on physical para- meters (the radius r) and is not determined by either charge or potential. This is true for all capacitors.

Example 1 (Cont.): What charge Q is needed to give a potential of 400 V? r = 0.08 m Capacitance, C +Q r Note: The farad (F) and the coulomb (C) are extremely large units for static electricity. The SI prefixes micro μ, nano n, and pico p are often used.

Parallel Plate Capacitance d Area A +Q -Q

Example 3. The plates of a parallel plate capacitor have an area of 0 Example 3. The plates of a parallel plate capacitor have an area of 0.4 m2 and are 3 mm apart in air. What is the capacitance? 3 mm d A 0.4 m2

Dielectric Materials Most capacitors have a dielectric material between their plates to provide greater dielectric strength and less probability for electrical discharge. + - Air Co Eo + - - + - + Dielectric reduced E + - - + C > Co E < Eo The separation of dielectric charge allows more charge to be placed on the plates—greater capacitance C > Co.

Dielectric constant: K = 1 for Air The dielectric constant K for a material is the ratio of the capacitance C with this material as compared with the capacitance Co in a vacuum. Dielectric constant: K = 1 for Air

Energy of Charged Capacitor The potential energy U of a charged capacitor is equal to the work (qV) required to charge the capacitor. If we consider the average potential difference from 0 to Vf to be V/2: Work = Q(V/2) = ½QV

Only, great theory but how does one use it?

Capacitors in Circuits Let’s start with the most basic circuit. Typically, we are solving for the charge on the capacitor and potential difference across the plates At times, we will also be asked about the stored energy.

Series Circuits The key here is - no choice Let’s find Q and ∆V for each capacitor.

Series Circuits The key here is - no choice Let’s find Q and ∆V for each capacitor.

Parallel Circuits The key here is - choice Let’s find Q and ∆V for each capacitor.

Complex Circuits The key here is - make it look either like series or parallel Let’s find Q and ∆V for each capacitor.

Complex Circuits The key here is - make it look either like series or parallel Let’s find Q and ∆V for each capacitor.