Capacitance PHY 2049 Chapter 25

Chapter 25 Capacitance In this chapter we will cover the following topics: -Capacitance C of a system of two isolated conductors. -Calculation of the capacitance for some simple geometries. -Methods of connecting capacitors (in series, in parallel). -Equivalent capacitance. -Energy stored in a capacitor. -Behavior of an insulator (a.k.a. dielectric) when placed in the electric field created in the space between the plates of a capacitor. -Gauss’ law in the presence of dielectrics. (25 - 1)

Capacitors

Capacitor  Composed of two metal plates.  Each plate is charged one positive one negative  Stores energy SYMBOL

A simple Capacitor TWO PLATES Battery WIRES

INSIDE THE DEVICE

What is STORED in the capacitor?  An Electric Field  Energy  Charge  All three  None of these

Two Charged Plates (Neglect Fringing Fields) d Air or Vacuum Area A - Q +Q E V=Potential Difference Symbol ADDED CHARGE

Where is the charge? d Air or Vacuum Area A - Q +Q E V=Potential Difference AREA=A  =Q/A

One Way to Charge:  Start with two isolated uncharged plates.  Take electrons and move them from the + to the – plate through the region between.  As the charge builds up, an electric field forms between the plates.  You therefore have to do work against the field as you continue to move charge from one plate to another.

Capacitor

More on Capacitors d Air or Vacuum Area A - Q +Q E V=Potential Difference Gaussian Surface Same result from other plate!

DEFINITION - Capacity  The Potential Difference is APPLIED by a battery or a circuit.  The charge q on the capacitor is found to be proportional to the applied voltage.  The proportionality constant is C and is referred to as the CAPACITANCE of the device.

UNITS  A capacitor which acquires a charge of 1 coulomb on each plate with the application of one volt is defined to have a capacitance of 1 FARAD  One Farad is one Coulomb/Volt

Continuing…  The capacitance of a parallel plate capacitor depends only on the Area and separation between the plates.  C is dependent only on the geometry of the device!

After the switch is closed, how much charge passed through the capacitor?  C/V  V/C  CV  C+V V

S P N (25 - 6)

Units of  0 pico

Simple Capacitor Circuits  Batteries Apply potential differences  Capacitors  Wires Wires are METALS. Continuous strands of wire are all at the same potential. Separate strands of wire connected to circuit elements may be at DIFFERENT potentials.

NOTE  Work to move a charge from one side of a capacitor to the other is = qEd.  Work to move a charge from one side of a capacitor to the other is qV  Thus qV = qEd  E=V/d As before

TWO Types of Connections SERIES PARALLEL

Parallel Connection V C Equivalent =C E

Series Connection V C 1 C 2 q -q The charge on each capacitor is the same !

Series Connection Continued V C 1 C 2 q -q

More General

Example C 1 C 2 V C3C3 C1=12.0  f C2= 5.3  f C3= 4.5  d (12+5.3)pf series (12+5.3)pf

More on the Big C  We move a charge dq from the (-) plate to the (+) one.  The (-) plate becomes more (-)  The (+) plate becomes more (+).  dW=Fd=dq x E x d +q -q E=  0 A/d +dq

So….

Not All Capacitors are Created Equal  Parallel Plate  Cylindrical  Spherical

Spherical Capacitor

Calculate Potential Difference V (-) sign because E and ds are in OPPOSITE directions.

Continuing… Lost (-) sign due to switch of limits.