PHY PHY 184 184 Spring 2007 Lecture 11 Title Capacitors 1 25 07 184 Lecture 11 1 Announcements Homework Set 3 is due Tuesday Jan 30 at 8 00 am Homework Set 4 opened this morning Today we will finish up the electric potential and start with capacitors 1 25 07 184 Lecture 11 2 Review Electric Potential V x If a charge q moves in an electric field U Uf Ui We V U q potential energy electric potential definition For reference V 0 at infinity V x x E ds E is the gradient of V E E E x 1 25 07 y z V V V x y z 184 Lecture 11 3 Review Electric Potential 2 For a point source Q kQ V r r For many sources V x n Vi x superposition i 1 1 25 07 184 Lecture 11 4 Problem Solving Strategies Given a charge distribution calculate the electric field and the electric potential i Use Gauss Law to derive the electric field E ii For the potential V 0 at infinity use For an arrangement of charges remember the superposition principle principle This holds for the electric force the electric field the electric potential and the electric potential energy 1 25 07 184 Lecture 11 0 E dA qenclosed V x E ds x Note a b abcos 5 Capacitors 1 25 07 184 Lecture 11 6 Capacitors Capacitors are devices that store energy in an electric field Capacitors are used in many every day applications Heart defibrillators Camera flash units Capacitors are an essential part of electronics Capacitors can be micro sized on computer chips or super sized for high power circuits such as FM radio transmitters 1 25 07 184 Lecture 11 7 Capacitance Capacitors come in a variety of sizes and shapes Concept A capacitor consists of two separated conductors usually called plates even if these conductors are not simple planes We will define a simple geometry and generalize from there We will start with a capacitor consisting of two parallel conducting plates each with area A separated by a distance d We assume that these plates are in vacuum air is very close to a vacuum 1 25 07 184 Lecture 11 8 Parallel Plate Capacitor q q We charge the capacitor by placing a charge q on the top plate a charge q on the bottom plate e g using a battery Because the plates are conductors the charge will distribute itself evenly over the surface of the conducting plates The electric potential V is proportional to the amount of charge on the plates More precisely potential difference V V V 1 25 07 184 Lecture 11 9 Parallel Plate Capacitor 2 q q The proportionality constant between the charge q and the electric potential difference V is the capacitance C We will call the electric potential difference V the potential or the voltage across the plates The capacitance of a device depends on the area of the plates and the distance between the plates but does not depend on the voltage across the plates or the charge on the plates The capacitance of a device tells us how much charge is required to produce a given voltage across the plates 1 25 07 184 Lecture 11 q CV C q V 10 Clicker Question q q What is the NET CHARGE on the charged capacitor A q q 0 B q q 2q C q D none of the above 1 25 07 184 Lecture 11 11 Clicker Question q q What is the NET CHARGE on the charged capacitor A q q 0 Charges are added with their signs However we refer to the charge of a capacitor as being q the charge of a capacitor is not the net charge 1 25 07 184 Lecture 11 12 Definition of Capacitance q q The definition of capacitance is C C V V The units of capacitance are coulombs per volt The unit of capacitance has been given the name farad abbreviated F named after British physicist Michael Faraday 1791 1867 1C 1F 1V A farad is a very large capacitance Typically we deal with F 10 6 F nF 10 9 F or pF 10 12 F 1 25 07 184 Lecture 11 13 Charging Discharging a Capacitor We can charge a capacitor by connecting the capacitor to a battery or to a DC power supply A battery or DC power supply is designed to supply charge at a given voltage When we connect a capacitor to a battery charge flows from the battery until the capacitor is fully charged If we then disconnect the battery or power supply the capacitor will retain its charge and voltage A real life capacitor will leak charge slowly but here we will assume ideal capacitors that hold their charge and voltage indefinitely 1 25 07 184 Lecture 11 14 Charging Discharging a Capacitor 2 Illustrate the charging processing using a circuit diagram Lines represent conductors The battery or power supply is represented by The capacitor is represented by the symbol This circuit has a switch ab open When the switch is between positions a and b the circuit is open not connected pos a When the switch is in position a the battery is connected across the capacitor Fully charged q CV pos b When the switch is in position b the two plates of the capacitor are connected Electrons will move around the circuit a current will flow and the capacitor will discharge 1 25 07 184 Lecture 11 15 Demo Big Spark Energy stored in this particular capacitor 90 J This is equivalent to the kinetic energy of a mass of 1 kg moving at a velocity of 13 4 m s E 12 mv2 2E 2 90 J v 13 4 m s m 1 kg 1 25 07 184 Lecture 11 16 Parallel plate capacitor a simple ideal model 1 25 07 184 Lecture 11 17 Parallel Plate Capacitor Consider two parallel conducting plates separated by a distance d This arrangement is called a parallel plate capacitor The upper plate has q and the lower plate has q The electric field between the plates points from the positively charged plate to the negatively charged plate We will assume ideal parallel plate capacitors in which the electric field is constant between the plates and zero elsewhere Real life capacitors have fringe field near the edges 1 25 07 184 Lecture 11 18 Parallel Plate Capacitor 2 We can calculate the electric field between the plates using Gauss Law 0 E dA qenclosed We take a Gaussian surface shown by the red dashed line Flux through bottom surface of the top plate only EA Enclosed charge q 1 25 07 184 Lecture 11 q E 0A 19 Parallel Plate Capacitor 3 Now we calculate the electric potential across the plates of the capacitor in terms of the electric field We define the electric potential across the capacitor to be V and we carry out the integral in the direction of the blue arrow f V Vf Vi i E ds E ds cos 180 …
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