Lecture 8 Lecture 8 --Electrostatic PotentialElectrostatic PotentialChapter 28 Chapter 28 --Thursday February 1stThursday February 1st•Review of electrostatic potential •More example problems•Equipotential surfaces•Conductors (time permitting)Reading: pages 635 thru 670 (chapter 28/29) in HRKReading: pages 635 thru 670 (chapter 28/29) in HRKRead and understand the sample problemsRead and understand the sample problemsWebAssignWebAssignhomework: set 3, due Sun. 4th at 11:59pmhomework: set 3, due Sun. 4th at 11:59pmGraded problems (Ch. 28) Graded problems (Ch. 28) ––Ex. 2, 12, 16, 24; Prob. 6, 13Ex. 2, 12, 16, 24; Prob. 6, 13Practice problems (Ch. 28):Practice problems (Ch. 28):Ex. 29, 31, 43, 47; Prob. 15Ex. 29, 31, 43, 47; Prob. 15••Exam 1, Feb 6Exam 1, Feb 6thth, 8:30 , 8:30 ––10:10 am (10:10 am (ChsChs. 25. 25--29)29)••Review in here tonight and Monday at 5:30pmReview in here tonight and Monday at 5:30pmElectrostatic Potential EnergyElectrostatic Potential Energy••The potential energy is a property of both of the The potential energy is a property of both of the charges, not one or the other.charges, not one or the other.••If we choose a reference such that U = If we choose a reference such that U = 00when the when the charges are infinitely far apart, then we can simplify charges are infinitely far apart, then we can simplify the expression for the potential energy as follows.the expression for the potential energy as follows.12121111()44rooqqUr d qqrrπε πε∞⎛⎞=− ⋅ = − =⎜⎟∞⎝⎠+∫FsGGAgain, the sign of Again, the sign of UUis not a problem. It is is not a problem. It is taken care of by the signs of the charges taken care of by the signs of the charges qq11andandqq22..Potential Energy of a System of ChargesPotential Energy of a System of Chargesq10U =q2r12121214oqqUrπε=q3r1312 1312 131144ooqq qqUrrπε πε=+r2312 13 2312 13 23111444oooqq qq qqUrrrπε πε πε=++Assumes that the Assumes that the superposition superposition principle is validprinciple is validThe Electrostatic PotentialThe Electrostatic Potential••We can eliminate the test charge from the problem in We can eliminate the test charge from the problem in exactly the same fashion as we did in chapter 26 when exactly the same fashion as we did in chapter 26 when defining the electric field. In doing so, we define a new defining the electric field. In doing so, we define a new quantity known as the Electrostatic Potential quantity known as the Electrostatic Potential difference difference ΔΔV:V:UqV WΔ=Δ =−••This This ‘‘scalar potentialscalar potential’’depends only on the details of depends only on the details of the static charge distribution.the static charge distribution.0114bbaaob adQVdqrrπε⎛⎞⋅Δ=− =− ⋅ = −⎜⎟⎝⎠∫∫FsEsGGGG()14roQVr drπε∞=− ⋅ =∫EsGGThe main subject of this courseThe main subject of this courseTest chargeSourcechargesElectrostaticsElectrostatics14iioiqVrπε=∑14odqVrπε=∫Some more sophisticated vector calculusSome more sophisticated vector calculus21andVd V⇒Δ =− ⋅ =−∇∫rrEr EGGGGGˆˆˆVVVVxyz⎡⎤∂∂∂=−∇ =− + +⎢⎥∂∂∂⎣⎦EijkGQVE()14oQVrrπε=ˆˆˆVVVVxyz⎡⎤∂∂∂=−∇ =− + +⎢⎥∂∂∂⎣⎦EijkG21VdΔ=− ⋅∫rrErGGGG21ˆ4oqrπε=ErGCoulomb’s law:enclosedoqdε⋅=∫EAGGvGauss’ law:Summary of first four chaptersSummary of first four chaptersEquipotential surfacesEquipotential surfacesSurfaces of Surfaces of constant constant electrostatic electrostatic potentialpotential()14oqVrrπε=Equipotential surfacesEquipotential surfaces()00bbaaWqV d qVV=− Δ = ⋅ =− −∫EsGG••EE--field lines perpendicular to equipotential surfacesfield lines perpendicular to equipotential surfaces••Positive work when qPositive work when q00is accelerated by the fieldis accelerated by the fieldEquipotential surfacesEquipotential surfacesHere are some example surfaces including field lines Here are some example surfaces including field lines ((point charge, infinite charged plane and a dipolepoint charge, infinite charged plane and a dipole).).••By spacing the equipotential surfaces by the same By spacing the equipotential surfaces by the same potential difference potential difference ((ΔΔVV)), one can get a feel for the , one can get a feel for the electric field strength electric field strength ((E = E = −−ΔΔV/V/ΔΔrr)), i.e. the closer the , i.e. the closer the spacing, the stronger the field.spacing, the stronger the field.Conductors in EConductors in E--fields: dynamic conditionsfields: dynamic conditionsDynamicDynamic StaticConductors in EConductors in E--fields: dynamic conditionsfields: dynamic conditions••If the EIf the E--field is maintained, field is maintained, then the dynamics persist, then the dynamics persist, i.e. charge continues to flow i.e. charge continues to flow indefinitely.indefinitely.••This is no longer strictly the This is no longer strictly the domain of electrostatics.domain of electrostatics.••Note the direction of flow of Note the direction of flow of the charge carriers the charge carriers (electrons).(electrons).iElectrical current:dqidt=SI unit: 1 ampere (A) = 1 coulomb per second (C/s)Conductors in EConductors in E--fields: dynamic conditionsfields: dynamic conditions••If the EIf the E--field is maintained, field is maintained, then the dynamics persist, then the dynamics persist, i.e. charge continues to flow i.e. charge continues to flow indefinitely.indefinitely.••This is no longer strictly the This is no longer strictly the domain of electrostatics.domain of electrostatics.••Note the direction of flow of Note the direction of flow of the charge carriers the charge carriers (electrons).(electrons).oridijAdA=id=⋅∫jAGGCurrent density:Current density and drift speedCurrent density and drift speedΔL()()11/ddenA LiqjenvAAt ALvΔΔ== = =ΔΔn = # electrons per unit volumedne=−jvGGStarting pointfor Ohm’s LawOhmic materials (OhmOhmic materials (Ohm’’s law)s law)••We will see in a minute We will see in a minute that vthat vddis proportional is proportional to E. Thus, j to E. Thus, j ∝∝E, i.e.E, i.e.σ=jEGGOhm’s Lawρ=EjGGσis the electrical conductivityρis the electrical resistivity1σρ=SI unit for resistivity is ohm⋅ meter: 1 ohm = 1 volt/ampereSI unit for conductivity is siemens per meter:1 siemens = 1 ampere/volt = (1 ohm)-1Ohmic
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