Electric Potential

1 P04 -Class 04: OutlineHour 1:Working In GroupsExpt. 1: VisualizationsHour 2:Electric PotentialPick up Group Assignment at Back of Room2 P04 -GroupsP04 -Advantages of Groups•Three heads are better than one•Don’t know? Ask your teammates•Do know? Teaching reinforces knowledgeLeave no teammate behind!•Practice for real life –science and engineering require teamwork; learn to work with othersP04 -What Groups Aren’t•A Free RideWe do much group based work (labs & Friday problem solving). Each individual must contribute and sign name to workIf you don’t contribute (e.g. aren’t in class) you don’t get creditP04 -Group Isn’t Working Well?1.Diagnose problem and solve it yourself --Most prevalent MIT problem: free rider.2.Talk to Grad TA3.Talk to the teamwork consultantDon’t wait: Like most problems,teamwork problems get worse the longer you ignore themP04 -Introduce YourselvesPlease discuss:•What is your experience in E&M?•How do you see group working?•What do you expect/want from class?•What if someone doesn’t participate?•What if someone doesn’t come to class?Try to articulate solutions to foreseeable problems now (write them down)7 P04 -Experiment 1: VisualizationsNeed experiment write-up from course packet.Turn in tear sheet at end of classEach GROUP hands in ONE tear sheet signed by each member of group8 P04 -Last Time:Gravitational & Electric Fields9 P04 -Gravity -ElectricityMass MCharge q(±)2ˆMGr=−gr􀁇2ˆeqkr=Er􀁇CREATE:gm=Fg􀁇􀁇Eq=FE􀁇􀁇This is easiest way to picture fieldFEEL:10 P04 -Potential Energyand PotentialStart with Gravity11 P04 -Gravity: Force and Work Gravitational Force on m due to M:2ˆgMmGr=−Fr􀁇Work done by gravity moving m from A to B:gBgAWd=⋅∫Fs􀁇􀁇PATHINTEGRAL12 P04 -Work Done by Earth’s GravityWork done by gravity moving m from Ato B:ggWd=⋅∫Fs􀁇􀁇()2ˆˆˆBAGMmrdrrdθ−⎛⎞=⋅+⎜⎟⎝⎠∫rrθ11BAGMmrr⎛⎞=−⎜⎟⎝⎠2BArrGMmdrr=−∫BArrGMmr=⎡⎤⎢⎥⎣⎦What is the sign moving from rAto rB?13 P04 -Work Near Earth’s Surface2ˆˆEGMgr≈−=−gyy􀁇G roughly constant:Work done by gravity moving m from Ato B:()ˆBAmgd=−⋅∫ys􀁇ggWd=⋅∫Fs􀁇􀁇ByyAmgdy=−∫()BAmgyy=−−Wgdepends onlyon endpoints–noton path taken –Conservative Force14 P04 -Potential Energy (Joules)BgBAggextAUUUdWW∆=−=−⋅=−=+∫Fs􀁇􀁇02ˆ(1)ggGMmGMmUUrr=−→=−+Fr􀁇•U0: constant depending on reference point•Only potential difference ∆Uhas physical significance 0ˆ(2)ggmgUmgyU=−→=+Fy􀁇15 P04 -Gravitational Potential(Joules/kilogram)Define gravitational potential difference:g(/)BBggAAUVmddm∆∆==−⋅=−⋅∫∫Fsgs􀁇􀁇􀁇􀁇􀁎􀁎􀁎􀁎gFieldForceEnergyPotentialJust as , ggUV→∆→∆Fg􀁇􀁇That is, two particle interaction 􀃆single particle effect16 P04 -PRS Question:Masses in Potentials17 P04 -Move to Electrostatics18 P04 -Gravity -Electrostatics2ˆMGr=−gr􀁇Mass MCharge q(±)2ˆeqkr=Er􀁇gm=Fg􀁇􀁇Eq=FE􀁇􀁇Both forces are conservative, so…BggAUd∆=−⋅∫Fs􀁇􀁇BEAUd∆=−⋅∫Fs􀁇􀁇BAVd∆=−⋅∫Es􀁇􀁇BgAVd∆=−⋅∫gs􀁇􀁇19 P04 -Potential & EnergyBAVd∆≡−⋅∫Es􀁇􀁇Units: Joules/Coulomb = VoltsWork done to move qfrom A to B:extBAWUUU=∆=−qV=∆Joules20 P04 -Potential: Summary Thus FarCharges CREATEPotential Landscapes00"0"()VVVVd=+∆≡−⋅∫rrEs􀁇􀁇􀁇􀁇21 P04 -Potential LandscapePositive ChargeNegative Charge22 P04 -Potential: Summary Thus FarCharges CREATEPotential Landscapes00"0"()VVVVd=+∆≡−⋅∫rrEs􀁇􀁇􀁇􀁇Charges FEELPotential Landscapes()()UqV=rr􀁇􀁇We work with ∆U (∆V) because only changes matter23 P04 -Potential LandscapePositive ChargeNegative Charge24 P04 -3 PRS Questions:Potential & Potential Energy25 P04 -Creating Potentials:Two Examples26 P04 -Potential Created by Pt Chargeθrsdˆˆθdrdr+=􀁇BBAAVVVd∆=−=−⋅∫Es􀁇􀁇2ˆrkQrE=􀁇22ˆBBAAdrkQdkQrr=−⋅=−∫∫rs􀁇11BAkQrr⎛⎞=−⎜⎟⎝⎠Take V= 0 at r= ∞:rkQrV=)(ChargePoint 27 P04 -2 PRS Questions:Point Charge Potential28 P04 -Potential LandscapePositive ChargeNegative Charge29 P04 -Deriving E from V30 P04 -Deriving E from Vˆx∆=∆si􀁇A = (x,y,z), B=(x+∆x,y,z)BAVd∆=−⋅∫Es􀁇􀁇(,,)(,,)xxyzxyzVd+∆∆=−⋅∫Es􀁇􀁇≅−⋅∆Es􀁇􀁇ˆ()xxEx=−⋅∆=−∆Ei􀁇xVVExx∆∂≅−→−∆∂Ex= Rate of change in V with y and z held constant31 P04 -Deriving E from VIf we do all coordinates:V=−∇E􀁇ˆˆˆVVVxyz⎛⎞∂∂∂=−+⎜⎟∂∂∂⎝⎠Ei+jk􀁇ˆˆˆVxyz⎛⎞∂∂∂=−+⎜⎟∂∂∂⎝⎠i+jkGradient (del) operator:ˆˆˆxyz∂∂∂∇≡+∂∂∂ij+k32 P04 -In Class ProblemFrom this plot of potential vs. position, create a plot of electric field vs. positionBonus: Is there charge somewhere? Where?33 P04 -Configuration Energy34 P04 -Configuration EnergyHow much energy to put two charges as pictured?1)First charge is free2)Second charge sees first:12121222114oqqUWqVrπε===35 P04 -Configuration EnergyHow much energy to put three charges as pictured?1)Know how to do first two2)Bring in third:()3312WqVV=+312013234qqqrrπε⎛⎞=+⎜⎟⎝⎠Total configuration energy:13231223121323012132314qqqqqqUWWUUUrrrπε⎛⎞=+=++=++⎜⎟⎝⎠36 P04 -In Class ProblemHow much energy in joules is required to put the three charges in the configuration pictured if they start out at infinity?What is the electric potential in volts at point P?Suppose you move a fourth change +3Q from infinity in to point P. How much energy does that require (joules)?