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MIT 8 02T - Lecture Notes

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Class 04: OutlineGroupsAdvantages of GroupsWhat Groups Aren’tGroup Isn’t Working Well?Introduce YourselvesExperiment 1: VisualizationsLast Time:Gravitational & Electric FieldsGravity - ElectricityPotential Energyand PotentialGravity: Force and WorkWork Done by Earth’s GravityWork Near Earth’s SurfacePotential Energy (Joules)Gravitational Potential(Joules/kilogram)PRS Question:Masses in PotentialsMove to ElectrostaticsGravity - ElectrostaticsPotential & EnergyPotential: Summary Thus FarPotential LandscapePotential: Summary Thus FarPotential Landscape3 PRS Questions:Potential & Potential EnergyCreating Potentials:Two ExamplesPotential Created by Pt Charge2 PRS Questions:Point Charge PotentialPotential LandscapeDeriving E from VDeriving E from VDeriving E from VIn Class ProblemConfiguration EnergyConfiguration EnergyConfiguration EnergyIn Class Problem1P04 -Class 04: OutlineHour 1:Working In GroupsExpt. 1: VisualizationsHour 2:Electric PotentialPick up Group Assignment at Back of Room2P04 -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)7P04 -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 group8P04 -Last Time:Gravitational & Electric Fields9P04 -Gravity - ElectricityMass M Charge q (±)2ˆMGr=−grG2ˆeqkr=ErGCREATE:gm=FgGGEq=FEGGThis is easiest way to picture fieldFEEL:10P04 -Potential Energyand PotentialStart with Gravity11P04 -Gravity: Force and Work Gravitational Force on m due to M:2ˆgMmGr=−FrGWork done by gravity moving m from A to B:gBgAWd=⋅∫FsGGPATHINTEGRAL12P04 -Work Done by Earth’s GravityWork done by gravity moving m from A to B:ggWd=⋅∫FsGG()2ˆˆˆBAGMmrdr rdθ−⎛⎞=⋅+⎜⎟⎝⎠∫r r θ11BAGMmrr⎛⎞=−⎜⎟⎝⎠2BArrGMmdrr=−∫BArrGMmr=⎡⎤⎢⎥⎣⎦What is the sign moving from rAto rB?13P04 -Work Near Earth’s Surface2ˆˆEGMgr≈−=−gyyGG roughly constant:Work done by gravity moving m from A to B:()ˆBAmg d=−⋅∫ysGggWd=⋅∫FsGGByyAmgdy=−∫()BAmgy y=−−Wgdepends only on endpoints– not on path taken –Conservative Force14P04 -Potential Energy (Joules)BgBA g g extAUUU d W W∆ = − =− ⋅ =− =+∫FsGG02ˆ(1)ggGMm GMmUUrr=− → =− +FrG• U0: constant depending on reference point• Only potential difference ∆U has physical significance 0ˆ(2)ggmg U mgy U=− → = +FyG15P04 -Gravitational Potential(Joules/kilogram)Define gravitational potential difference:g(/)BBggAAUVmddm∆∆= =− ⋅ =− ⋅∫∫FsgsGGGGNNNNgFieldForce Energy PotentialJust as , ggUV→∆→∆FgGGThat is, two particle interaction Æ single particle effect16P04 -PRS Question:Masses in Potentials17P04 -Move to Electrostatics18P04 -Gravity - Electrostatics2ˆMGr=−grGMass M Charge q (±)2ˆeqkr=ErGgm=FgGGEq=FEGGBoth forces are conservative, so…BggAUd∆=− ⋅∫FsGGBEAUd∆=− ⋅∫FsGGBAVd∆=− ⋅∫EsGGBgAVd∆=− ⋅∫gsGG19P04 -Potential & EnergyBAVd∆≡− ⋅∫EsGGUnits: Joules/Coulomb = VoltsWork done to move q from A to B:ext B AWUUU=∆ = −qV=∆Joules20P04 -Potential: Summary Thus FarCharges CREATE Potential Landscapes00"0"()VVVV d=+∆≡− ⋅∫rrEsGGGG21P04 -Potential LandscapePositive ChargeNegative Charge22P04 -Potential: Summary Thus FarCharges CREATE Potential Landscapes00"0"()VVVV d=+∆≡− ⋅∫rrEsGGGGCharges FEEL Potential Landscapes()()UqV=rrGGWe work with ∆U (∆V) because only changes matter23P04 -Potential LandscapePositive ChargeNegative Charge24P04 -3 PRS Questions:Potential & Potential Energy25P04 -Creating Potentials:Two Examples26P04 -Potential Created by Pt Chargeθrsdˆˆθdrdr +=GBBAAVV V d∆= − =− ⋅∫EsGG2ˆrkQrE =G22ˆBBAAdrkQ d kQrr=− ⋅ =−∫∫rsG11BAkQrr⎛⎞=−⎜⎟⎝⎠Take V = 0 at r = ∞:rkQrV =)(ChargePoint27P04 -2 PRS Questions:Point Charge Potential28P04 -Potential LandscapePositive ChargeNegative Charge29P04 -Deriving E from V30P04 -Deriving E from Vˆx∆=∆siGA = (x,y,z), B=(x+∆x,y,z)BAVd∆=− ⋅∫EsGG(,,)(,,)xxyzxyzVd+∆∆=− ⋅∫EsGG≅−⋅∆EsGGˆ()xxEx=−⋅∆ =− ∆EiGxVVExx∆∂≅− →−∆∂Ex= Rate of change in V with y and z held constant31P04 -Deriving E from VIf we do all coordinates:V=−∇EGˆˆˆVVVxy z⎛⎞∂∂ ∂=− +⎜⎟∂∂ ∂⎝⎠Ei+jkGˆˆˆVxy z⎛⎞∂∂∂=− +⎜⎟∂∂ ∂⎝⎠i+ j kGradient (del) operator:ˆˆˆxyz∂∂∂∇≡ +∂∂∂ij+k32P04 -In Class ProblemFrom this plot of potential vs. position, create a plot of electric field vs. positionBonus: Is there charge somewhere? Where?33P04 -Configuration Energy34P04 -Configuration EnergyHow much energy to put two charges as pictured?1) First charge is free2) Second charge sees first:121212 2 2 114oqqUWqVrπε== =35P04 -Configuration EnergyHow much energy to put three charges as pictured?1) Know how to do first two2) Bring in third:()3312WqVV=+312013 234qqqrrπε⎛⎞=+⎜⎟⎝⎠Total configuration energy:13 23122 3 12 13 23012 13 2314qq qqqqUWW U U Urrrπε⎛⎞=+= + + = + +⎜⎟⎝⎠36P04 -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


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