Last time… Fields, forces, work, and potentialPotential from electric fieldEquipotential linesTopographic mapElectric field from potentialQuick QuizElectric Potential - Uniform FieldCheck of basic casesPotential ( V ) of spherical conductorQuick quizConnected spheresVarying E-fields on conductorPotential and chargeSlide 14CapacitanceCapacitorsDefinition of CapacitanceHow was charge transferred?Work done to charge a capacitorExample: Parallel plate capacitorSlide 21What is potential difference?What is the capacitance?Tues. Oct. 7, 2008 Physics 208 Lecture 11 1Last time… Fields, forces, work, and potentialElectric forces and work++Electric potential energy and electric potentialTues. Oct. 7, 2008 Physics 208 Lecture 11 2Potential from electric fielddV largest in direction of E-field.dV smallest (zero) perpendicular to E-field € dV = −r E • dr l € dr l € r E V=Vo € V = Vo−r E dr l € V = Vo+r E dr l € dr l € dr l € V = VoTues. Oct. 7, 2008 Physics 208 Lecture 11 3Equipotential linesLines of constant potentialIn 3D, surfaces of constant potentialTues. Oct. 7, 2008 Physics 208 Lecture 11 4Topographic mapEach lines is constant elevationSame as constant gravitational potentialgh (energy = mgh)Height interval between lines constantTues. Oct. 7, 2008 Physics 208 Lecture 11 5Electric field from potentialSpell out the vectors: € dW =r F ext• dr s = −r F Coulomb• dr s ⇒dV = −r E • dr s € dV = − Exdx + Eydy + Ezdz( )€ Ex= −dVdx, Ey= −dVdy, Ez= −dVdzUsually written € r E = −r ∇V = −dVdx,dVdy,dVdz ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ € r E = −r ∇V = −dVdx,dVdy,dVdz ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Said before thatThis works forTues. Oct. 7, 2008 Physics 208 Lecture 11 6Quick QuizSuppose the electric potential is constant everywhere. What is the electric field?A) PositiveB) NegativeC) IncreasingD) DecreasingE) ZeroTues. Oct. 7, 2008 Physics 208 Lecture 11 7Electric Potential - Uniform FieldConstant E-field corresponds to linearly decreasing (in direction of E) potential € dV = −r E • dr s € =− EdxAB∫= −E dxAB∫= −E xB− xA( )ABxHere V depends only on x, not on y € ⇒ VB−VA= −r E AB∫• dr s = −Eˆ x AB∫• dr sTues. Oct. 7, 2008 Physics 208 Lecture 11 8Check of basic casesPrevious quick quiz: uniform potential corresponds to zero electric fieldLinear potential corresponds to constant electric field€ E = −∇V = −∇ constant( )= 0€ E = −∇V = −∇ −Ex( )=∂∂xEx,∂∂yEx,∂∂zEx ⎛ ⎝ ⎜ ⎞ ⎠ ⎟= Eˆ xTues. Oct. 7, 2008 Physics 208 Lecture 11 9Potential ( V ) of spherical conductorWhat is V of spherical conductor relative to infinity? Charge on surface spherical charge shellGauss’ law E = keQ / r2 in the radial directionV is work / Coulomb to bring point charge from ∞€ V R( )−V ∞( )= −E∞R∫• ds€ E • ds = E dr = Edr € r E € dr s € = ER∞∫dr = kQr2R∞∫dr= −kQrR∞= kQRTues. Oct. 7, 2008 Physics 208 Lecture 11 10Quick quizTwo conducting spheres of diff radii connected by long conducting wire. What is approximately true of Q1, Q2?€ kQ/RA) Q2>Q1B) Q2<Q1C) Q2=Q1R1R2Q1Q2Previous result says conducting sphere of radius R carrying charge Q is at a potentialTues. Oct. 7, 2008 Physics 208 Lecture 11 11Connected spheresSince both must be at the same potential,€ kQ1R1=kQ2R2⇒ Q2=R2R1Q1Surface charge densities? € η =Q4πR2⇒ η2=R1R2η1Smaller radius sphere has smaller chargeSmaller radius sphere has smaller chargeSmaller sphere has larger surface charge densitySmaller sphere has larger surface charge densityElectric field? € E =ηεo⇒ E2=R1R2η1Local E-field bigger at more sharply curved (smaller R) regionsLocal E-field bigger at more sharply curved (smaller R) regionsTues. Oct. 7, 2008 Physics 208 Lecture 11 12Varying E-fields on conductorLarger electric fields near smaller radii surfaces. Large electric fields at sharp points, Strong fields can ionize air atoms.Tues. Oct. 7, 2008 Physics 208 Lecture 11 13Potential and chargeHave shown that a conductor has an electric potential, and that potential depends on its chargeFor a charged conducting sphere:+++++++++++€ V R( )−V ∞( )= kQR=kRQElectric potential proportional to total chargeElectric potential proportional to total chargeTues. Oct. 7, 2008 Physics 208 Lecture 11 14Quick QuizConsider this conducting object. When it has total charge Qo, its electric potential is Vo. When it has charge 2Qo, its electric potential A. is VoB. is 2VoC. is 4VoD. depends on shapeTues. Oct. 7, 2008 Physics 208 Lecture 11 15CapacitanceElectric potential of any conducting object proportional to its total charge. € V =1CQC = capacitanceLarge capacitance: need lots of charge to change potentialSmall capacitance: small charge can change potential.Tues. Oct. 7, 2008 Physics 208 Lecture 11 16CapacitorsWhere did the charge come from?Usually transferred from another conducting object, leaving opposite charge behindA capacitor consists of two conductorsConductors generically called ‘plates’Charge transferred between platesPlates carry equal and opposite chargesPotential difference between platesproportional to charge transferred QTues. Oct. 7, 2008 Physics 208 Lecture 11 17Definition of CapacitanceSame as for single conductorbut V = potential difference between platesQ = charge transferred between platesSI unit of capacitance is farad (F) = 1 Coulomb / VoltThis is a very large unit: typically use F = 10-6 F, nF = 10-9 F, pF = 10-12 F€ ΔV =1CQTues. Oct. 7, 2008 Physics 208 Lecture 11 18How was charge transferred?Battery has fixed electric potential difference across its terminals Conducting plates connected to battery terminals by conducting wires. Vplates = Vbattery across platesElectrons movefrom negative battery terminal to -Q platefrom +Q plate to positive battery terminalThis charge motion requires workThe battery supplies the workV€ Q = CΔVTues. Oct. 7, 2008 Physics 208 Lecture 11 19Requires work to transfer charge dq from one plate:Work done to charge a capacitor€ dW = ΔVdq =qCdq€ dW = ΔVdq =qCdq€ W =qC0Q∫dq =Q22C€ W =qC0Q∫dq =Q22CWork done stored as potential energy in capacitorTotal work = sum of incremental work€ U =Q22C=12QΔV =12C ΔV( )2€ U =Q22C=12QΔV =12C ΔV( )2Tues. Oct. 7, 2008 Physics 208
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