Application of Gauss's Law to Capacitors

1 P8-Class 08: OutlineHour 1:Last Time: ConductorsExpt. 3: Faraday Ice PailHour 2:Capacitors & Dielectrics2 P8-Last Time:Conductors3 P8-Conductors in EquilibriumConductors are equipotential objects:1) E = 0 inside2) Net charge inside is 03) E perpendicular to surface 4) Excess charge on surface0εσ=E4 P8-Conductors as Shields5 P8-Hollow ConductorsCharge placed INSIDE induces balancing charge INSIDE+q----+++-------+++++6 P8-Hollow ConductorsCharge placed OUTSIDE induces charge separation on OUTSIDE+q----+++E=07 P8-PRS SetupI1I2O1O2What happens if we put Q in the center?8 P8-PRS Questions:Point Charge Inside Conductor9 P8-Demonstration:Conductive Shielding10 P8-Visualization: Inductive Charginghttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/40-chargebyinduction/40-chargebyinduction.html11 P8-Experiment 3:Faraday Ice Pail12 P8-Last Time:CapacitorsP8-Capacitors: Store Electric EnergyQCV=∆To calculate:1) Put on arbitrary ±Q 2) Calculate E3) Calculate ∆VParallel Plate Capacitor:0ACdε=14 P8-Batteries & Elementary Circuits15 P8-Ideal BatteryFixes potential difference between its terminalsSources as much charge as necessary to do soThink: Makes a mountain16 P8-Batteries in Series∆V1∆V2Net voltage change is ∆V = ∆V1 + ∆V2Think: Two Mountains Stacked17 P8-Batteries in ParallelNet voltage still ∆VDon’t do this!18 P8-Capacitors in Parallel19 P8-Capacitors in ParallelSame potential!1212,QQCCVV==∆∆20 P8-Equivalent Capacitance?()121212QQQCVCVCCV=+=∆+∆=+∆12eqQCCCV==+∆21 P8-Capacitors in SeriesDifferent Voltages NowWhat about Q?22 P8-Capacitors in Series23 P8-Equivalent Capacitance1212QQV,VCC∆=∆=12VVV∆=∆+∆(voltage adds in series)12eqQQQVCCC∆==+12111eqCCC=+24 P8-PRS Question:Capacitors in Series and Parallel25 P8-Dielectrics26 P8-Demonstration:Dielectric in Capacitor27 P8-DielectricsA dielectric is a non-conductor or insulatorExamples: rubber, glass, waxed paperWhen placed in a charged capacitor, the dielectric reduces the potential difference between the two platesHOW???28 P8-Molecular View of DielectricsPolar Dielectrics : Dielectrics with permanent electric dipole moments Example: Water 29 P8-Molecular View of DielectricsNon-Polar DielectricsDielectrics with induced electric dipole momentsExample: CH430 P8-Dielectric in CapacitorPotential difference decreases because dielectric polarization decreases Electric Field!31 P8-Gauss’s Law for DielectricsUpon inserting dielectric, a charge density σ’is induced at its surface0SinsideqdEAε⋅==∫∫EA􀁇􀁇􀁷0'εσσ−=E()0'Aσσε−=What is σ’?32 P8-Dielectric Constant κDielectric weakensoriginal field by a factor κ000'EEσσσεκκε−=≡=1'1σσκ⎛⎞=−⎜⎟⎝⎠⇒Gauss’sLaw with dielectrics:0Sfreeinsideqdκε⋅=∫∫EA􀁇􀁇Dielectric constantsVacuum1.0Paper 3.7Pyrex Glass 5.6Water 8033 P8-Dielectric in a CapacitorQ0= constant after battery is disconnected0VVκ=Upon inserting a dielectric: 00000/QQQCCVVVκκκ====34 P8-Dielectric in a CapacitorV0 = constant when battery remains connectedUpon inserting a dielectric: 0QQκ=000QQCCVVκκ===35 P8-PRS Questions:Dielectric in a Capacitor36 P8-Group: Partially Filled Capacitor What is the capacitance of this capacitor?