Conductors, Insulators and Capacitors

1 P07 -Class 07: OutlineHour 1:Conductors & Insulators Expt. 2: Electrostatic ForceHour 2:Capacitors2 P07 -Last Time:Gauss’s Law3 P07 -Gauss’s Law 0closedsurfaceSencEQdεΦ=⋅=∫∫EA􀁇􀁇􀁷In practice, use symmetry:•Spherical (r)•Cylindrical (r, 􀁁)•Planar (Pillbox, A)4 P07 -Conductors5 P07 -Conductors and InsulatorsA conductor contains charges that are free to move (electrons are weakly bound to atoms)Example: metalsAn insulator contains charges that are NOT free to move (electrons are strongly bound to atoms)Examples: plastic, paper, wood6 P07 -ConductorsConductors have free charges􀃆E must be zero inside the conductor􀃆Conductors are equipotential objectsENeutral Conductor----++++7 P07 -Equipotentials8 P07 -Topographic Maps9 P07 -Equipotential CurvesAll points on equipotential curve are at same potential.Each curve represented by V(x,y) = constant10 P07 -PRS Question:Walking down a mountain11 P07 -Direction of Electric Field EEis perpendicular to all equipotentialsConstant E fieldPoint ChargeElectric dipole12 P07 -Properties of Equipotentials•E field lines point from high to low potential•E field lines perpendicular to equipotentials•Have no component along equipotential•No work to move along equipotential13 P07 -Conductors in EquilibriumConductors are equipotential objects:1) E = 0 inside2) Net charge inside is 03) E perpendicular to surface 4) Excess charge on surface0εσ=E14 P07 -Conductors in EquilibriumPut a net positive charge anywhere inside a conductor, and it will move to the surface to get as far away as possible from the other charges of like sign. http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/34-pentagon/34-pentagon320.html15 P07 -Expt. 2: Electrostatic Force16 P07 -Expt. 2: Electrostatic Forcehttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/36-electrostaticforce/36-esforce320.html17 P07 -Experiment 2:Electrostatic ForceP07 -Capacitors and CapacitanceP07 -Capacitors: Store Electric EnergyCapacitor: two isolated conductors with equal and opposite charges Q and potential difference ∆V between them.QCV=∆Units: Coulombs/Volt or Farads20 P07 -Parallel Plate Capacitor0=E0=EQAσ+=QAσ−=−d?E=21 P07 -Parallel Plate CapacitorWhen you put opposite charges on plates, charges move to the inner surfaces of the plates to get as close as possible to charges of the opposite signhttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/35-capacitor/35-capacitor320.html22 P07 -Calculating E (Gauss’s Law)0Sinqdε⋅=∫∫EA􀁇􀁇􀁷00εεσAQE==()0GaussGaussAEAσε=Note: We only “consider” a single sheet! Doesn’t the other sheet matter?23 P07 -Alternate Calculation Method+ + + + + + + + + + + + + +Top Sheet:02Eσε=−02Eσε=--------------Bottom Sheet:02Eσε=02Eσε=−000022QEAσσσεεεε=+==24 P07 -Parallel Plate CapacitortopbottomVd∆=−⋅∫ES􀁇􀁇0QdAε=Ed=dAVQC0ε=∆=Cdepends only on geometric factors Aand d25 P07 -Demonstration:Big Capacitor26 P07 -Spherical CapacitorTwo concentric spherical shells of radii aand bWhat is E?Gauss’s Law 􀃆E≠0 only for a < r < b, where it looks like a point charge:rEˆ420rQπε=􀁇27 P07 -Spherical CapacitorFor an isolated spherical conductor of radius a:20ˆˆ4baQdrrπε=−⋅∫rr()1104−−−=∆=baVQCπεoutsideinsideVd∆=−⋅∫ES􀁇􀁇0114Qbaπε⎛⎞=−⎜⎟⎝⎠Is this positive or negative? Why?aC04πε=28 P07 -Capacitance of EarthFor an isolated spherical conductor of radius a:aC04πε=mF1085.8120−×=εm104.66×=amF7.0F1074=×=−CA Farad is REALLY BIG! We usually use pF(10-12) or nF(10-9)29 P07 -1 Farad CapacitorHow much charge?()()1F12V12CQCV=∆==30 P07 -PRS Question:Changing C Dimensions31 P07 -Demonstration:Changing C Dimensions32 P07 -Energy Stored in Capacitor33 P07 -Energy To Charge Capacitor1.Capacitor starts uncharged.2.Carry +dqfrom bottom to top. Now top has charge q = +dq, bottom -dq3.Repeat4.Finish when top has charge q = +Q, bottom -Q+q-q34 P07 -Work Done Charging CapacitorAt some point top plate has +q, bottom has –qPotential difference is ∆V= q /CWork done lifting another dqis dW = dq ∆V+q-q35 P07 -Work Done Charging CapacitorSo work done to move dqis:dWdqV=∆1qdqqdqCC==Total energy to charge to q = Q:01QWdWqdqC==∫∫+q-q212QC=36 P07 -Energy Stored in CapacitorQCV=∆Since2221212VCVQCQU∆=∆==Where is the energy stored???37 P07 -Energy Stored in CapacitorEnergy stored in the E field!andoACVEddε==Parallel-plate capacitor:212UCV=()221()22ooAEEdAddεε==×()Euvolume=×2fieldenergydensity2oEEuEε==38 P07 -1 Farad Capacitor -EnergyHow much energy?()()221211F12V272JUCV=∆==Compare to capacitor charged to 3kV:()()()()2224311100µF3kV221110F310V450J2UCV−=∆==××=39 P07 -PRS Question:Changing C DimensionsEnergy Stored40 P07 -Demonstration:Dissectible Capacitor