Physics 2102Physics 2102Lecture: 12 MON FEBLecture: 12 MON FEBCapacitance IICapacitance IIPhysics 2102Jonathan Dowling25.4–5Capacitors in Parallel: V=ConstantCapacitors in Parallel: V=Constant• An ISOLATED wire is anequipotential surface:V=Constant• Capacitors in parallel haveSAME potential difference butNOT ALWAYS same charge!•VAB = VCD = V•Qtotal = Q1 + Q2•CeqV = C1V + C2V•Ceq = C1 + C2• Equivalent parallelcapacitance = sum ofcapacitancesA BCDC1C2Q1Q2CeqQtotalV!=!VAB =!VA –VB V!=!VCD =!VC –VD VAVBVCVDΔV=V! Cparallel= C1+ C2Capacitors in Series: Q=ConstantCapacitors in Series: Q=Constant• Q1 = Q2 = Q = Constant• VAC = VAB + VBCAB CC1C2Q1Q2CeqQ = Q1 = Q2 21CQCQCQeq+=! 1Cseries=1C1+1C2SERIES: • Q is same for all capacitors• Total potential difference = sum of VIsolated Wire:Q=Q1=Q2=ConstantCapacitors in parallel and in seriesCapacitors in parallel and in series• In series :1/Cser = 1/C1 + 1/C2Vser!=!V1 !+!V2Qser=!Q1!=!Q2C1C2Q1Q2C1C2Q1Q2• In parallel :Cpar = C1 + C2Vpar!=!V1!=!V2Qpar!=!Q1!+!Q2CeqQeqExample: Parallel or Series?Example: Parallel or Series?What is the charge on each capacitor?C1=10 µFC3=30 µFC2=20 µF120V• Qi = CiV• V = 120V = Constant• Q1 = (10 µF)(120V) = 1200 µC• Q2 = (20 µF)(120V) = 2400 µC• Q3 = (30 µF)(120V) = 3600 µCNote that:• Total charge (7200 mC) is sharedbetween the 3 capacitors in theratio C1:C2:C3 — i.e. 1:2:3Parallel: Circuit Splits Cleanly in Two (Constant V)! Cpar= C1+ C2+ C3= 10 + 20 + 30( )µF = 60µFExample: Parallel or SeriesExample: Parallel or SeriesWhat is the potential difference across each capacitor?C1=10µFC3=30µFC2=20µF120V• Q = CserV• Q is same for all capacitors• Combined Cser is given by:! 1Cser=1(10µF )+1(20µF )+1(30µF)• Ceq = 5.46 µF (solve above equation)• Q = CeqV = (5.46 µF)(120V) = 655 µC• V1= Q/C1 = (655 µC)/(10 µF) = 65.5 V• V2= Q/C2 = (655 µC)/(20 µF) = 32.75 V• V3= Q/C3 = (655 µC)/(30 µF) = 21.8 VNote: 120V is shared in theratio of INVERSEcapacitances i.e.(1):(1/2):(1/3) (largest C gets smallest V)Series: Isolated Islands (Constant Q)Example: Series or Parallel?Example: Series or Parallel?In the circuit shown,what is the charge onthe 10µF capacitor?10 µF10µF10V10µF5µF5µF10V• The two 5µF capacitors are inparallel• Replace by 10µF• Then, we have two 10µFcapacitors in series• So, there is 5V across the 10 µFcapacitor of interest• Hence, Q = (10µF )(5V) = 50µCNeither: Circuit Compilation Needed!Energy U Stored in a CapacitorEnergy U Stored in a Capacitor• Start out with unchargedcapacitor• Transfer small amount ofcharge dq from one plate tothe other until charge on eachplate has magnitude Q• How much work was needed?dq!=QVdqU0!==QCQdqCq02222CV=Energy Stored in Electric Field ofEnergy Stored in Electric Field of CapacitorCapacitor• Energy stored in capacitor: U = Q2/(2C) = CV2/2• View the energy as stored in ELECTRIC FIELD• For example, parallel plate capacitor: EnergyDENSITY = energy/volume = u =! u =Q22CAd==!"#$%&AddAQ022'2022 AQ!2220200EAQ!!!=""#$%%&'=volume = AdGeneralexpression forany region withvacuum (or air)ExampleExample• 10µF capacitor is initially charged to 120V.• 20µF capacitor is initially uncharged.• Switch is closed, equilibrium is reached.• How much energy is dissipated in the process?10µF (C1)20µF (C2)Initial energy stored = (1/2)C1Vinitial2 = (0.5)(10µ F)(120)2 = 72mJ Final energy stored = (1/2)(C1 + C2)Vfinal2 = (0.5)(30µ F)(40)2 = 24mJ Energy lost (dissipated) = 48mJInitial charge on 10µF = (10µF)(120V)= 1200µCAfter switch is closed, let charges = Q1 and Q2.Charge is conserved: Q1 + Q2 = 1200µCAlso, Vfinal is same: 2211CQCQ=221QQ =• Q1 = 400µC• Q2 = 800µC• Vfinal= Q1/C1 = 40 VSummarySummary• Any two charged conductors form a capacitor.• Capacitance : C= Q/V• Simple Capacitors:Parallel plates: C = ε0 A/dSpherical : C = 4π ε0 ab/(b-a)Cylindrical: C = 2π ε0 L/ln(b/a)• Capacitors in series: same charge, not necessarily equalpotential; equivalent capacitance 1/Ceq=1/C1+1/C2+…• Capacitors in parallel: same potential; not necessarilysame charge; equivalent capacitance Ceq=C1+C2+…• Energy in a capacitor: U=Q2/2C=CV2/2; energy density u=ε0E2/2• Capacitor with a dielectric: capacitance increases
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