Slide 1Capacitors and CapacitanceCapacitanceSlide 4Capacitors and CapacitanceParallel Plate CapacitorParallel Plate Capacitor -- exampleIsolated Parallel Plate CapacitorParallel Plate Capacitor & BatterySpherical CapacitorSpherical CapacitorCylindrical CapacitorCylindrical CapacitorSummaryCapacitors in ParallelCapacitors in seriesCapacitors in parallel and in seriesExample 1Example 2Example 3Energy Stored in a CapacitorEnergy Stored in Electric FieldExampleDielectric ConstantExampleExample (soln)SummaryPhysics 2102 Capacitors Physics 2102Gabriela GonzálezCapacitors and CapacitanceCapacitor: any two conductors, one with charge +Q, other with charge -Q+Q-QUses: storing and releasing electric charge/energy. Most electronic capacitors:micro-Farads (mF),pico-Farads (pF) -- 10-12 FNew technology: compact 1 F capacitorsPotential DIFFERENCE between conductors = VQ = CV -- C = capacitanceUnits of capacitance: Farad (F) = Coulomb/VoltCapacitance•Capacitance depends only on GEOMETRICAL factors and on the MATERIAL that separates the two conductors•e.g. Area of conductors, separation, whether the space in between is filled with air, plastic, etc.+Q-Q(We first focus on capacitorswhere gap is filled by AIR!)Electrolytic (1940-70)Electrolytic (new)Paper (1940-70)Tantalum (1980 on)Ceramic (1930 on)Mica(1930-50)Variableair, micaCapacitorsCapacitors and CapacitanceCapacitor: any two conductors, one with charge +Q, other with charge -Q+QUses: storing and releasing electric charge/energy. Most electronic capacitors:micro-Farads (mF),pico-Farads (pF) -- 10-12 FNew technology: compact 1 F capacitorsPotential DIFFERENCE between conductors = VQ = CV C = capacitanceUnits of capacitance: Farad (F) = Coulomb/Volt-QParallel Plate Capacitor+Q-QWhat is the capacitance C?Area of each plate = ASeparation = dcharge/area = s= Q/ARelate E to potential difference V:dxdEV0AQddxAQd000dAVQC0E field between the plates: (Gauss’ Law)AQE00We want capac i t ance: C=Q/VParallel Plate Capacitor -- example•A huge parallel plate capacitor consists of two square metal plates of side 50 cm, separated by an air gap of 1 mm •What is the capacitance?•C = e0A/d = (8.85 x 10-12 F/m)(0.25 m2)/(0.001 m)= 2.21 x 10-9 F(small!!)Lesson: difficult to get large valuesof capacitance without specialtricks!Isolated Parallel Plate Capacitor•A parallel plate capacitor of capacitance C is charged using a battery. •Charge = Q, potential difference = V.•Battery is then disconnected. •If the plate separation is INCREASED, does potential difference V:(a) Increase?(b) Remain the same?(c) Decrease?+Q-Q• Q is fixed!• C decreases (=e0A/d)• Q=CV; V increases.€ C =QV=QEd=ε0AdParallel Plate Capacitor & Battery•A parallel plate capacitor of capacitance C is charged using a battery. •Charge = Q, potential difference = V.•Plate separation is INCREASED while battery remains connected.+Q-Q• V is fixed by battery!• C decreases (=e0A/d)• Q=CV; Q decreases• E = Q/ e0A decreasesDoes the electric field inside:(a) Increase?(b) Remain the same?(c) Decrease?€ C =QV=QEd=ε0AdSpherical CapacitorWhat is the electric field insidethe capacitor? (Gauss’ Law)Radius of outer plate = bRadius of inner plate = aConcentric spherical shells:Charge +Q on inner shell,-Q on outer shellRelate E to potential differencebetween the plates:204 rQEbardEVbabarkQdrrkQ2bakQ11Spherical CapacitorWhat is the capacitance?C = Q/V =Radius of outer plate = bRadius of inner plate = aConcentric spherical shells:Charge +Q on inner shell,-Q on outer shellIsolated sphere: let b >> a,baQQ1140)(40ababaC04Cylindrical CapacitorWhat is the electric field in between the plates?Relate E to potential differencebetween the plates:Radius of outer plate = bRadius of inner plate = acylindrical surface of radius rLength of capacitor = L+Q on inner rod, -Q on outer shellrLQE02bardEVbabaLrQdrrLQ002ln2abLQln20Cylindrical CapacitorWhat is the capacitance C?C = Q/V =Radius of outer plate = bRadius of inner plate = aLength of capacitor = LCharge +Q on inner rod,-Q on outer shellabLQQln20abLln20Example: co-axial cable.Summary• Any two charged conductors form a capacitor.•Capacitance : C= Q/V•Simple Capacitors:Parallel plates: C = e0 A/dSpherical : C = 4p e0 ab/(b-a)Cylindrical: C = 2p e0 L/ln(b/a)Capacitors in Parallel•A wire is a conductor, so it is an equipotential.•Capacitors in parallel have SAME potential difference but NOT ALWAYS same charge.•VAB = VCD = V •Qtotal = Q1 + Q2•CeqV = C1V + C2V•Ceq = C1 + C2•Equivalent parallel capacitance = sum of capacitancesA BCDC1C2Q1Q2CeqQtotalPARALLEL: • V is same for all capacitors• Total charge in Ceq = sum of chargesCapacitors in series•Q1 = Q2 = Q (WHY??) •VAC = VAB + VBCAB CC1C2Q1Q2CeqQ21CQCQCQeq21111CCCeqSERIES: • Q is same for all capacitors• Total potential difference in Ceq = sum of VCapacitors in parallel and in series•In series : –1/Ceq = 1/C1 + 1/C2–Veq=V1 +V2–Qeq=Q1=Q2C1C2Q1Q2C1C2Q1Q2•In parallel : –Ceq = C1 + C2–Veq=V1=V2–Qeq=Q1+Q2CeqQeqExample 1What is the charge on each capacitor?10 mF30 mF20 mF120V• Q = CV; V = 120 V• Q1 = (10 mF)(120V) = 1200 mC • Q2 = (20 mF)(120V) = 2400 mC• Q3 = (30 mF)(120V) = 3600 mCNote that:•Total charge (7200 mC) is shared between the 3 capacitors in the ratio C1:C2:C3 -- i.e. 1:2:3Example 2What is the potential difference across each capacitor?10 mF30 mF20 mF120V• Q = CV; Q is same for all capacitors• Combined C is given by:)30(1)20(1)10(11FFFCeq• Ceq = 5.46 mF• Q = CV = (5.46 mF)(120V) = 655 mC• V1= Q/C1 = (655 mC)/(10 mF) = 65.5 V• V2= Q/C2 = (655 mC)/(20 mF) = 32.75 V• V3= Q/C3 = (655 mC)/(30 mF) = 21.8 VNote: 120V is shared in the ratio of INVERSE capacitances i.e.1:(1/2):(1/3) (largest C gets smallest V)Example 3In the circuit shown, what is the charge on the 10F capacitor?10 mF10 mF10V10 mF5 mF5 mF10V•The two 5F capacitors are in parallel•Replace by 10F •Then, we have two 10F capacitors in series•So, there is 5V across the 10F capacitor of interest•Hence, Q = (10F )(5V) = 50CEnergy Stored in a Capacitor•Start out with uncharged
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