Physics 2102 Lecture: 11 FRI 06 FEBTutoring Lab Now Open TuesdaysCapacitors and CapacitanceCapacitancePowerPoint PresentationParallel Plate CapacitorSlide 7Parallel Plate Capacitor — ExampleIsolated Parallel Plate CapacitorParallel Plate Capacitor & BatterySpherical CapacitorSlide 12Cylindrical CapacitorSummaryPhysics 2102 Physics 2102 Lecture: 11 FRI 06 FEB Lecture: 11 FRI 06 FEB Capacitance I Capacitance I Physics 2102Jonathan Dowling25.1–3QuickTime™ and a decompressorare needed to see this picture.Tutoring Lab Now Open Tuesdays•Lab Location: 102 Nicholson (across the hall from class)•Lab Hours:MTWT: 12:00N–5:00PMF: 12:N–3:00PMCapacitors and Capacitors and CapacitanceCapacitanceCapacitor: any two conductors, one with charge +Q, other with charge –Q+Q–QUses: storing and releasing electric charge/energy. Most electronic capacitors:micro-Farads (F),pico-Farads (pF) — 10–12 FNew technology: compact 1 F capacitorsUses: storing and releasing electric charge/energy. Most electronic capacitors:micro-Farads (F),pico-Farads (pF) — 10–12 FNew technology: compact 1 F capacitorsPotential DIFFERENCE between conductors = VQ = CCV where C = capacitanceUnits of capacitance: Farad (F) = Coulomb/VoltCapacitanceCapacitance•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, micaCapacitorsCapacitorsParallel Plate Parallel Plate CapacitorCapacitor-QE field between the plates: (Gauss’ Law)Relate E to potential difference V:What is the capacitance C?Area of each plate = ASeparation = dcharge/area = = Q/A∫⋅=dxdEV0rrAQddxAQd000εε∫==dAVQC0ε==dAVQC0ε==AQE00εεσ==AQE00εεσ==We want capacitance: CA=AQ/VA€ Units :C2Nm2m2m ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=C2Nm ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=CCJ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=CV ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ √€ Units :C2Nm2m2m ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=C2Nm ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=CCJ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=CV ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ √+QCapacitance and Your iPhone!Capacitance and Your iPhone!dAVQC0ε==Parallel Plate Capacitor — Parallel Plate Capacitor — ExampleExample•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?Lesson: difficult to get large valuesof capacitance without specialtricks! Lesson: difficult to get large valuesof capacitance without specialtricks! C = 0A/d = (8.85 x 10–12 F/m)(0.25 m2)/(0.001 m)= 2.21 x 10–9 F(Very Small!!)€ Units :C2Nm2m2m ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=C2Nm ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=CCJ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=CV ⎡ ⎣ ⎢ ⎤ ⎦ ⎥≡ F[ ]= Farad€ Units :C2Nm2m2m ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=C2Nm ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=CCJ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=CV ⎡ ⎣ ⎢ ⎤ ⎦ ⎥≡ F[ ]= FaradIsolatedIsolated Parallel Plate Parallel Plate CapacitorCapacitor•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 (=0A/d)• V=Q/C; V increases.Parallel Plate Capacitor & Parallel Plate Capacitor & BatteryBattery•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/0A decreasesDoes the Electric Field Inside:(a) Increase?(b) Remain the Same?(c) Decrease?Spherical CapacitorSpherical 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 rQEπε=∫⋅=bardEVrrbabarkQdrrkQ⎥⎦⎤⎢⎣⎡−==∫2⎥⎦⎤⎢⎣⎡−=bakQ11Spherical CapacitorSpherical 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,⎥⎦⎤⎢⎣⎡−=baQQ1140πε)(40abab−=πεaC04πε=Cylindrical CapacitorCylindrical CapacitorWhat is the electric field in between the plates? Gauss’ Law!Relate E to potential differencebetween the plates:Radius of outer plate = bRadius of inner plate = acylindrical Gaussiansurface of radius rLength of capacitor = L+Q on inner rod, –Q on outer shellrLQE02πε=∫⋅=bardEVrrbabaLrQdrrLQ⎥⎦⎤⎢⎣⎡==∫002ln2 πεπε€ =Q2πε0Llnba ⎛ ⎝ ⎜ ⎞ ⎠ ⎟QuickTime™ and a decompressorare needed to see this picture.€ C = Q /V =2πε0Llnba ⎛ ⎝ ⎜ ⎞ ⎠ ⎟€ C = Q /V =2πε0Llnba ⎛ ⎝ ⎜ ⎞ ⎠ ⎟SummarySummary• Any two charged conductors form a capacitor.•Capacitance : C= Q/V•Simple Capacitors:Parallel plates: C = 0 A/dSpherical: C = 4e0 ab/(b-a)Cylindrical: C = 20
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