PHY2049: Chapter 251Î Capacitors in parallelÎ Capacitors in seriesIt is foolish to connect capacitors in series.⋅⋅⋅++=21eq111CCC⋅⋅⋅++=21eqCCCExample: 100 μF in series with 10 μF is 9 μF (check yourself). The 100 μF capacitor will be totally wasted.Parallel and series capacitors—summaryPHY2049: Chapter 252Î Two capacitor in series Ceqeven smaller than smaller of the twoÎ n capacitors in series Ceqeven smaller than smallest of allsmaller of two(continued)21111CCCeq+=21222121/1CCCCCCCCCeq<+=+=L+++=3211111CCCCeqless than 1111CCeq>1CCeq<smallestPHY2049: Chapter 253Î Four 1 μF in parallel. Find Ceq.Î Four 1 μF in series. Find Ceq.Î 1.3 μF and 2.0 μF in series. Ceqis: (a) 0.79 μF (b) 1.65 μF (c) 3.3 μFExamples4 μF0.25 μFPHY2049: Chapter 254Î Equivalent capacitance? 1 and 2 in parallel Together, in series with 3Example: parallel-series combo1.0 μF2.0 μF3.0 μF1.0 + 2.0 = 3.0 μF 5.110.320.310.311==+=eqC1.5 μFPHY2049: Chapter 255Î Charge on C1? Total charge Charge on 1 and 2 Charge on 1(continued)1.0 μF2.0 μF3.0 μF10 V x 1.5 μF = 15 μCSame as the total! 10 V+q+q–q–q+q–qq=q1+ q2 Potential differences across 1 and 2 are the same. q1and q2 in proportion with C1and C2. q1=q x 1.0/(1.0+2.0) =5.0 μCPHY2049: Chapter 256Î In terms of charge Derived by considering work dW’ done by a fictitious process which moves infinitesimally small amount of charge +dq’ from conductor 1 to conductor 2 of capacitor, leaving behind –dq’on conductor 1:Î In terms of potential Since q=CV (definition of C)Î Compare with(kinetic energy) (spring) Energy stored in capacitor221CVU =CqU22=221mvK =221kxU =PHY2049: Chapter 2572021Eεu =Î Two alternative views Energy is stored in charge configuration in capacitor Energy is stored in E field in capacitorÎ Second view (will be important later in dealing with electromagnetic waves) Define energy density Show for parallel-plate capacitorÎ This equation holds for any E field produced at any point in space by any source Derivation requires vector calculusEnergy stored in electric fieldvolumeUu =PHY2049: Chapter 258Equivalence of two views (by example)ÎView 1 Energy is stored in capacitor’s charge configuration Generalize definition of capacitance to single conductor and find (see page 661)ÎView 2 Energy is stored in E field Inside Outside2041rQπεE =Checks!Spherical conductor+ ++ ++ ++ ++ ++ ++ ++ ++ +QRRπεC04=CQU22=RπεQU028=()RπεQrdrπεQdrrπrQπεεudVURR0220222200outside8844121∫∫∫∞∞====2021Eεu =0=EPHY2049: Chapter 259DielectricsÎDielectric is insulator. In E field, it becomes partly polarized. For microscopic view, read page 671. If dielectric fills the gap of charged capacitor, E0due to charges +q and –q partly polarizes it, inducing charges –q’ and +q’ near surfaces. These in turn produce field that partly cancels E0. Net field E proportional to, and less than, E0.ÎWhat’s the point? E0 → E = E0/κ less than E0 V0 → V = V0/κ from definition of V C0→ C = κ C0since C=q/VLarger than C0, which means capacitorstores more charge for given potential difference applied by battery. Beneficialto fill gap with dielectric.PHY2049: Chapter 2510(continued)ÎΚis called dielectric constant. Larger than 1.ÎInduced charge q’. So far, general to any capacitor. Now restrict ourselves to parallel-plate capacitor 00εqAE =0)(εqqEA′−+=qqqEE′−=0κqκqq <−=′11Gauss’ lawInduced charge q’ is less than q. κ=1 (vacuum, no dielectric) → q’=0 No induced charge.κ large (strong dielectric) → q’→qPHY2049: Chapter 2511REMINDERÎNext WebAssign due Tomorrow, ThursdayÎTest 1 on Monday in Class (Chapters 21–25) Study Sample Exams Must bring Gator1 ID card (Will take away points if you forget ;< ) Calculator (No formulae allowed on calculator) One 8.5”x11” formula sheet (May use both sides) Blank scratch paper (Department has no money to provide) Pencil, eraser, and sharpner, as
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