EXAM REVIEW ON MONDAY

Transcription

1 EXAM REVIEW ON MONDAY 6:5 8:5 PM McCarty A Room G86 By JJ Stankowicz Also, formula sheet has been posted. PHY049: Chapter 5

2 Capacitance calculation review +q q Why do we always consider only +q and q pairs? Why not just any q and Q? PHY049: Chapter 5

3 Why do we always consider only +q and q pairs? Why not just any q and Q? Charging capacitor Battery just moves electrons from one side to the other until potential difference across capacitor reaches battery s emf (aka voltage ) V. Does not add or remove charge. In general In electronic circuits, capacitors are used in such ways that +q and q occur as a pair. PHY049: Chapter 5 3

4 Capacitors in parallel (derivation of formula) Three capacitors in parallel are charged by battery or power supply For given V applied by battery/supply, the combo stores more charge, q + q + q 3, than a single capacitor. Larger capacitance: C eq = C + C + C 3 Generalize to more than three capacitors C eq = C + C + C 3 + PHY049: Chapter 5 4

5 Capacitors in series (derivation of formula) Two capacitors in series are charged by battery or power supply no charge +q no charge no charge before after Induced charges appear immediately +q still no net charge q after q attracted to +q +q attracted to q q PHY049: Chapter 5 5

6 Potential difference (continued) +q C q C +q q V V Electric field is absent in the connecting wire, a conductor. No potential difference between two ends of the wire. Potential differences are only across the capacitors. V=V + V =q/c + q/c V=q/C eq Definition of C eq /C eq =/C + /C Generalize to more than two capacitors in series /C eq =/C + /C + PHY049: Chapter 5 6

7 Parallel and series capacitors summary Capacitors in parallel C eq = C + C + Capacitors in series C eq = C + C + It is foolish to connect capacitors in series. Example: 00 μf in series with 0 μf is 9 μf (check yourself). The 00 μf capacitor will be totally wasted. PHY049: Chapter 5 7

8 Examples Four μf in parallel. Find C eq. 4 μf Four μf in series. Find C eq. 0.5 μf.3 μf and.0 μf in series. C eq is: (a) 0.79 μf (b).65 μf (c).6 μf (d) 3.3 μf PHY049: Chapter 5 8

9 (continued) Capacitors in series = L > C eq C C eq C C C3 C eq < C smallest C eq is smaller than the smallest of all PHY049: Chapter 5 9

10 Example: parallel-series combo Equivalent capacitance?.0 μf.0 μf 6.0 μf and in parallel = 3.0 μf Together, in series with 3 C eq = = =.0.0 μf PHY049: Chapter 5 0

11 Charge on C? (See Sample Problem 5- for an alternative solution) (continued).0 μf.0 μf 0 V 6.0 μf q = C V a (q +q ) Must find potential difference (aka voltage) Va across C (also across C.) Note: V a + V b = V (applied voltage ) Need one more equation to relate Va to Vb. Must be through the fact that charge stored in capacitor 3 is q +q. q + q = C V a + C V a, V a +q +q V b q + q =C 3 V b C V a + C V a =C 3 V b, i.e., V a :V b =C 3 :(C +C ) V a V a =V C 3 /[C 3 +(C +C )] q =VC C 3 /[C 3 +(C +C )] 6.7 μc PHY049: Chapter 5

12 In terms of charge Derived by considering work dw done by a fictitious process which moves infinitesimally small amount of charge +dq from conductor to conductor of capacitor, leaving behind dq on conductor : In terms of potential Since q=cv (definition of C) Similarities with Energy stored in capacitor q U = C U = CV mv U = kx K = (kinetic energy) (spring) PHY049: Chapter 5

13 Energy stored in electric field Two alternative views Energy is stored in charge configuration in capacitor Energy is stored in E field Second view (will be important later in dealing with electromagnetic waves) Define energy density Show for parallel-plate capacitor u = ε This equation holds for any E field produced at any point in space by any source Derivation requires vector calculus u = U volume 0 E PHY049: Chapter 5 3

14 Equivalence of two views (by example) View Energy is stored in capacitor s charge configuration q U = C Capacitance: C = πε 0 L / ln( b / a) View U Cylindrical capacitor Energy is stored in E field u = ε0e q In the gap E = Elsewhere πε Lr = 0 U = q ln( b / a) 4πε L b q q dr udv ε0 a 4πε0 Lr = 4πε0L = a r gap 0 E = 0 q 4πε b ( πrldr) = ln( b / a) Agrees! PHY049: Chapter L

15 Dielectrics Dielectric is insulator. In E field, it becomes partly polarized. For microscopic view, read Section 5-7. If dielectric fills the gap of charged capacitor, E 0 due to charges +q and q partly polarizes it, inducing charges q and +q near surfaces. These in turn produce field that partly cancels E 0. Net field E proportional to, and less than, E 0. What s the point? E 0 E = E 0 /κ less than E 0 V 0 V = V 0 /κ from definition of V C 0 C = κ C 0 since C=q/V Larger than C 0, which means capacitor stores more charge for given potential difference applied by battery. Beneficial to fill gap with dielectric. PHY049: Chapter 5 5

16 (continued) Κis called dielectric constant. Larger than. Induced charge q. So far, general to any capacitor. Now restrict ourselves to parallel-plate capacitor E A = 0 q ε 0 q + ( q ) EA = ε0 E q q = E q 0 κ Gauss law q q = < κ q Induced charge q is always less than q. κ= (vacuum, no dielectric) q =0 No induced charge. κ large (strong dielectric) q q PHY049: Chapter 5 6

17 Concept Question All capacitors are identical. Across each combo, the same voltage (potential difference) is applied. Which combo stores the highest energy? (a) (b) (c) (d) PHY049: Chapter 5 7

18 REMINDER Exam on Wednesday in Class (Chapters 5) Study sample exams posted Must bring Gator ID card (Will take away points if you forget ;< ) Calculator (No formulae allowed on calculator) Pencil, eraser, and sharpner, as usual PHY049: Chapter 5 8