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Chapter 24

Chapter 24. Capacitance and Dielectrics. Goals for Chapter 24. To understand capacitors and calculate capacitance To analyze networks of capacitors To calculate the energy stored in a capacitor To examine dielectrics and how they affect capacitance. Introduction.

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Chapter 24

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  1. Chapter 24 Capacitance and Dielectrics

  2. Goals for Chapter 24 • To understand capacitors and calculate capacitance • To analyze networks of capacitors • To calculate the energy stored in a capacitor • To examine dielectrics and how they affect capacitance

  3. Introduction • How do camera flash units store energy? • Capacitors are devices that store electric potential energy. • Energy of capacitor is stored in E field.

  4. Introduction

  5. Introduction

  6. Capacitors and capacitance • Any two conductors separated by an “insulator” form a capacitor. • Insulator willallow E field between the conductors, • Insulator will not allow charge to flow from one conductor through itself to the other. • The more charge you can hold, the larger the capacitor! “capacity” • Charge a capacitor by pushingit there with a potential voltage “pressure”

  7. Capacitors and capacitance • The definition of capacitance is C = Q/Vab. • Q = “charge stored” • Q is held symmetrically (same +Q as –Q) • Vab = pressure that pushes and keeps charge there • C increases as Q increases • more capacity! • C decreases as Vab increases • more pressure required to hold the charge there, so less effective in storing it temporarily!

  8. Capacitors and capacitance • The definition of capacitance is C = Q/Vab. • Units of Capacitance: [Coulombs/Volt] = FARADs • But [Volt] = [Joules/Coulomb] • Farads = Coulombs2/Joule • How much charge can you store per joule of work to keep them there?

  9. Parallel-plate capacitor • TWO parallel conducting plates • Separated by distance that is small compared to their dimensions.

  10. Evolution in Capacitors • Film dielectrics • “Wet” electrolytic capacitors • Ceramic Capacitors • Polymer Capacitors • http://www.capacitorlab.com/capacitor-types-polymer/

  11. Parallel-plate capacitor • The capacitance of a parallel-plate capacitor is C = 0A/d

  12. Parallel-plate capacitor • The capacitance of a parallel-plate capacitor is C = 0A/d. • Note! C is engineered! You control Area & distance by design! • C increases with Area • C decreases with separation

  13. Parallel-plate capacitor • The capacitance of a parallel-plate capacitor is C = 0A/d. • 0 has units of [Farads-meters/area] = [Farads/meter] • Recall “0” from Coulomb’s Law: Force = (1/4p 0) q1q2/r2 • Units of 0 were Coulomb2/Newton-meter2 • So [Farads/meter] = [Coulomb2/Newton-meter2] • [Farads] = [Coulomb2/Newton-meter] • [Farads] = [Coulomb2/Joule]

  14. Parallel-plate capacitor • Example 24.2: • Plates 2.00 m2 in area; 5.00 mm apart; 10 kV applied Potential Difference. • C=? • Q on each plate = ? • E field between plates = ?

  15. Parallel-plate capacitor • Example 24.2: • Plates 2.00 m2 in area; 5.00 mm apart; 10 kV applied Potential Difference. • C= e0A/d= 8.85 x 10-12 F/m *2.00 m2/.005 m = 3.54 x 10-9 F • Q on each plate from C = Q/V so Q = CV = 3.54 x 10-5 C • E field between plates from V = Ed so E = V/d = 2.00 x 10-6 V/m • Or… E = s/e0 = (Q/Area)/e0 = 3.54 x 10-5C/2.00m2 / e0

  16. A spherical capacitor • Example 24.3 – Two concentric spherical shells separated by vacuum.. What is C?

  17. A spherical capacitor • Example 24.3 – Two concentric spherical shells separated by vacuum.. What is C? • C = Q/V so we need V! • Get V from E field! • V = kq/r in general for spherical charge distribution • Vab = Va – Vb = “Voltage of a relative to b”

  18. A spherical capacitor • Example 24.3 – Two concentric spherical shells separated by vacuum.. What is C? • C = Q/V so we need V! • Get V from E field! • V = kq/r in general for spherical charge distribution • Vab = Va – Vb = kQ/ra – kQ/rb = kQ (1/ra – 1/rb) • Vab = Q/4pe0 (1/ra – 1/rb) • C = Q/V = 4pe0 (1/ra – 1/rb) = 4pe0 (ra rb)/ (rb – ra)

  19. A spherical capacitor • Example 24.3 – Two concentric spherical shells separated by vacuum. What is C? • C = Q/V = 4pe0 (1/ra – 1/rb) = 4pe0 (ra rb)/ (rb – ra) • Check! • C increases as Area increases? • YES! • C decreases as separation increases? • YES!!

  20. A cylindrical capacitor • Example 24.4 – Linear charge density +l on outer cylinder of radius rb, - lon inner cylinder of radius ra. What is C?

  21. A cylindrical capacitor

  22. A cylindrical capacitor • Example 24.4 – Linear charge density +l on outer cylinder of radius rb, - l on inner cylinder of radius ra. What is C? • C = Q/Vab; find Q and find Vab! • Q = l L • V = [l/2pe0] ln (r0/r) • r0 = distance where V was defined to be zero! • Say r0 = rb here, so Vab = (l/2pe0)ln (rb/ra) • Does this make sense?? YES!

  23. A cylindrical capacitor • Example 24.4 – Linear charge density +l on outer cylinder of radius rb, - l on inner cylinder of radius ra. What is C? • C = Q/Vab = l L/[(l/2pe0) ln (rb/ra)] • C = 2pe0L/ ln (rb/ra) • Check: • Units = Farads/Meter x Meters = Farads • C increases as L increases, • and C increaes as rb closer to ra!

  24. Capacitors in series • Capacitors are in seriesif connected one after the other

  25. Capacitors in series • Capacitors are in seriesif connected one after the other • NOTE: • Charges are the same on all plates in the series (even with different capacitances!)

  26. Capacitors in series • Capacitors are in seriesif connected one after the other • Equivalent capacitance of a series combination:1/Ceq = 1/C1 + 1/C2 + 1/C3 + …

  27. Capacitors in series • Capacitors are in seriesif connected one after the other • Equivalent capacitance of a series combination:1/Ceq = 1/C1 + 1/C2 + 1/C3 + … • NOTE • Ceq is always LESS than the smallest capacitor in series! • Say C1 = 10 mF, C2 = 20 mF, C3 = 30 mF • 1/Ceq = 1/10 + 1/20 + 1/30 = 6/60 + 3/60 + 2/60 = 11/60 • Ceq = 60/11 = 5.45 mF (< C1 !!) • WHY?

  28. Capacitors in parallel • Capacitors are connected in parallel between a and b if potential difference Vab is the same for all the capacitors.

  29. Capacitors in parallel • Capacitors are connected in parallel between a and b if potential difference Vab is the same for all the capacitors. • The equivalent capacitance of a parallel combination is the sum of the individual capacitances: Ceq = C1 + C2 + C3 + … .

  30. Capacitors in parallel • Capacitors are connected in parallel between a and b if potential difference Vab is the same for all the capacitors. • The equivalent capacitance of a parallel combination is the sum of the individual capacitances: Ceq = C1 + C2 + C3 + … . • Note! • Ceq is always MORE than largest capacitor in parallel! • Say C1 = 10 mF, C2 = 20 mF, C3 = 30 mF • Ceq = 10 + 20 + 30 = 60 mF > C3 • WHY??

  31. Calculations of capacitance • Example 24.6, a capacitor network: • Find C eq?

  32. Calculations of capacitance • Example 24.6, a capacitor network: • Find C eq?

  33. Calculations of capacitance • Example 24.6, a capacitor network: • Find C eq?

  34. Energy stored in a capacitor • The potential energy stored in a capacitor is • U = Q2/2C • U increases with more stored charge; • U is less for fixed charge on a larger capacitor • U = Q2/2C but C = Q/V…. so • U = 1/2 QV. • U = Q2/2C and Q = CV … so • U = 1/2 CV2

  35. Energy stored in a capacitor • The potential energy stored in a capacitor is U = Q2/2C = 1/2 CV2 = 1/2 QV. • The capacitor energy is stored in the electric field between the plates. The energy density is u = 1/2 0E2.

  36. Pull capacitors apart when connected to a battery? • Start with a capacitor C = e0A/d, connected to voltage source V, storing charge Q +Q +V - d -Q

  37. Pull capacitors apart when connected to a battery? • Pull plates apart - this takes work against field. +Q +V - -Q

  38. Pull capacitors apart when connected to a battery? • One way to see this…..Capacitance decreases • C = Q/V but V stays the same (fixed by battery) so Q +q* +V - d* -q*

  39. Pull capacitors apart when connected to a battery? +q* +q +V - d* -q* -q

  40. Pull capacitors apart when connected to a battery? +q* +V E - d* -q* -q

  41. Pull capacitors apart when connected to a battery? • d increases, A constant => C decreases • V fixed, so C = Q/V => Q = CV decreases • Q decreases so s decreases and E decreases • Energy = ½ QV decreases +q* E +V - d* -q* -q

  42. Pull capacitors apart when NOT connected?? • Start with a capacitor C = e0A/d, charged by voltage source V, storing charge Q +Q +V - d -Q

  43. Pull capacitors apart when NOT connected?? • Start with a capacitor C = e0A/d, charged by voltage source V, storing charge Q, then disconnected +Q d -Q

  44. Pull capacitors apart when connected to a battery? • Pull plates apart - this takes work against field. • But charge can’t go anywhere! +Q -Q

  45. Pull capacitors apart when connected to a battery? • One way to see this…..Capacitance decreases • C = Q/V but Q stays the same so V increase +Q d* -Q

  46. Pull capacitors apart when connected to a battery? • Another way to see this…..Q same, so s same, so E the same! +q* E d* -q*

  47. Pull capacitors apart when connected to a battery? • Another way to see this…..Q same, so s same, so E the same! +q* E constant V* d* -q* Energy density increases! (more volume in the capacitor)

  48. Energy stored in a capacitor • The potential energy stored in a capacitor is U = Q2/2C = 1/2 CV2 = 1/2 QV. • The capacitor energy is stored in the electric field between the plates. The energy density is u = 1/2 0E2. • The Z machine shown below can produce up to 2.9  1014 W using capacitors in parallel!

  49. Example 24.7 • C1 = 8.0 mF, charged to 120 V. Switch is open. • What is Q on C1? Energy stored?

  50. Example 24.7 • What is Q on C1? • C = Q/V so Q = CV = 960 8.0 mC • What is energy stored? • U = ½ CV2 or ½ QV = 0.58 J

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