Lesson 6 Capacitors and Capacitance

1. Lesson 6Capacitors and Capacitance

2. Today, we will: • learn what a capacitor is. • learn the definition of capacitance. • find the electric field and voltage inside a parallel-plate capacitor. • find the capacitance of the capacitor. • learn that a dielectric is a material with polar molecules. • learn how dielectrics increase capacitance. • find the energy stored in a capacitor and in the electric field. Class 16

3. Section 1Capacitance, Charge, and Voltage

4. What is a Capacitor? Conductors that can hold charge. Cables, hands, etc. all have capacitance. For our purposes: two conductors, one with charge +Q and one with charge −Q.

5. What is a Capacitor? We “charge” a capacitor by connecting it to a battery. +

6. What is a Capacitor? • When we disconnect the battery, charge remains on the conductors. We “charge” a capacitor by connecting it to a battery.

7. What is a Capacitor? • When we disconnect the battery, charge remains on the conductors. We “charge” a capacitor by connecting it to a battery. • If we connect the conductors, charge will then flow from one to the other.

8. Why Are Capacitors Useful? Capacitors can provide uniform electric fields. We use them to accelerate or deflect charged beams, etc. We can store charge for later use. We can charge many capacitors and then discharge them at one time to produce very large currents for a short time. Capacitors are important in AC (alternating current = sinusoidal) circuits, but we’ll study that later.

9. Charging a Capacitor When we attach a capacitor to a battery: Charge builds up on the conductors. The charge on the + conductor is equal and opposite the charge on the − conductor. We call +Q the “charge on the capacitor” Voltage builds up on the capacitor until it has the same voltage as the battery. Electric field builds up in the capacitor.

10. Charging a Capacitor Q We find that voltage is proportional to charge. V

11. Charging a Capacitor Q We find that voltage is proportional to charge. V

12. Q=CV Capacitance If capacitance is large - - the capacitor holds a large charge at a small voltage.

13. Section 2Parallel Plate Capacitors

14. Parallel-plate Capacitors made of two plates each of area A (the shape doesn’t matter) plates are separated by a distance d.

15. Parallel-plate Capacitors The electric field is the sum of the electric fields of a positively charged palate …

16. Parallel-plate Capacitors … and a negatively charged plate.

17. Parallel-plate Capacitors … and a negatively charged plate.

18. Parallel-plate Capacitors The electric fields outside the plates cancel out.

19. Parallel-plate Capacitors The electric fields outside the plates cancel out. Make the outside fields disappear.

20. Parallel-plate Capacitors The electric fields between the plates add. Just make the arrows align…

21. Parallel-plate Capacitors The charges move to the inside of the plates. Move the + and – symbols toward the center.

22. Parallel-plate Capacitors The electric field inside is uniform. The electric field outside is small.

23. Section 3Electric Field, Voltage, and Capacitance in a Parallel-Plate Capacitor

24. Electric Field of a Capacitor We can find the electric field in a capacitor from Coulomb’s law and our knowledge of field lines!

25. Electric Field of a Capacitor The field lines inside a capacitor:

26. Electric Field of a Capacitor The field lines inside a capacitor:

27. Electric Field of a Capacitor capacitor with a charge Q and plate area A point charge with a charge Q. + + + + + + + +

28. Electric Field of a Capacitor • Field lines begin on the positive charge in both cases. • Since the positive charge is the same, the number of field lines is the same. + + + + + + + +

29. Electric Field of a Capacitor ←same N → N lines between the plates! + + + + + + + +

30. Electric Field of a Capacitor + + + + + + + +

31. d We know how the voltage relates to the electric field because the electric field is constant. Parallel-plate Capacitors We always ignore the minus sign, so V will be positive:

32. d Now we can find the capacitance: Parallel-plate Capacitors

33. d Now we can find the capacitance: Parallel-plate Capacitors • If the plate area is large, the capacitor • can hold more charge. • If the plate separation is small, the • charges on the two plates attract each • other with a stronger force, so the • capacitor can hold more charge.

34. Parallel-plate CapacitorEquations

35. Section 4Dielectrics

36. Dielectrics A dielectric is an insulator with polar molecules that is placed between the plates of a capacitor.

37. Dielectrics Polar molecules rotate in the electric field of the capacitor.

38. Dielectrics The net charge inside the dielectric is zero.

39. Dielectrics But there is leftover charge on the surfaces of the dielectric.

40. Dielectrics E of plates This charge produces an electric field that opposes the electric field of the plates. E of dielectric

41. Problem Type 1:Fixed Charge With the dielectric: Without the dielectric: A capacitor is charged with a battery to a charge Q. The battery is removed and a dielectric is inserted.

42. Problem Type 1:Fixed Charge With the dielectric: A capacitor is charged with a battery to a charge Q. The battery is removed and a dielectric is inserted.

43. Problem Type 1:Fixed Charge The electric field of the dielectric reduces the voltage across the capacitor, causing the capacitance to rise. A capacitor is charged with a battery to a charge Q. The battery is removed and a dielectric is inserted.

44. Problem Type 2:Fixed Voltage With the dielectric: Without the dielectric: A capacitor is connected to a battery with voltage V and remains connected as a dielectric is inserted.

45. Problem Type 2:Fixed Voltage With the dielectric: A capacitor is connected to a battery with voltage V and remains connected as a dielectric is inserted.

46. Problem Type 2:Fixed Voltage The charge on the dielectric pulls additional charge from the battery to the plates, causing the capacitance to rise. A capacitor is connected to a battery with voltage V and remains connected as a dielectric is inserted.

47. Section 5Energy in Capacitors and Electric Fields

48. Energy in a Capacitor Start with two parallel plates with no charge. Move one charge from one plate to the other. There is no electric field and no force, so it requires no work.

49. Energy in a Capacitor After the charge is transferred, the capacitor has a small charge and a small field. The field causes a force on the next charge we move, forcing us to do work.

50. Energy in a Capacitor When the charge on a capacitor is q, the voltage is q/C and the electric field is V/d=q/Cd. The force on a small charge dq is