Chapter 26

1. Chapter 26 Capacitance and Dielectrics

2. Intro • In this chapter we will introduce the first of the 3 simple electric circuit elements that we will discuss in AP Physics • Capacitor • Resistor • Inductor • Capacitors are commonly used devices often as form of energy storage in a circuit.

3. 26.1 • Capacitor- two conductors separated by an insulator called a dielectric. • Often times the two conductors of a capacitor are called plates. • If the plates carry a charges of equal magnitude and opposite sign, there will exist a potential difference (ΔV) or a voltage between the two.

4. 26.1 • How much charge can we store on the plates? • Experiments have shown that the amount of charge stored increases linearly with voltage between the conductors. • We will call the constant of proportionality Capacitance .

5. 26.1 • Capacitance is defined as the ratio of the magnitude of the charge on either conductor to the magnitude of the potential difference between the conductors. • Capacitance is a constant for a given capacitor and has units of C/V or F (farad)

6. 26.1 • 1 Farad is a Coulomb of charge per Volt which is a huge capacitance. • More commonly in the 10-6 to 10-12 range or microfarads (μF) to picofarads (pF) • Actual capacitors may often be marked mF (microfarad) or mmF (micromicrofarad)

7. 26.1 • Consider two parallel plates attached to a battery (potential difference source) • The battery creates an Electric field within the wires, moving electrons onto the negative plate. • This continues until the negative terminal, wire and plate are equipotential.

8. 26.1 • The opposite occurs with the positive terminal, pulling electrons from the plate until the plate/wire/+ terminal are equipotential. • Example- • A 4 pF capacitor will be able to store 4 pC of charge for every volt of potential difference between the plates. • If we attach a 1.5 V battery, one of the conductors will have a +6 pC charge, the other will have -6 pC. • 12 V battery?

9. 26.1 • Quick Quiz p 797

10. 26.2 Calculating Capacitance • We can derive expressions for the capacitance of pairs of oppositely charged conductors by calculating ΔV using techniques from the previous chapters. • The calculations are generally straightforward for simple capacitors with symmetrical geometry.

11. 26.2 • A single conductor • Sphere (w/infinite imaginary shell) • Since at the sphere, V = kQ/R, and at ∞, V = 0

12. 26.2 • Parallel Plate Capacitors • For plates whose separation is much smaller than their size. • From earlier (Ch 24) the E field between the plates is • So the potential difference is

13. 26.2 • And the capacitance is therefore • The capacitance is proportional to area and inversely proportional to the plate separation. • True in the middle of the plates, but not near the edges.

14. 26.2 • The capacitor stores electrical potential energy as well as charge due to the separation of the positive and negative charges on the plates. Quick Quiz p. 800 Examples 26.1-26.3

15. 26.2 • Example 26.2 Cylindrical Capacitor

16. 26.2 • Example 26.3 Spherical Capacitor

17. 26.3 Combinations of Capacitors • Often two or more capacitors are combined in electric circuits. • We can calculate the equivalent capacitance of a circuit, based on how the capacitors are connected. • We will use circuit diagrams (schematics) as pictorial representations of the circuit.

18. 26.3 • Connecting wires- straight lines • Capacitors- parallel lines of equal length • Batteries- parallel lines of unequal length • Switch- swinging line representing “open” or “closed” circuits

19. 26.3 • Parallel Combination- two capacitors connected with their own conducting path to the battery. • The potential difference across each capacitor is the same, and its equal to the potential across the combination.

20. 26.3 • When the battery is connected electrons are removed from the positive plates and deposited on the negative plates. • This flow of charge ceases when the potential across the plates reaches that of the battery. • The capacitors are then at maximum charge Q1 and Q2, with a total charge given by

21. 26.3 • Because the voltages across the capacitors are the same the charges are • If we wanted to replace the two capacitors with a single equivalent capacitor the total charge stored must be

22. 26.3 • Therefore the equivalent capacitance must be • The equivalent capacitance for any number of parallel combination of capacitors is

23. 26.3

24. 26.3 • Series Combination- two or more capacitors connected along the same conducting path

25. 26.3 • As the battery charges the capacitors the electrons leaving the positive plate of C1 end up on the negative plate of C2. • The electrons from the positive plate of C2 move to the negative plate of C1. • All capacitors hold the same charges.

26. 26.3 • The voltage of the battery is split across the capacitors. • The total potential difference across a series combination of capacitors is the sum of the potential difference across each individual capacitor.

27. 26.3 • If we wanted to find one capacitor equivalent to the series combination, the total potential difference is • And each individual is

28. 26.3 • So from • We get • And finally

29. 26.3 • The inverse of the equivalent capacitance is equal to the sum of the inverses of the individual capacitances in series combination.

30. 26.3 • Quick Quizzes p. 805 • Example 26.4 p. 806

31. 26.4 Energy Stored in a Charged Capacitor • To determine the energy in a capacitor, we’re going to look at an atypical charging process. • We’re going to imagine moving the charge from one plate to the other mechanically, through the space in between.

32. 26.4 • Assume we currently have a charge q on our capacitor, giving the current potential difference to be ΔV = q/C. • The work it will take to move a small increment of charge across the gap is

33. 26.4 • The total work W, required to charge the capacitor from q = 0 to q = Q is

34. 26.4 • The work done in charging the capacitor is the Electrical Potential Energy stored and applies to any capacitor regardless of geometry. • Energy increases as both Charge and Voltage increase, within a limit. At high enough potential difference, discharge will occur between the plates.

35. 26.4 • We can describe the energy as being stored in the electric field between the plates. • For parallel plate caps • Therefore

36. 26.4 • We use this expression to derive a new quantity called Energy Density (uE) • Since the volume occupied by the field is Ad, the energy U per unit volume is (U/Ad) • The energy density of an E-field is proportional the square of the magnitude of the E-field at a given point.

37. 26.4 • Quick Quizzes p 808 • Example 26.5

38. 26.4 • Defibrillation- Capacitors store 360 J of energy at a potential difference that will deliver the energy in a time of 2 ms. • Circuitry allows the capacitor to be charged (to a much higher voltage than the battery) over several seconds. • Similar technology to camera flashes

39. 26.5 Capacitors and Dielectrics • A dielectric is a non-conducting material (rubber/glass/waxed paper) that can be placed between the plates of a capacitor to increase its capacitance. • If the space is entirely filled with the dielectric material, C will increase by a dimensionless factor κ, the dielectric constant of the material.

40. 26.5 • The new capacitance voltage and charge will be given by (for constant charges) (for constant voltage) • Or specifically for a parallel plate capacitor

41. 26.5

42. 26.5 • Again we see that capacitance still increases with decreasing d. • In practice though, there is a lower limit to d before discharge across the plates will occur for a given voltage. • So for any given capacitor of separation d, there is a maximum voltage limit.

43. 26.5 • This limit depends on a factor called the Dielectric Strength. • This is the maximum Electric Field (V/m) that the material can withstand before its insulating properties break down and the material becomes a conductor. • Similar in concept to the spark touching a doorknob and also corona discharge.

44. 26.5 • For parallel plates, the maximum voltage (AKA “working voltage,” “breakdown voltage,” and “rated voltage” is determined by • Where Emax is the dielectric strength

45. 26.5 • Table of Dielectric Constants/Strengths p. 812

46. 26.5 • Types of Commercial Capacitors • Tubular Capacitors- metallic foil interlaced with wax paper/mylar, rolled into a tube.

47. 26.5 • High voltage Capacitors- interwoven metallic plates immersed in an insulating (silicon) oil.

48. 26.5 • Electrolytic Capacitors • Designed for large charges at low voltages. • One conducting foil immersed in an conducting fluid (electrolyte) • The metal forms a thin insulating oxide layer when voltage is applied.

49. 26.5 • Variable Capacitors- interwoven sets of plates with one set fixed and one set able to be rotated. • Typical used for tuning dial circuits (radios, power supplies etc)

50. 26.5 • Quick Quizzes p 813-814 • Examples 26.6, 26.7