1. Ch 26 – Capacitance and Dielectrics The capacitor is the first major circuit component we’ll study…

2. All conductors display some degree of capacitance. Usually, the term “ capacitor ” refers to two separate pieces of metal acting together. Each piece of metal is referred to as a plate. Ch 26.1 – Capacitance

3. Electric field lines generated by a real parallel plate capacitor. Notice, the field is essentially uniform between the plates.

4. Fig 26-4b, p.800 Charging a capacitor:

5. Fig 26-4b, p.800 Charging a capacitor: Chemical potential energy maintains charge separation in the battery  the battery generates an E-field.

6. Fig 26-4b, p.800 Charging a capacitor: Chemical potential energy maintains charge separation in the battery  the battery generates an E-field. The battery’s E-field accelerates charges in the wires. Electrons flow off the orange plate and toward the blue plate.

7. Fig 26-4b, p.800 Charging a capacitor: Chemical potential energy maintains charge separation in the battery  the battery generates an E-field. The battery’s E-field accelerates charges in the wires. Electrons flow off the orange plate and toward the blue plate. Now, there is a charge imbalance across the capacitor’s plates. So… what must exist in between the plates of the capacitor? … an electric field.

8. Fig 26-4b, p.800 Charging a capacitor: Chemical potential energy maintains charge separation in the battery  the battery generates an E-field. The battery’s E-field accelerates charges in the wires. Electrons flow off the orange plate and toward the blue plate. Now, there is a charge imbalance across the capacitor’s plates. So… what must exist in between the plates of the capacitor? … an electric field. Charge continues to flow until the E- field in the capacitor is strong enough to cancel the E-field in the battery.

9. A capacitor stores electrical potential energy by virtue of separating charges. The stored energy is “in” the capacitor’s E-field. Based on geometry and materials, some capacitors are better at storing energy than others. Ch 26.1 – Capacitance

10. Called a capacitor because the device has some “capacity” to store electrical charge, given a particular applied potential difference (voltage). The ability of a capacitor to store charge given a certain applied voltage is called its “ capacitance.” Ch 26.1 – Capacitance

11. Size of the capacitor (A, d) Geometric arrangement –Plates –Cylinders Material between conductors –Air –Paper –Wax Ch 26.1 – Capacitance – factors affecting capacitance

12. The “capacitance,” C, of a capacitor is the ratio of the charge on either conductor to the potential difference between the conductors: Ch 26.1 – Capacitance

13. magnitude of charge on one plate voltage across the capacitor Units: 1 F = 1 C/V Ch 26.1 – Capacitance The “farad”

14. (a)How much charge is on each plate of a 4.00μF capacitor when it is connected to a 12.0-V battery? (b)If this same capacitor is connected to a 1.50-V battery, what charge is stored? EG – Definition of Capacitance

15. - one plate has +Q, the other -Q - for each plate σ = Q/A Ch 26.2 – Parallel Plate Capacitors

16. - one plate has +Q, the other -Q - for each plate σ = Q/A Ch 26.2 – Parallel Plate Capacitors - Gauss’s Law  the E-field just outside one of the plates

17. Uniform E-field

18. Ch 26.2 – Parallel Plate Capacitors Working backwards from the uniform E-field, the magnitude of the voltage between the plates is

19. Ch 26.2 – Parallel Plate Capacitors Working backwards from the uniform E-field, the magnitude of the voltage between the plates is But:   Capacitance of parallel plate capacitor

20. Capacitance of a parallel-plate capacitor is directly proportional to the area of the plates and inversely proportional to the distance between the plates. Think about: Ch 26.2 – Parallel Plate Capacitors

21. A solid, cylindrical conductor of radius a and charge Q is coaxial with a cylindrical shell of negligible thickness, radius b>a, and charge –Q. Find the capacitance of this capacitor if its length is l. EG 26.1 – Cylindrical Capacitor

22. A spherical capacitor consists of a spherical conducting shell of radius b and charge –Q concentric with a smaller conducting sphere of radius a and charge Q. Find the capacitance of this device. EG 26.2 – Spherical Capacitor

23. Capacitors are intentionally used in circuits to alter the rates of change of voltages. Capacitors can be hooked up in two ways: - networked in parallel - networked in series Ch 26.3 – Combinations of Capacitors

24. Connecting wires are conductors in electrostatic equilibrium  E in =0 Left plates at same electric potential as the positive terminal of the battery. Right plates at same electric potential as the negative terminal of the battery. Therefore, all capacitors in a parallel network experience the same potential difference, in this case, ΔV. ΔV1=ΔV2=ΔVΔV1=ΔV2=ΔV ΔVΔV Ch 26.3 – Combinations of Capacitors – parallel network

25. ΔV1=ΔV2=ΔVΔV1=ΔV2=ΔV ΔVΔV The individual voltages across parallel capacitors are equal, and they are equal to the voltage applied across the network.

26. When battery is attached to circuit  capacitors quickly reach maximum charge, Q 1 and Q 2.  total charge stored by the circuit is Q tot = Q 1 + Q 2. So, for a given applied voltage, this network has some “capacity” to store charge. In other words, the network itself can be thought of as a single capacitor, even though it has many components. Ch 26.3 – Combinations of Capacitors – parallel network ΔV1=ΔV2=ΔVΔV1=ΔV2=ΔV ΔVΔV

27. Let’s replace the two-capacitor network with a single equivalent capacitor that has capacitance C eq. Based on the definition of capacitance, C eq = Q tot /ΔV.  Q tot = C eq ΔV But, Q tot = Q 1 + Q 2, so  C eq ΔV = C 1 ΔV 1 + C 2 ΔV 2 In conclusion:  C eq = C 1 + C 2 (parallel network) ΔV1=ΔV2=ΔVΔV1=ΔV2=ΔV ΔVΔV Ch 26.3 – Combinations of Capacitors – parallel network

28. In general,  C eq = C 1 + C 2 +… (parallel network) The equivalent capacitance of a parallel network is: -the algebraic sum of the individual capacitances - greater than any of the individual capacitances composing the network ΔVΔV Q tot C eq Ch 26.3 – Combinations of Capacitors – parallel network

29. Left plate of capacitor 1 is at same potential as positive terminal of the battery. Right plate of capacitor 2 is at same potential as negative terminal of battery. “Middle leg” has no net charge. ΔVΔV Ch 26.3 – Combinations of Capacitors – series network

30. When battery is connected, electrons flow off the left plate of C 1 and onto the right plate of C 2. Electrons accumulate on right plate of C 2, establishing an electric field. E-field forces electrons off the left plate of C 2 and onto right plate of C 1. All right plates end up with –Q, and all left plates end up with +Q. Ch 26.3 – Combinations of Capacitors – series network ΔVΔV

31. In other words, the magnitude of charge on all the plates is equal. Q 1 = Q 2 = Q Additionally, once the circuit reaches electrostatic equilibrium, the voltage across the network must cancel the battery’s voltage. ΔV tot = ΔV 1 + ΔV 2 ΔV1ΔV1 ΔV2ΔV2 { { ΔVΔV Ch 26.3 – Combinations of Capacitors – series network

32. This series network has some ability to store charge given an applied voltage, ie., it has some capacitance In other words, even though the network has multiple components, it can be modeled using a single equivalent capacitor. Lets build the same circuit using a single equivalent capacitor. ΔV1ΔV1 ΔV2ΔV2 { { ΔVΔV Ch 26.3 – Combinations of Capacitors – series network

33. Q 1 = Q 2 = Q (previous result) ΔV tot = ΔV 1 + ΔV 2 (previous result) From the definition of capacitance, ΔV tot = Q/C eq Substituting for Δ V tot, Q/C eq = Q 1 /C 1 + Q 2 /C 2 Cancelling Q, 1/C eq = 1/C 1 + 1/C 2 (series combination) ΔV1ΔV1 ΔV2ΔV2 { { ΔVΔV Ch 26.3 – Combinations of Capacitors – series network

34. Q 1 = Q 2 = Q (previous result) ΔV tot = ΔV 1 + ΔV 2 (previous result) From the definition of capacitance, ΔV tot = Q/C eq Substituting for Δ V tot, Q/C eq = Q 1 /C 1 + Q 2 /C 2 Cancelling Q, 1/C eq = 1/C 1 + 1/C 2 (series combination) Ch 26.3 – Combinations of Capacitors – series network ΔVΔV ΔVΔV { Q C eq

35. Ch 26.3 – Combinations of Capacitors – series network ΔVΔV ΔVΔV { Q C eq In general, 1/C eq = 1/C 1 + 1/C 2 +… (series combination) The inverse of the equivalent capacitance is the algebraic sum of the inverses of the individual capacitances. The equivalent capacitance is always less than any individual capacitances in the network.

36. Find the equivalent capacitance between a and b for the combination of capacitors shown. All capacitances are in microfarads. EG 26.3 – Equivalent Capacitance