1. Ch. 24 Capacitance & Dielectrics
AP Physics C

2. Capacitance Ability of a capacitor to store charge
Ratio of charge stored to potential difference
1 C/V = 1 Farad (F)

3. Capacitor A device that stores electric charge
Used with resistors in timing circuits because it takes time for a capacitor to fill with charge
Used to smooth varying DC supplies by acting as a reservoir of charge
Used in filter circuits because capacitors easily pass AC (changing) signals but they block DC (constant) signals

4. Types of Capacitors
Polarized
Unpolarized

5. Polarized Capacitors
Electrolytic capacitors are polarized and they must be connected the correct way.
Examples:
Schematic Symbol

6. Un-polarized Capacitors
Small value capacitors are un-polarized and may be connected either way round.
Examples:
Schematic Symbol

7. Charging/Discharging

8. Factors Affecting Capacitance:
The capacitance of a capacitor is affected by three factors:
The area of the plates
The distance between the plates
The dielectric constant of the material between the plates

9. Area of the plates:
Larger plates provide greater capacity to store electric charge. Therefore, as the area of the plates increase, capacitance increases.

10. Distance between the plates:
Capacitance is directly proportional to the electrostatic force field between the plates. This field is stronger when the plates are closer together. Therefore, as the distance between the plates decreases, capacitance increases. As the distance between the plates increases, capacitance decreases.

11. The dielectric between the plates:
The ability of the dielectric to support electrostatic forces is directly proportional to the dielectric constant. Therefore, as the dielectric constant increases, capacitance increases.

12. Capacitance:

13. Sample Problem 1
An air-filled parallel-plate capacitor has a plate area of 2.0 cm2 and a plate separation of 1.00 mm.
Find its capacitance.
If a potential difference of 100 V is applied across the plates, what is
The charge on each plate and
The electric field between the plates?
What is its capacitance for a plate separation of 3.00 mm?

14. Sample Problem 2
A spherical conductor 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.
What is the magnitude of the electric field outside the spherical capacitor?

15. Sample Problem 3
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 cylindrical capacitor if its length is l.
What is the magnitude of the electric field outside the cylindrical capacitor?

16. Energy Storage
The work it takes to charge up a capacitor is equal to the electrical energy stored in a capacitor
Alternate forms:

17. Capacitors in Circuits:
Series Circuit-
Same charge
Potential differences add

18. Capacitors in Circuits:
Parallel Circuit-
Charges add
Potential differences are the same

19. Sample Problem 4
Two capacitors, 6 μF and 10 μF, are connected in series across a potential difference of 24 V.
Determine:
Equivalent capacitance,
Charge stored on each capacitor,
Potential difference across each capacitor, and
Energy stored on each capacitor.

20. Sample Problem 5
Two capacitors, 6 μF and 10 μF, are connected in parallel across a potential difference of 24 V.
Determine:
Equivalent capacitance,
Charge stored on each capacitor,
Potential difference across each capacitor, and
Energy stored on each capacitor.

21. Sample Problem 6 Given the following values: C1 = 10 pF C2 = 20 pF
ΔV = 120 V
Find the equivalent capacitance, charge stored on each capacitor, energy stored in each capacitor and the potential difference across each capacitor.

22. Sample Problem 7
A capacitor with a capacitance of 8.0 μF is charged by connecting it to a source of potential difference 120 V. Once the capacitor is charged fully, it is disconnected from the source.
What is the charge stored on the 8.0 μF capacitor?
What is the energy stored in the capacitor?

23. Sample Problem 8
The charged capacitor in the previous problem is connected to an uncharged 4.0 μF capacitor.
Explain how the charge is distributed once the connection is made.
How much charge is stored on each capacitor?
What is the potential difference across each capacitor?
What is the total energy stored on this combination?
Compare this to the energy stored before the connection was made.

24. Conceptual Questions:
A parallel-plate capacitor is fully charged and then disconnected from the source. When the plates are pulled apart, do the following quantities increase, decrease, or stay the same?
Capacitance?
Charge stored?
Energy stored?
Potential difference?
Electric field strength?

25. Conceptual Questions:
A parallel-plate capacitor is fully charged and remains connected from the source. When the plates are pulled apart, do the following quantities increase, decrease, or stay the same?
Capacitance?
Charge stored?
Energy stored?
Potential difference?
Electric field strength?

26. Dielectrics: What are they?
Non-conducting materials
What are the three functions of a dielectric?
Increases capacitance
Increases maximum operating voltage
Mechanical support that prevents the plates from touching; Allows the plates to be even closer together

27. Dielectric Strength:
The Dielectric Strength is the maximum Electric Field that the material can withstand before the material breaks down as an insulator and permits current to flow through the material.

28. Dielectric Strength (V/m)
Material
Dielectric Constant k = e / eo
Dielectric Strength (V/m)
Vacuum 1
Air (1 atm)
1-3 x10 6
Air (100 atm)
1.0548
Bakelite (typical)
4.9
24x106
Glass (Pyrex)
15x106
Polystyrene (typical)
2.6
25x106
Mylar (typical)
3.5
6-13 x10 6
Paper (typical)
14x106
Porcelain (typical) 7
4x106
Teflon (typical) 2
60x106
Mineral Oil (typical)
4.5
12x106
Water (20 oC)
80.4
Conductor

29. Dielectric Constant:
Depends on the permittivity-The permittivity of a substance is a characteristic which describes how it affects any electric field set up in it. A high permittivity tends to reduce any electric field present. We can increase the capacitance of a capacitor by increasing the permittivity of the dielectric material.

30. Voltage remains constant:
A fully-charged parallel-plate capacitor remains connected to a battery while a dielectric is slide between the plates. Do the following quantities increase, decrease, or stay the same?
Capacitance
Charge stored
Electric field strength between the plates
Potential difference
Energy stored in the capacitor

31. Charge remains constant:
A fully-charged parallel-plate capacitor is disconnected from a battery and then a dielectric is slid between the plates. Do the following quantities increase, decrease, or stay the same?
Capacitance
Charge stored
Electric field strength between the plates
Potential difference
Energy stored in the capacitor

32. Sample Problem 9:
A parallel-plate capacitor has plates of dimensions 2.0 cm by 3.0 cm separated by a 1.0 mm thickness of paper. The dielectric constant of paper is 3.7.
Find its capacitance.
What is the maximum charge that can be placed on the capacitor? The dielectric strength of paper is 16 x 106 V/m.
What is the maximum energy that can be stored in the capacitor?

33. Sample Problem 10
Suppose that the capacitance in the absence of a dielectric is 8.50 pF and that the capacitor is charged to a potential difference of 12.0 V. If the battery is disconnected and a slab of polystyrene (dielectric constant of 2.56) is inserted between the plates, what is the Uo – U?

34. Polarization of a Dielectric:
If a material contains polar molecules, they will generally be in random orientations when no electric field is applied. An applied electric field will polarize the material by orienting the dipole moments of polar molecules.

35. Parallel-Plate with Dielectric:
The insertion of a dielectric will cause the effective electric field to decrease and the capacitance to increase.

36. Sample Problem 11
A parallel-plate capacitor with a plate separation of d has a capacitance Co in the absence of a dielectric. What is the capacitance when a slab of dielectric material of dielectric constant K and thickness 1/3d is inserted between the plates?

37. Gauss’s law in dielectrics
The enclosed charge,
Qenclosed = (σ – σinduced)A
Applying Gauss’s law, gives
EA = (σ – σinduced)A/εo

38. Gauss’s law in dielectrics
Using: E = Eo/K
Eo = σ/εo
E = (σ – σinduced)/εo
You get that: (σ – σinduced) = σ/K

39. EA = (σ – σinduced)A/εo
Becomes: EA = σA/Kεo
KEA = σA/εo
Gauss’s law in dielectrics:
∫KE·dA = Qencl-free/ εo

40. Sample Problem 12
In the spherical capacitor in sample problem #2, the volume between the concentric spherical conducting shells is filled with an insulating oil with dielectric constant K. Use Gauss’s law to find the capacitance.

41. Works Cited: http://www.kpsec.freeuk.com/capacit.htm