1. Capacitance and Dielectrics
Chapter 26
Capacitance
and
Dielectrics

2. Capacitance
The capacitance, C, is a measure of the amount of electric charge stored (or separated) for a given electric potential
SI unit – Farad (F): 1 F = 1 C / V
A 1 Farad capacitance is very large – µF or pF capacitances are more common
Michael Faraday
1791 – 1867

3. Capacitor
A capacitor is a device used in a variety of electric circuits to store electric charge
Capacitance of a capacitor is the ratio of the magnitude of the charge on either conductor (plate) to the magnitude of the potential difference between the conductors (plates)
This capacitance of a device depends on the geometric arrangement of the conductors

4. Parallel-Plate Capacitor
This capacitor consists of two parallel plates (each of area A) separated by a distance d each carrying equal and opposite charges
Each plate is connected to a terminal of the battery
The battery is a source of potential difference
If the capacitor is initially uncharged, the battery establishes an electric field in the connecting wires

5. Parallel-Plate Capacitor
The field applies a force on electrons in the wire just outside of the plates, causing the electrons to move onto the negative plate until equilibrium is achieved
The plate, the wire and the terminal are all at the same potential, and there is no field present in the wire and the movement of the electrons ceases
At the other plate, electrons are moving away from the plate and leaving it positively charged

6. Parallel-Plate Capacitor
The electric field due to one plate
The total electric field between the plates is given by
The field outside the plates is zero

7. Parallel-Plate Capacitor
In the final configuration, the potential difference across the capacitor plates is the same as that between the terminals of the battery
For a parallel-plate capacitor whose plates are separated by air:

8. Electric Field in a Parallel-Plate Capacitor
The electric field between the plates is uniform near the center and nonuniform near the edges
The field may be taken as constant throughout the region between the plates

9. Spherical Capacitor

10. Chapter 26 Problem 5
An air-filled capacitor consists of two parallel plates, each with an area of 7.60 cm2, separated by a distance of 1.80 mm. A 20.0-V potential difference is applied to these plates. Calculate (a) the electric field between the plates, (b) the surface charge density, (c) the capacitance, and (d) the charge on each plate.

11. Electric Circuits
A circuit is a collection of objects usually containing a source of electrical energy (such as a battery) connected to elements that convert electrical energy to other forms
A circuit diagram – a simplified representation of an actual circuit – is used to show the path of the real circuit
Circuit symbols are used to represent various elements (e.g., lines are used to represent wires, battery’s positive terminal is indicated by a longer line)

12. Capacitors in Parallel
When capacitors are first connected in parallel in the circuit, electrons are transferred from the left plates through the battery to the right plates, leaving the left plates positively charged and the right plates negatively charged
The flow of charges ceases when the voltage across the capacitors equals that of the battery
The capacitors reach their maximum charge when the flow of charge ceases

13. Capacitors in Parallel
The total charge is equal to the sum of the charges on the capacitors: Qtotal = Q1 + Q2
The potential differences across the capacitors is the same and each is equal to the voltage of the battery
A circuit diagram for two
capacitors in parallel

14. Capacitors in Parallel
The capacitors can be replaced with one capacitor with a equivalent capacitance Ceq – the equivalent capacitor must have exactly the same external effect on the circuit as the original capacitors

15. Capacitors in Parallel
For more than two capacitors in parallel:
The equivalent capacitance of a parallel combination of capacitors is greater than any of the individual capacitors

16. Capacitors in Series
When a battery is connected to the circuit, electrons are transferred from the left plate of C1 to the right plate of C2 through the battery
As this negative charge accumulates on the right plate of C2, an equivalent amount of negative charge is removed from the left plate of C2, leaving it with an excess positive charge
All of the right plates gain charges of –Q and all the left plates have charges of +Q

17. Capacitors in Series
An equivalent capacitor can be found that performs the same function as the series combination
The potential differences add up to the battery voltage

18. Capacitors in Series For more than two capacitors in series:
The equivalent capacitance is always less than any individual capacitor in the combination

19. Problem-Solving Strategy
Be careful with the choice of units
Combine capacitors:
When two or more unequal capacitors are connected in series, they carry the same charge, but the potential differences across them are not the same
The capacitances add as reciprocals and the equivalent capacitance is always less than the smallest individual capacitor

20. Problem-Solving Strategy
Be careful with the choice of units
Combine capacitors:
When two or more capacitors are connected in parallel, the potential differences across them are the same
The charge on each capacitor is proportional to its capacitance
The capacitors add directly to give the equivalent capacitance

21. Problem-Solving Strategy
Redraw the circuit and continue
Repeat the process until there is only one single equivalent capacitor
To find the charge on, or the potential difference across, one of the capacitors, start with your final equivalent capacitor and work back through the circuit reductions

22. Equivalent Capacitance

23. Chapter 26 Problem 25
Find the equivalent capacitance between points a and b in the combination of capacitors shown in the figure

24. Energy Stored in a Capacitor
Before the switch is closed, the energy is stored as chemical energy in the battery
When the switch is closed, the energy is transformed from chemical to electric potential energy
The electric potential energy is related to the separation of the positive and negative charges on the plates
A capacitor can be described as a device that stores energy as well as charge

25. Energy Stored in a Capacitor
Assume the capacitor is being charged and, at some point, has a charge q on it
The work needed to transfer a charge from one plate to the other:
The total work required:
The work done in charging the capacitor appears as electric potential energy U:

26. Energy Stored in a Capacitor
This applies to a capacitor of any geometry
The energy stored increases as the charge increases and as the potential difference increases
The energy can be considered to be stored in the electric field
For a parallel-plate capacitor, the energy can be expressed in terms of the field as U = ½ (εoAd)E2

27. Capacitors with Dielectrics
A dielectric is an insulating material (e.g., rubber, plastic, etc.)
When placed between the plates of a capacitor, it increases the capacitance: C = κ Co = κ εo (A/d)
κ - dielectric constant
The capacitance is
multiplied by the factor κ
when the dielectric
completely fills the region
between the plates

28. Capacitors with Dielectrics
Tubular – metallic foil interlaced with thin sheets of paraffin-impregnated paper rolled into a cylinder
Oil filled (for high-V capacitors) – interwoven metallic plates are immersed in silicon oil
Electrolytic (to store large amounts of charge at relatively low voltages) – electrolyte is a solution that conducts electricity via motion of ions in the solution

29. Dielectric Strength
For any given plate separation, there is a maximum electric field that can be produced in the dielectric before it breaks down and begins to conduct
This maximum electric field is called the dielectric strength

31. Chapter 26 Problem 37
Determine (a) the capacitance and (b) the maximum voltage that can be applied to a Teflon®-filled parallel-plate capacitor having a plate area of 175 cm2 and an insulation thickness of mm.

32. Electric Dipole
An electric dipole consists of two charges of equal magnitude and opposite signs separated by 2a
The electric dipole moment p is directed along the line joining the charges from –q to +q and has a magnitude of p ≡ 2aq
Assume the dipole is placed in a uniform field, external to the dipole (it is not the field produced by the dipole) and makes an angle θ with the field
Each charge has a force of F = Eq acting on it

33. Electric Dipole The net force on the dipole is zero
The forces produce a net torque on the dipole:
t = 2Fa sin θ = pE sin θ
The torque can also be expressed as the cross product of the moment and the field:
The potential energy can be expressed as

34. An Atomic Description of Dielectrics
Molecules are said to be polarized when a separation exists between the average position of the negative charges and the average position of the positive charges
Polar molecules are those in which this condition is always present
Molecules without a permanent polarization are called nonpolar molecules
The average positions of the positive and negative charges act as point charges, thus polar molecules can be modeled as electric dipoles

35. An Atomic Description of Dielectrics
A linear symmetric molecule has no permanent polarization (a)
Polarization can be induced by placing the molecule in an electric field (b)
Induced polarization is the effect that predominates in most materials used as dielectrics in capacitors

36. An Atomic Description of Dielectrics
The molecules that make up the dielectric are modeled as dipoles; in the absence of an electric field they are randomly oriented
An external electric field produces a torque on the molecules partially aligning them with the electric field

37. An Atomic Description of Dielectrics
The presence of the positive (negative) charge on the dielectric effectively reduces some of the negative (positive) charge on the metal
This allows more charge on the plates for a given applied voltage and the capacitance increases

38. Answers to Even Numbered Problems
Chapter 26:
Problem 12
17.0 μF
9.00V
45.0 μC and 108 μC

39. Answers to Even Numbered Problems
Chapter 26:
Problem 36
13.3 nC
272 nC

40. Answers to Even Numbered Problems
Chapter 26:
Problem 50
(b) 40.0 μF
(c) 6.00 V across 50 μF with charge 300 μF; 4.00 V across 30 μF with charge 120 μF; 2.00 V across 20 μF with charge 40 μF; 2.00 V across 40 μF with charge 80 μF