1. Lesson 6 Capacitors and Capacitance

2. Class 16 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.

3. Section 1 Capacitance, 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. We “charge” a capacitor by connecting it to a battery.
What is a Capacitor?
We “charge” a capacitor by connecting it to a battery. +

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

7. We “charge” a capacitor by connecting it to a battery.
What is a Capacitor?
We “charge” a capacitor by connecting it to a battery.
When we disconnect the battery, charge remains on the conductors.
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. We find that voltage is proportional to charge.
Charging a Capacitor
We find that voltage is proportional to charge. Q V

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

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

13. Section 2 Parallel 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 3 Electric 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!
A factor of 2 different than a plane…

30. Electric Field of a Capacitor
A factor of 2 different than a plane…

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

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

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

34. Parallel-plate Capacitor Equations

35. Section 4 Dielectrics

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

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

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

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

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

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

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

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

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

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

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

47. Section 5 Energy 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

51. The work done in moving the charge is
Energy in a Capacitor
The work done in moving the charge is

52. The work done in charging the capacitor to its final charge Q is:
Energy in a Capacitor
The work done in charging the capacitor to its final charge Q is:

53. Energy in a Capacitor

54. Energy Density Energy per unit volume in a an electric field.
In a parallel-plate capacitor of volume v=Ad :

55. Energy Density
The density of the energy stored in any electric field, not just a capacitor, is:

56. Class 17
Today, we will:
learn how to combine capacitors in series and parallel
find that circuits RC circuits have charges and currents that depend on exponential functions
learn the meaning of the exponential time constant
find that the exponential time constant for an RC circuit is τ=RC

57. Section 6 Capacitors in DC Circuits

58. Capacitors in Circuits
In DC circuits, capacitors just charge or discharge.
No current flows after a capacitor is fully charged or discharged.

59. Capacitors in Circuits
Describe what happens in this circuit after the switch is closed.
20 μF
12 V
5 Ω
1 Ω
2 Ω

60. Capacitors in Circuits
Initially positive charge on the right plate of the capacitor pushes charge off the left plate. It is as if the capacitor were replaced by a wire.
20 μF
12 V
5 Ω
1 Ω
2 Ω

61. Capacitors in Circuits
When the capacitor starts charging, it behaves like a battery that opposes the flow of current.
20 μF
12 V
5 Ω
1 Ω
2 Ω + ─

62. Capacitors in Circuits
Eventually, the capacitor becomes fully charged. No more current flows. What is the final voltage on the capacitor?
20 μF
12 V
5 Ω
1 Ω
2 Ω + ─

63. Capacitors in Circuits
First, ignore the branch with the capacitor.
Rtotal=3 Ω. I = 4 A. V across the 1 Ω resistor is IR = 4 V.
20 μF
12 V
5 Ω
1 Ω
2 Ω + ─

64. Capacitors in Circuits
V across the 5 Ω resistor is 0. Why?
V across the capacitor is 4 V.
Q on the capacitor CV = 80 μC
20 μF
12 V
5 Ω
1 Ω
2 Ω + ─

65. Capacitors in Circuits
Summary:
In steady state, no current flows through the capacitor. Just find the voltage across the capacitor and you can determine the charge.

66. Section 7 Capacitors in Series and Parallel

67. Adding Capacitors Resistors: Capacitors:
R and 1/C enter the voltage equations in a similar way.
If you replace R with 1/C in series-parallel equations for
resistors, you get the correct result for capacitors!

68. Adding Capacitors
Series:
Parallel:

69. Let’s look at the voltage and charge equations…
As with resistors, the voltages across two capacitors in series add to get the total voltage.
As with resistors, the voltages across two capacitors in parallel are the same.
When we discharge two capacitors in parallel, the total charge that leaves the capacitors is the sum of the charges. (Recall that with resistors the sum of the currents is the total current in parallel.)

70. Why are the charges the same on capacitors in series?
To begin with, there is no charge on either capacitor.

71. Why are the charges the same on capacitors in series?
Before we start charging the two capacitors, the charge within the dashed box is zero.

72. Why are the charges the same on capacitors in series?
A the capacitors charge, the charge within the dashed box remains zero.

73. Why are the charges the same on capacitors in series?+Q –Q
When the left plate of the left capacitor acquires its final charge +Q, the right plate’s charge is –Q.

74. Why are the charges the same on capacitors in series?+Q –Q +Q –Q
The charge within the box must remain zero, so the right capacitor must have the same charge as the left capacitor.

75. Section 8 Charging and Discharging Capacitors and the Time Constant

76. RC Discharging I Charge a capacitor with a battery to a voltage V.
Disconnect the capacitor and attach it to a resistor.
The initial charge is Q=CV.
The charge decays to zero – but what is Q(t)? Q0
Q(t) t

77. RC Discharging I
Look at the voltage around the circuit. We can use Kirchoff’s loop rule: I
Voltage

78. RC Discharging The minus sign comes from: I > 0
Voltage
The minus sign comes from:
I > 0
Q is the charge on the capacitor
The capacitor is discharging so

79. RC Discharging I
Voltage

80. RC Discharging
This is a differential equation, but it is a really easy one to solve.
Usually we’ll just give you the solutions to differential equations.

81. RC Discharging

82. RC Time Constant τ (tau) is called the “RC time constant.” τ = RC.
τ has units of seconds.
When τ is big, capacitors charge and discharge slowly.
If R is large, not much current flows, so τ is big.
If C is large, there is a lot of charge that has to flow, so τ is big.

83. RC Discharging τ=3 s τ=2 s τ=1 s
Discharging capacitors with three different
time constants.
The time constant is the time it takes the
charge to drop to 1/e of its original value.
τ=3 s
τ=2 s
τ=1 s

84. RC Charging A capacitor is initially uncharged. C
Use a battery with voltage V0 to charge the capacitor.
The voltage increases to V0.
The charge increases to Q=CV0. V0 R

85. RC Charging I
We again use Kirchoff’s Loop rule:
Voltage

86. RC Charging I
We again use Kirchoff’s Loop rule:
Voltage

87. RC Charging This differential equation has the solution:
Try plugging the solution into the differential equation and see if it works!

88. RC Charging τ=1 s τ=2 s τ=3 s Charging capacitors with three different
time constants.
The time constant is the time it takes the
charge to rise to 1-1/e of its final value.