1. CHAPTER 17
CAPACITOR
&
DIELECTRICS
(PST :3 hours)
(PDT : 7 hours)
17.1 Capacitors
17.2 Capacitors in series and parallel
17.3 Charging and discharging of capacitors
17.4 Capacitors with dielectrics

2. 17.1 CAPACITORS LEARNING OUTCOMES :
At the end of this lesson, the students should be able to :
Define capacitance.
Use formulae,
Calculate the capacitance of parallel plate capacitor.

3. The electrical symbol for a capacitor is
17.1 Capacitors
A capacitor , sometimes called a condenser, is a device that can store electric charge.
It is consists of two conducting plates separated by a small air gap or a thin insulator (called a dielectric such as mica, ceramics, paper or even oil).
The electrical symbol for a capacitor is or

4. Capacitance, C
17.1 Capacitors
The ability of a capacitor to store charge is measured by its capacitance.
Capacitance is defined as the ratio of the charge on either plate to the potential difference between them.

5. The unit of capacitance is the farad (F).
17.1 Capacitors
The unit of capacitance is the farad (F).
1 farad is the capacitance of a capacitor if the charge on either of the plates is 1C when the potential difference across the capacitor is 1V.
i.e.
By rearranging the equation from the definition of capacitance, we get
where the capacitance of a capacitor, C is constant then
(The charges stored, Q is directly proportional to the potential difference, V across the conducting plate.)

6. One farad (1F) is a very large unit.
17.1 Capacitors
One farad (1F) is a very large unit.
Therefore in many applications the most convenient units of capacitance are microfarad and the picofarad where the unit conversion can be shown below :

7. Parallel-plate Capacitors
A parallel–plate capacitor consists of a pair of parallel plates of area A separated by a small distance d.
If a voltage is applied to a capacitor (connected to a battery), it quickly becomes charged.
One plate acquires a negative charge, the other an equal amount of positive charge and the full battery voltage appears across the plates of the capacitor (12 V).

8. 17.1 Capacitors
The capacitance of a parallel-plate capacitor, C is proportional to the area of its plates and inversely proportional to the plate separation.
Parallel-plate capacitor separated by a vacuum or
Parallel-plate capacitor separated by a dielectric material
0 : 8.85 x C2 N-1 m-2

9. Example 17.1
17.1 Capacitors
Calculate the capacitance of a capacitor whose plates are 20 cm x 3.0 cm and are separated by a 1.0-mm air gap.
What is the charge on each plate if the capacitor is connected to a 12-V battery?
What is the electric field between the plates?
Answer :

10. Example 17.2
17.1 Capacitors
An electric field of 2.80 x 105 V m-1 is desired between two parallel plates each of area 21.0 cm2 and separated by 250 cm of air. Find the charge on each plate.
(Given permittivity of free space, 0 = 8.85 x C2 N-1 m-2)
Answer :

11. Exercise 17.1
17.1 Capacitors
The plates of a parallel-plate capacitor are 8.0 mm apart and each has an area of 4.0 cm2. The plates are in vacuum. If the potential difference across the plates is 2.0 kV, determine
a) the capacitance of the capacitor.
b) the amount of charge on each plate.
c) the electric field strength was produced.

12. 17.2 Capacitors in series and parallel
LEARNING OUTCOMES :
At the end of this lesson, the students should be able to :
Deduce and use the effective capacitance of capacitors in series and parallel.
b) Derive and use equation of energy stored in a capacitor.

13. 17.2 Capacitors in series and parallel
17.2 (i) Capacitors connected in series
17.2 Capacitors in series and parallel +Q
- Q
Ceq,V V1 V2 V3 Q1 Q2 Q3
equivalent to
Figure above shows 3 capacitors connected in series to a battery of voltage, V.
When the circuit is completed, the electron from the battery (-Q) flows to one plate of C3 and this plate become negatively charge.

14. 17.2 Capacitors in series and parallel
This negative charge induces a charge +Q on the other plate of C3 because electrons on one plate of C3 are repelled to the plate of C2. Hence this plate is charged –Q, which induces a charge +Q on the other plate of C2.
This in turn produces a charge –Q on one plate of C1 and a charge of +Q on the other plate of capacitor C1.
Hence the charges on all the three capacitors are the same, Q.
The potential difference across capacitor C1,C2 and C3 are

15. 17.2 Capacitors in series and parallel
The total potential difference V is given by
If Ceq is the equivalent capacitance, then
Therefore the equivalent (effective) capacitance Ceq for n capacitors connected in series is given by
capacitors connected in series

16. 17.2 (ii) Capacitors connected in parallel+Q -Q
Ceq,V
equivalent to
Figure above shows 3 capacitors connected in parallel to a battery of voltage V.
When three capacitors are connected in parallel to a battery, the capacitors are all charged until the potential differences across the capacitors are the same.

17. 17.2 Capacitors in series and parallel
If not, the charge will flow from the capacitor of higher potential difference to the other capacitors until they all have the same potential difference, V.
The potential difference across each capacitor is the same as the supply voltage V.
Thus the total potential difference (V) on the equivalent capacitor is
The charge on each capacitor is

18. 17.2 Capacitors in series and parallel
The total charge is
and
Therefore the equivalent (effective) capacitance Ceq for n capacitors connected in parallel is given by
capacitors connected in parallel

19. 17.2 Capacitors in series and parallel
Example 17.3
17.2 Capacitors in series and parallel
50 V
C1 = 1µF
C2 = 2µF
In the circuit shown above, calculate the
charge on each capacitor
b) equivalent capacitance

20. 17.2 Capacitors in series and parallel
Example 17.4
17.2 Capacitors in series and parallel
In the circuit shown below, calculate the
equivalent capacitance
b) charge on each capacitor c) the pd across each capacitor
C1 = 1µF
C2 = 2µF V1 V2
50 V

21. Example 17.5 In the circuit shown below, calculate the
equivalent capacitance
charge on each capacitor
c) the pd across each capacitor
C1 = 6.0µF
C3 = 8.0µF V1
V2 =V3
12 V
C1 = 6.0µF
C23 = 12.0µF V1 V2
12 V

22. 17.2 Capacitors in series and parallel
Example 17.6
17.2 Capacitors in series and parallel
Find the equivalent capacitance between points a and b for the group of capacitors connected as shown in figure below.
Take
C1 = 5.00 F,
C2 = 10.0 F
C3 = 2.00 F.

23. 17.2 Capacitors in series and parallel
Solution 17.6
17.2 Capacitors in series and parallel
C1 = 5.00 F, C2 = 10.0 F and C3 = 2.00 F.
Series a
and
Series b
Series a
Series b
C12
C12
Parallel
parallel
C22

24. 17.2 Capacitors in series and parallel
Solution 17.6
17.2 Capacitors in series and parallel
C1 = 5.00 F, C2 = 10.0 F and C3 = 2.00 F.
• a
Parallel
Parallel
C12 C3
C12 Ca
C22
• b

25. 17.2 Capacitors in series and parallel
Solution 17.6
17.2 Capacitors in series and parallel
• a
Series
series Ca
Ceq
C22
• b

26. 17.2 Capacitors in series and parallel
Example 17.7
17.2 Capacitors in series and parallel
Determine the equivalent capacitance of the configuration shown in figure below. All the capacitors are identical and each has capacitance of 1 F.
1 F

27. 17.2 Capacitors in series and parallel
Solution 17.7
17.2 Capacitors in series and parallel
series
series
1 F Ca Ca
1 F
series
1 F
series
1 F Cb

28. 17.2 Capacitors in series and parallel
Solution 17.7
17.2 Capacitors in series and parallel
parallel
Ceq Cb
1 F
parallel

29. 17.2 Capacitors in series and parallel
Exercise 17.2
17.2 Capacitors in series and parallel C2 a C1 C3 b d
1. In the circuit shown in figure above, C1= 2.00 F, C2 = 4.00 F and C3 = 9.00 F. The applied potential difference between points a and b is Vab = 61.5 V. Calculate
a) the charge on each capacitor.
b) the potential difference across each capacitor.
c) the potential difference between points a and d.

30. 17.2 Capacitors in series and parallel
2. Four capacitors are connected as shown in figure below.
Calculate
a) the equivalent capacitance between points a and b.
b) the charge on each capacitor if Vab=15.0 V.
5.96 F, 89.5 C on 20 F, 63.2 C on 6 F, 26.3 C on 15 F and on 3 F.

31. 17.2 Capacitors in series and parallel
3. A 3.00-µF and a 4.00-µF capacitor are connected in series and this combination is connected in parallel with a 2.00-µF capacitor.
a) What is the net capacitance?
b) If 26.0 V is applied across the whole network, calculate the voltage across each capacitor.
3.71-µF, 26.0 V, 14.9 V, 11.1 V

32. 17.2 Capacitors in series and parallel
Energy stored in a capacitor, U
17.2 Capacitors in series and parallel
A charged capacitor stores electrical energy.
The energy stored in a capacitor will be equal to the work done to charge it.
A capacitor does not become charged instantly. It takes time.
Initially, when the capacitor is uncharged , it requires no work to move the first bit of charge over.
When some charge is on each plate, it requires work to add more charge of the same sign because of the electric repulsion.
The work needed to add a small amount of charge dq, when a potential difference V is across the plates is,

33. 17.2 Capacitors in series and parallel
Since V=q/C at any moment , where C is the capacitance, the work needed to store a total charge Q is
Thus the energy stored in a capacitor is or or

34. 17.2 Capacitors in series and parallel
Example 17.8
17.2 Capacitors in series and parallel
A camera flash unit stores energy in a 150 µF capacitor
at 200 V. How much energy can be stored?

35. 17.2 Capacitors in series and parallel
Example 17.9
17.2 Capacitors in series and parallel
A 2 µF capacitor is charged to 200V using a battery.
Calculate the
charge delivered by the battery
energy supplied by the battery.
energy stored in the capacitor.
Solution 17.9

36. 17.2 Capacitors in series and parallel
Exercise 17.3
17.2 Capacitors in series and parallel
Two capacitors, C1= 3.00 F and C2 = 6.00 F are connected in series and charged with a 4.00 V battery as shown in figure below.
Calculate
a) the total capacitance for the circuit above.
b) the charge on each capacitor.
c) the potential difference across each capacitor.
d) the energy stored in each capacitor.
e) the area of the each plate in capacitor C1 if the distance between two plates is 0.01 mm and the region between plates is vacuum.
2.00 µF
8.00 µC
V1 = 2.67 V, V2 = 1.33 V
U1 = 1.07 x J, U2 = 5.31 x 10-6 J
3.39 m 2

37. 17.3 Charging and discharging of
capacitors (1 hour)
LEARNING OUTCOMES :
At the end of this lesson, the students should be able to :
Define and use time constant, τ = RC.
Sketch and explain the characteristics of Q-t and I-t graph for charging and discharging of a capacitor.
Use formula for discharging and
for charging.

38. Charging a capacitor through a resistor
17.3 Charging and Discharging of Capacitor
Figure below shows a simple circuit for charging a capacitor.
When the switch S is closed, current Io immediately begins to flow through the circuit.
Electrons will flow out from the negative terminal of the battery, through the resistor R and accumulate on the plate B of the capacitor.
Then electrons will flow into the positive terminal of the battery, leaving a positive charge on the plate A.

39. 17.3 Charging and Discharging of Capacitor
As charge accumulates on the capacitor, the potential difference across it increases and the current is reduced until eventually the maximum voltage across the capacitor equals the voltage supplied by the battery, Vo.
At this time, no further current flows (I = 0) through the resistor R and the charge Q on the capacitor thus increases gradually and reaches a maximum value Qo.

40. 17.3 Charging and Discharging of Capacitor
The charge on the capacitor increases exponentially with time
The current through the resistor decreases exponentially with time
Charge on charging capacitor :
Current in resistor :
where

41. Discharging a capacitor through a resistor
Figure below shows a simple circuit for discharging a capacitor.
When a capacitor is already charged to a voltage Vo and it is allowed to discharge through the resistor R as shown in figure below.
When the switch S is closed, electrons from plate B begin to flow through the resistor R and neutralizes positive charges at plate A. C

42. 17.3 Charging and Discharging of Capacitor
Initially, the potential difference (voltage) across the capacitor is maximum, V0 and then a maximum current I0 flows through the resistor R.
When part of the positive charges on plate A is neutralized by the electrons, the voltage across the capacitor is reduced.
The process continues until the current through the resistor is zero.
At this moment, all the charges at plate A is fully neutralized and the voltage across the capacitor becomes zero.

43. 17.3 Charging and Discharging of Capacitor
Charge on discharging capacitor :
Current in resistor :
The current through the resistor decreases exponentially with time.
The charge on the capacitor decreases exponentially with time.
The negative sign indicates that as the capacitor discharges, the current direction opposite its direction when the capacitor was being charged.
For calculation of current in discharging process, ignore the negative sign in the formula.

44. 17.3 Charging and Discharging of Capacitor
Time constant, 
17.3 Charging and Discharging of Capacitor
It is a measure of how quickly the capacitor charges or discharges.
Its formula,
Its unit is second (s).
Charging process
The time constant is defined as the time required for the capacitor to reach 0.63 or 63% of its maximum charge (Qo).
The time constant is defined as the time required for the current to drop to 0.37 or 37% of its initial value(I0).
when t=RC
when t=RC

45. 17.3 Charging and Discharging of Capacitor
Discharging Process
The time constant is defined as the time required for the charge on the capacitor/current in the resistor decrease to 0.37 or 37% of its initial value.
when t=RC
when t=RC

46. 17.3 Charging and Discharging of Capacitor
Example 17.10
17.3 Charging and Discharging of Capacitor
Consider the circuit shown in figure below, where C1= 6.00 F, C2 = 3.00 F and V = 20.0 V.
Capacitor C1 is first charged by the closing of switch S1. Switch S1 is then opened, and the charged capacitor is connected to the uncharged capacitor by the closing of S2. Calculate the initial charge acquired by C1 and the final charge on each capacitor.

47. 17.3 Charging and Discharging of Capacitor
Solution 17.10
17.3 Charging and Discharging of Capacitor
After the switch S1 is closed. The capacitor C1 is fully charged and the
charge has been placed on it is given by S1 V C1 + + + + + - - - - -
After the switch S2 is closed and S1 is opened. The capacitors C1 and
C2 (uncharged) are connected in parallel and the equivalent capacitance is S2 C1 C2
The total charge Q on the circuit is given by + + - - + -

48. 17.3 Charging and Discharging of Capacitor
Solution 17.10
17.3 Charging and Discharging of Capacitor
The charge from capacitor C1 flows to the capacitor C2 until the
potential difference V’ across each capacitor is the same (parallel) and
given by S2 C1 C2 + + + - - -
Therefore the final charge accumulates
- on capacitor C1 :
- on capacitor C2 :

49. 17.3 Charging and Discharging of Capacitor
Example 17.11
17.3 Charging and Discharging of Capacitor
In the RC circuit shown in figure below, the battery has fully charged the capacitor.
Then at t = 0 s the switch S is thrown from position a to b. The battery voltage is 20.0 V and the capacitance C = 1.02 F. The current I is observed to decrease to 0.50 of its initial value in 40 s. Determine
a. the value of R.
b. the time constant, 
b. the value of Q, the charge on the capacitor at t = 0.
c. the value of Q at t = 60 s

50. 17.3 Charging and Discharging of Capacitor
Solution 17.11
17.3 Charging and Discharging of Capacitor

51. At the end of this lesson, the students should be able to :
Capacitors With Dielectrics
LEARNING OUTCOMES :
At the end of this lesson, the students should be able to :
Define dielectric constant.
Describe the effect of dielectric on a parallel plate capacitor.
Use formula

52. 17. 4 Capacitors with Dielectrics
A dielectric is an insulating material. Hence no free electrons are available in it.
When a dielectric (such as rubber, plastics, ceramics, glass or waxed paper) is inserted between the plates of a capacitor, the capacitance increases.
The capacitance increases by a factor  or r which is called the dielectric constant (relative permittivity) of the material.

53. 17.4 Charging With Dielectrics
Two types of dielectric :
i) non-polar dielectric
For an atom of non-polar dielectric, the center of the negative charge of the electrons ‘coincides’ with the center of the positive charge of the nucleus.
* It does not become a permanent dipole. + -
ii) polar dielectric
- Consider the molecule of waters.
- Its two positively charge hydrogen ions are ‘attracted’ to a negatively charged oxygen ion.
- Such an arrangement of ions causes the center of the negative charge to be permanently separated slightly away from the center of the positive charge, thus forming a permanent dipole.

54. 17.4 Charging With Dielectrics
Dielectric constant,  (r) is defined as the ratio between the capacitance of given capacitor with space between plates filled with dielectric, C with the capacitance of same capacitor with plates in a vacuum, C0.

55. 17.4 Charging With Dielectrics
From the definition of the capacitance,
and
Q is constant
where
From the relationship between E and V for uniform electric field,
and
where

56. 17.4 Charging With Dielectrics
Material
Dielectric constant, εr 
Dielectric Strength (106 V m-1)
Air 3
Mylar
3.2 7
Paper
3.7 16
Silicone oil
2.5 15
Water 80 -
Teflon
2.1 60
The dielectric strength is the maximum electric field before dielectric breakdown (charge flow) occurs and the material becomes a conductor.

57. 17.4 Charging With Dielectrics
Example 17.12
17.4 Charging With Dielectrics
A parallel-plate capacitor has plates of area A = 2x10-10 m2 and separation d = 1 cm. The capacitor is charged to a potential difference V0 = 3000 V. Then the battery is disconnected and a dielectric sheet of the same area A is placed between the plates as shown in figure below.
dielectric

58. 17.4 Charging With Dielectrics
Example 17.12
17.4 Charging With Dielectrics
In the presence of the dielectric, the potential difference across the plates is reduced to 1000 V. Determine
a) the initial capacitance of the air-filled capacitor.
b) the charge on each plate before the dielectric is inserted.
the capacitance after the dielectric is in place.
the relative permittivity.
the permittivity of dielectric sheet.
the initial electric field.
the electric field after the dielectric is inserted.
(Given permittivity of free space, 0 = 8.85 x F m-1)

59. 17.4 Charging With Dielectrics
Solution 17.12
17.4 Charging With Dielectrics

60. Dielectric effect on the parallel-plate capacitor
17.4 Charging With Dielectrics
Dielectric effect on the parallel-plate capacitor
In part a, the region between the charged plates is empty. The field lines point from the positive toward the negative plate

61. 17.4 Charging With Dielectrics
In part b, a dielectric has been inserted between the plates. Because of the electric field between the plates, the molecules of the dielectric (whether polar or non-polar) will tend to become oriented as shown in the figure, the negative ends are attracted to the positive plate and the positive ends are attracted to the negative plate. Because of the end-to-end orientation, the left surface of the dielectric become negatively charged, and the right surface become positively charged.

62. 17.4 Charging With Dielectrics
Because of the surface charges on the dielectric, not all the electric field lines generated by the charges on the plates pass through the dielectric.
As figure c shows, some of the field lines end on the negative surface charges and begin again on the positive surface charges.

63. 17.4 Charging With Dielectrics
Thus, the electric field inside the dielectric is less strong than the electric field inside the empty capacitor, assuming the charge on the plates remains constant.
This reduction in the electric field is described by the dielectric constant εr which is the ratio of the field magnitude Eo without the dielectric to the field magnitude E inside the dielectric:

64. Capacitor without dielectric Capacitor with dielectric Relationship
17.4 Charging With Dielectrics
Quantity
Capacitor without dielectric
Capacitor with dielectric
Relationship
Electric field Eo E
E < Eo
Potential difference Vo V
V < Vo
Charge Qo Q
Q = Qo
Capacitance Co C
C > Co