CHAPTER 2: CAPACITORS & DIELECTRICS

CHAPTER 2

CAPACITORS
&

DIELECTRICS

KMM

TOPIC 2

CAPACITOR & DIELECTRICS

2.1 Capacitance and Capacitors in Series and Parallel
2.2 Charging and Discharging of Capacitors
2.3 Capacitors with Dielectrics

2.1 CAPACITANCE AND CAPACITORS IN
SERIES AND PARALLEL

LEARNING OUTCOMES :
At the end of this lesson, the students should be able
to :

a) Define and use Capacitance,

b) Determine the effective capacitance
of capacitors in series and parallel.

c) Apply energy stored in a capacitor,

Capacitors

Capacitor
-- is a device that is capable of storing

charges (called a condenser).
-- come in different shapes & sizes.
-- used in a variety of electric circuits.
-- consists of 2 conductors separated by a

small air gap or insulator.
-- This insulator called a dielectric could be

made of mica, ceramics, paper or oil.

3

-- the conductors could be rigid metal plates

facing each other or a pair of insulated
metal foil rowed into a cylindrical casing.

-- circuit

symbol for
capacitor :

4

Capacitance ( C )

Definition: The capacitance of a capacitor is
defined as the ratio of the magnitude of the
charge on either plate to the potential
difference between them.

where 5
Q : charge on one of the plates
V : potential difference across them

• Unit of capacitance : C V-1 @ Farad ( F )
• SI unit : 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.

• If printed code on capacitor , “103” means 10000 pF or
10 nF)
i.e.

• Meaning: The ability of a capacitor to store
charge is measured by its capacitance.

***Remarks: Capacitance measures the charge on the capacitor for 6
an unit voltages across it.

The capacitance for a capacitor does not
change unless it is designed to be a variable
capacitor.

The charges stored ( Q ) is directly
proportional to the potential difference ( V )
across the conducting plate.

A capacitor with a large capacitance can
hold more charges than one with a
smaller capacitance for the same
potential difference applied across them.

7

CAPACITORS CONNECTED IN SERIES V
+Q - Q
equivalent to
Ceff

• Figure above shows 3 capacitors connected

in series to a battery of voltage, V.

• When the circuit is completed, the electrons
move away from the negative terminal of
the battery (-Q) to one plate of C3 and this
plate become negatively charge.

equivalent to V
+Q - Q

Ceff

• 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 each
capacitor C1, C2, C3, … Cn are V1, V2, V3, …Vn
respectively.

Hence

Q1 = Q2 = Q3 = Q 10

Since total potential difference :

V  V1 V2 V3  ... Vn (1)

If Ceff is the effective capacitance for a
single capacitor that could replace the
series combination & store the same
charge at the same voltage.

Ceff Q V Q
V Ceff

Substituting into (1) :

Q  Q  Q  Q  ...  Q
Ceff C1 C2 C3 Cn

Canceling the common Q’s, we get :

Value of CE is always smaller than the
smallest capacitance in the combination.

11

SUMMARY EQUATION V

+Q - Q

equivalent to Ceff

Capacitors connected in parallel

V+Q -Q

equivalent to Ceff

• 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.

The charges stored by each capacitor C1, C2,
C3, … Cn are Q1 , Q2 , Q3 , …Qn respectively.

The total charge ( Q ) is the sum of the
charges on each capacitor :

Q  Q1  Q2  Q3  ...  Qn (1)

A capacitor with the equivalent capacitance,
Ceff would hold this same total charge when
connected to the battery, so :

Ceff  Q  Q  Ceff V
V

14

Substituting into (1) :

Ceff V  C1V  C2V  C3V  ...  CnV

Canceling the common V’s, we get :

For this case, the Ceff is larger than the
largest individual capacitance.

15

SUMMARY EQUATION V

+Q - Q

equivalent to Ceff

Example 1
What is the total capacitance in a, b and c ?

(a) (b) (c)

17

(a) In parallel , the total capacitance is
given by

Ceff  C1  C2  C3

(b) In series, the total capacitance is given by

1 111
Ceff C1 C2 C3

1  11

Ceff 30 F

Ceff  30 F  2.73F

11 18

(c) The equivalent capacitor for the 2
capacitors connected in parallel :

C12  C1  C2

C12  15 F

The total capacitance for the parallel &
series connections is CE where

1  1 1 1  1  2

Ceff C12 C3 15 F 15 F 15 F

Ceff  15 F  7.5F

2 19

Example 2
Find the potential difference across the
capacitors X, Y and Z .
Find the charges reside on the capacitors X,
Y and Z .

20

The equivalent capacitance for Y & Z
is CYZ where

The equivalent capacitance for X, Y & Z :
1 11  1  1

Ceff CX CYZ 6 F 3 F

Ceff  2 F 21

The total charge for the equivalent
capacitance is

Q  Ceff V

The charge for the capacitor X = 2.4x10-5 C
Potential difference across X ,

22

Y & Z is wired in parallel, so
Thus :

23

Energy stored in a charged capacitor
When the switch is closed, charges begin to
accumulate on the plates.
A small amount of work ( ∆W ) is done in
bringing a small amount of charge ( ∆q ) from
the battery to the capacitor.

since

24

The total work ( W ) required to increase
the accumulated charge from zero to Q
is given by

25

The work done in charging the capacitor
appears as electric potential energy U
stored in the capacitor.

Energy stored
in capacitor

Since

and

26

Example 3

A 4 μF and 6 μF capacitor connected in
series are charged by a 240 V power supply.
Calculate

(a) The charge on each capacitor
(b) The potential difference across each

capacitor.
(c) The total energy stored in each capacitor.

27

Solution

The combined capacitance for 2 capacitors
in series :

1 11 5

Ceff 4 6 12

Ceff  2.4F

28

Q on each capacitor connected in series is the
same & is equal to the Q on the combined
capacitor.

Potential difference across 4μF capacitor :

29

Potential difference across 6μF capacitor :
The total energy stored in 4μF capacitor :
The total energy stored in 6μF capacitor :

30

2.2 Charging and discharging of capacitors

LEARNING OUTCOMES :
At the end of this lesson, the students should be able to :

a) State physical meaning of time constant and use τ = RC.

b) Sketch and explain the characteristics of Q - t and I - t
graph for charging and discharging of a capacitor.

c) Use:
i) for discharging
ii) for charging.

Time Constant,
• Time constant is a product of resistance,

R and capacitance, C.

where = time constant
R = resistance
C = capacitance

• It is a measure of how quickly the
capacitor charges or discharges.

• Its formula,

• Its unit is second (s).

For discharge circuit

If the time constant RC is small, then the
charge will diminish rapidly

If the time constant RC is high, then the
charge will diminish slowly

For Charging circuit

If the time constant RC is small, capacitor
charges rapidly.

If the time constant RC is high, it takes a long
time for capacitor to reach its final charge,
that is capacitor charges slowly.

33

Charging & Discharging of capacitors

Charging a capacitor
A capacitor in series with a resistor, switch &
battery.

V0 A
B

R

34

• If the switch is closed at t = 0, charge begin to
flow setting up a current in the circuit.

• Electrons move away from the negative terminal
of the battery, through the resistor R and
accumulate on the plate B of the capacitor.

• Then electrons from plate A are repelled and
travel to the positive terminal of the battery,

leaving a positive charge on the plate A.

• As the current flows (I decreases), the charge Q

continues accumulate (Q increases) on the plate of

capacitor and the potential difference across it also

increases.

• The current stops (I=0) flowing when the voltage

(maximum) across the capacitor equals the voltage

supplied by the battery, Vo, at which the capacitor
has maximum charge, Qo.
35

By that time, current, I stops flowing. The
maximum charge Q0 is reached.

V0 A
B

R

The value of the maximum charge depends on
the voltage of the battery.

36

Plots of capacitor charge, Q
vs time t :

The charge on the capacitor
increases exponentially with time

Charge on charging
capacitor :

where
RC is called time constant,
of the circuit.

unit : second ( s )

Definition time constant for charging process :
The time constant, τ is defined as the time required for the capacitor to

increase to 63% of its maximum charge (Q0). 37

Plots of current, I vs time t :

The current through the resistor
decreases exponentially with time

Current in resistor :

where
RC is called time constant,
of the circuit.

unit : second ( s )

Definition time constant for charging process : 38
The time constant, τ is defined as the time required for the current across

the resistor to decrease to 37% of its initial value (I0).

Discharging a capacitor through a resistor

V0 A
B

R
When the switch is closed, the capacitor discharges.
Electrons will flow from plate B through the
resistor R to plate A neutralizing the positive
charges on plate A.
When the capacitor is fully discharged, I = 0, V = 0. 39

• 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.

Plots of capacitor charge, Q
vs time t :

The charge on the capacitor
decreases exponentially with time.

Charge on discharging
capacitor :

Definition time constant for dicharging process :
The time constant, τ is defined as the time required for the charge of capacitor

to decrease to 637% of its maximum charge (Q0). 41

Plots of current, I vs time t :

The current through the resistor
decreases exponentially with time.

Current in resistor :

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.

Definition time constant for dicharging process :
The time constant, τ is defined as the time required for the current across the

resistor to decrease to 37% of its initial value (I0). 42

Example 5

An uncharged capacitor and a resistor are
connected in series to a battery. If V = 12.0 V,
C = 5.0 μF & R = 8.0x105 Ω. Find

(a) the time constant
(b) the maximum charge on the capacitor
(c) the maximum current in the circuit
(e) the charge on the capacitor after one time
constant has elapsed.

43

Solution
(a)
(b) The maximum charge on the capacitor :

(c) The maximum current in the circuit :

44

(d) t = τ = 4 s , Q = ?

45

Quick Check On Your Understanding

Referring to Figure below, describe what
happens to the light bulb after the switch is
closed. Assume that the capacitor has a
large capacitance and is initially
uncharged, and assume that the light
illuminates when connected directly across
the battery terminals.

46

47

Answer

The bulb will light up for a while
immediately after the switch is closed.

As the capacitor charges, the bulb gets
progressively dimmer.

When the capacitor is fully charged the

current in the circuit is zero and the bulb
does not glow at all.

If the value of RC is small, this whole

process might occupy a very short time
interval.

48

2.3 Capacitors With Dielectrics

LEARNING OUTCOMES :
At the end of this lesson, the students should be able
to :
a) Define and use dielectric constant,

b) Describe the effects of dielectric on a parallel
plate capacitor.

c) Apply capacitance of air-filled parallel plate
capacitor,

e) Determine capacitance with dielectric,