17.1 Capacitors

1
UNIT 17 CAPACITOR DIELECTRICS (3 hours)
17.1 Capacitors 17.2 Capacitors in series and
parallel 17.3 Charging and discharging of
capacitors 17.4 Capacitors with dielectrics
2
SUBTOPIC
17.1 Capacitors (1/2 hour)
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.

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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
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17.1 Capacitors
Capacitance, C

• 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.

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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.)
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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

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17.1 Capacitors
Parallel-plate Capacitors

• A parallelplate 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).

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17.1 Capacitors

• The capacitance of a parallel-plate capacitor,
  C is

The capacitance of a parallel-plate capacitor 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 10-12 C2 N-1 m-2
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17.1 Capacitors
Example 17.1

1. 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.
2. What is the charge on each plate if the capacitor
   is connected to a 12-V battery?
3. What is the electric field between the plates?

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17.1 Capacitors
Example 17.2
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 0.250 cm of air. Find
the charge on each plate. (Given permittivity of
free space, ?0 8.85 x 10-12 C2 N-1 m-2)
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17.1 Capacitors
Exercise
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.
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SUBTOPIC
17.2 Capacitors in series and parallel (1hour)
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.

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17.2 Capacitors in series and parallel
Capacitors connected in series
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.

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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 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

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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
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17.2 Capacitors in series and parallel
Capacitors connected in parallel
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

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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
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17.2 Capacitors in series and parallel
Example 17.3

• In the circuit shown above, calculate the
• charge on each capacitor
• equivalent capacitance

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17.2 Capacitors in series and parallel
Example 17.4

• In the circuit shown below, calculate the
• equivalent capacitance
  
• b) charge on each capacitor
• c) the pd across each capacitor

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17.2 Capacitors in series and parallel
Example 17.5

• In the circuit shown below, calculate the
• equivalent capacitance
• charge on each capacitor
• c) the pd across each capacitor

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17.2 Capacitors in series and parallel
Example 17.6
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.
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17.2 Capacitors in series and parallel
Solution 17.6
C1 5.00 ?F, C2 10.0 ?F and C3 2.00
?F.
Series b
Series a
C12
C12
parallel
C22
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17.2 Capacitors in series and parallel
Solution 17.6
C1 5.00 ?F, C2 10.0 ?F and C3 2.00
?F.
a
Parallel
C3
C12
C12
Ca
C22
b
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17.2 Capacitors in series and parallel
Solution 17.6
a
series
Ca
Ceq
C22
b
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17.2 Capacitors in series and parallel
Example 17.7
Determine the equivalent capacitance of the
configuration shown in figure below. All the
capacitors are identical and each has capacitance
of 1 ?F.
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17.2 Capacitors in series and parallel
Solution 17.7
series
Ca
1 ?F
1 ?F
series
1 ?F
Cb
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17.2 Capacitors in series and parallel
Solution 17.7
parallel
Cb
1 ?F
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17.2 Capacitors in series and parallel
Exercise
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.
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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 Vab15.0 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.
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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
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17.2 Capacitors in series and parallel
Energy stored in a capacitor, U

• 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,

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17.2 Capacitors in series and parallel

• Since Vq/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
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17.2 Capacitors in series and parallel
Example 17.8
A camera flash unit stores energy in a 150 µF
capacitor at 200 V. How much energy can be
stored?
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17.2 Capacitors in series and parallel
Example 17.9

• 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.

36
Exercise
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 10 -5 J, U2 5.31 x 10-6 J
3.39 m 2
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SUBTOPIC
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, t 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.
  

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17.3 Charging and discharging of capacitors
Charging a capacitor through a resistor

• 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.

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17.3 Charging and discharging of capacitors

• 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 capacitors
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
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17.3 Charging and discharging of capacitors
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.

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17.3 Charging and discharging of capacitors

• 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 capacitors
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.
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17.3 Charging and discharging of capacitors
Time constant, ?

• 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 tRC
when tRC
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17.3 Charging and discharging of capacitors
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 tRC
when tRC
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17.3 Charging and discharging of capacitors
Example 17.10
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.
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17.3 Charging and discharging of capacitors
Solution 17.10
48
17.3 Charging and discharging of capacitors
Solution 17.10
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17.3 Charging and discharging of capacitors
Example 17.11
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
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17.3 Charging and discharging of capacitors
Solution 17.11
51
SUBTOPIC
17.4 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 Capacitors 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.
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17.4 Capacitors 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 Capacitors 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
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17.4 Capacitors with dielectrics
Material Dielectric constant, er ? Dielectric Strength (106 V m-1)
Air 1.00059 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 Capacitors with dielectrics
Example17.12
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.
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17.4 Capacitors with dielectrics
Example 17.12

• 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
  10-12 F m-1)

59
17.4 Capacitors with dielectrics
Solution 17.12
60
17.4 Capacitors 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
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.
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17.4 Capacitors 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.

• 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 er which is
  the ratio of the field magnitude Eo without the
  dielectric to the field magnitude E inside the
  dielectric

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17.4 Capacitors with dielectrics
Quantity Capacitor without dielectric Capacitor with dielectric Relationship
Electric field Eo E E lt Eo
Potential difference Vo V V lt Vo
Charge Qo Q Q Qo
Capacitance Co C C gt Co