### 3.3 Capacitors

```Capacitors
Capacitors
• A capacitor is a device for storing electric charge.
• It can be any device which can store charges.
• Basically, capacitors consists of two metal plates
separated by an insulator. The insulator is called
dielectric. (e.g. polystyrene, oil or air)
• Circuit symbol:
+
Dielectric
_
Examples of Capacitors
• Paper, plastic, ceramic
and mica capacitors
• Electrolytic capacitors
• Air capacitors
Charging a capacitor
Q
R
R
t
Computer simulation 1
Charging a capacitor
I
I decreases
exponentially with t.
R
R
t
Charging and discharging capacitors
• Video
• Computer simulation 2
Charging a Capacitor (2)
• Voltage-charge
characteristics
Vc
or
Q
• Current flow
I
Vc ∝ Q
V C  V 0 (1  e
t
RC
)
t
I  Ioe
t
RC
t
Charging of capacitors
• When a capacitor is connected across a battery, electrons
flow from the negative terminal of the battery to a plate of
the capacitor connected to it. At the same rate, electrons
flow from the other plate of the capacitor to the positive
terminal of the battery. This gives a flow of current as the
capacitor is being charged.
• As charges accumulate on the plates of the capacitor,
electric potential built across the plates. This hinders
further accumulation of charges and makes the charge
up current decreasing. When the potential difference across
the plates equals that of the battery, the current becomes
zero.
Discharging of Capacitors (1)
Q
R
R
t
Computer simulation 1
Discharging of Capacitors (1)
I
R
R
t
Discharging a Capacitor (2)
• Voltage-charge
characteristics
• Current flow
I
VC
or
Q
t
RC
RC
t
Q  Q0e
I  Ioe
t
t
Capacitance (1)
• Consider any isolated pair of conductors with
charge Q
Capacitance is defined as
C 
Q
V
where Q = charge on one conductor
V = potential difference between two conductors
Capacitance of a Capacitor
C 
Q
V
• Note that Q is not the net charge on the capacitor, which is zero.
• Capacitance is a measure of a capacitor's ability to store charge.
• The more charge a capacitor can hold at a given potential
difference, the larger is the capacitance.
• Capacitance is also a measure of the energy storage capability
of a capacitor.
• Unit of capacitance: CV-1 or farad (F).
• Farad is a very large unit. Common units are 1mF = 10-6 F, 1nF
= 10-9 F and 1pF = 10-12 F
Markings of capacitor
• Consider a ‘6.3V 1500mF’
capacitor shown in the following
figure. Note that:
• (1) Maximum voltage across the
capacitor should not exceed 6.3 V,
otherwise (leakage or) breakdown
may occur.
• (2) Capacitance of 1500mF means
the capacitor holds 1500mC of
charge for every 1 V of voltage
across it.
Example 1
• Find the maximum charge stored by the capacitor shown in
the figure above.
• Solution:
Capacitance of an isolated conducting
sphere
Q
+
+
• Capacitance = Q/V
• For an isolated conducting
sphere,
+
+
+
V 
+
+
+
- - - - - - - -
1
4 

Q
a
• ∴ C = Q/V = 4a
Example 2
• Find the capacitance of the earth given that the radius of
the earth is 6 x 106 m.
• Solution
• Note:
• The earth’s capacitance is large compared
with that of other conductors used in
electrostatics. Consequently, when a
charged conductor is ‘earthed’, it loses most
of its charge to the earth or discharged.
Parallel Plate Capacitor
• Suppose two parallel plates of a capacitor
each have a charge numerically equal to Q.
area A
+Q
d
–Q
• As C = Q/V
where Q = EA and V = Ed
Electric field
strength
E 



Q
A
 C = A/d
• C depends on the geometry of the conductors.
• Geometrical properties of capacitor
• Parallel plate capacitor capacitance depends
on area and plate separation. For large C,
we need area A large and separation d small.
C 
A
d
Example 3
• The plates of parallel-plate capacitor in vacuum are 5 mm
apart and 2 m2 in area. A potential difference of 10 kV is
applied across the capacitor. Find
(a) the capacitance
• Solution
Example 3
• The plates of parallel-plate capacitor in vacuum are 5 mm
apart and 2 m2 in area. A potential difference of 10 kV is
applied across the capacitor. Find
(b) the charge on each plate, and
• Solution
Example 3
• The plates of parallel-plate capacitor in vacuum are 5 mm
apart and 2 m2 in area. A potential difference of 10 kV is
applied across the capacitor. Find
(c) the magnitude of the electric field between the plates.
• Solution
Application – variable capacitors
• A variable capacitor is a capacitor
whose capacitance may be
intentionally and repeatedly changed
mechanically or electronically
• Variable capacitors are often used in
circuits to tune a radio (therefore they
are sometimes called tuning
capacitors)
• In mechanically controlled variable
capacitors, the amount of plate
surface area which overlaps can be
changed as shown in the figure below.
simulation
Permittivity of dielectric between
the plates
C 
A
d
• A dielectric is an insulator
under the influence of an
E field. The following
table shows some
dielectrics and their
corresponding relative
permittivity.
• Capacitance can be
increased by replacing the
dielectric with one of
higher permittivity.
Dielectric
Relative
permittivity
Vacuum
1
Air
1.0006
Polythene
2.3
Waxed paper
2.7
Mica
5.4
Glycerin
43
Pure water
80
Strontium
titanate
310
Action of Dielectric (1)
• A molecule can be regarded as a collection of atomic
nuclei, positively charged, and surrounded by a cloud of
negative electrons.
- - + - no field
no net charge
net -ve
charge
- - +- -
net +ve
charge
Field
• When the molecule is in an electric field, the nuclei are
urged in the direction of the field, and the electrons in
the opposite direction.
• The molecule is said to be polarized.
Action of Dielectric (2)
• When a dielectric is in a charged capacitor, charges
appear as shown below.
• These charges are of opposite sign to the charges on
the plates.
• The charges reduce the electric
field strength E between the plates.
• The potential difference between
the plates is also reduced as E = V/d.
• From C = Q/V, it follows that C is
increased.
Capacitors in series and parallel
• Computer simulation 1
• Computer simulation 2
Formation of a Capacitor
• Capacitors are formed all
of the time in everyday
situations:
– when a charged
thunderstorm cloud
induces an opposite
charge in the ground
below,
– when you put your hand
near the monitor screen of
this computer.
Charged Capacitor
• A capacitor is said to be charged when
there are more electrons on one
conductor plate than on the other.
When a capacitor is
charged, energy is
stored in the
dielectric material in
the form of an
electrostatic field.
Functions of Dielectrics
• It solves the mechanical problem of
maintaining two large metal plates at a very
small separation without actual contact.
• Using a dielectric increases the maximum
possible potential difference between the
capacitor plates.
• With the dielectric present, the p.d. for a
given charge Q is reduced by a factor εr and
hence the capacitance of the capacitor is
increased.
Relative permittivity and Dielectric Strength
• The ratio of the capacitance with and without
the dielectric between the plates is called the
relative permittivity. or dielectric constant.
r 
Cd
Cv


o
• The strength of a dielectric
(electric field strength) at
which its insulation breakdown.
Relative permittivity of some dielectrics
Dielectric
Vacuum
Air
Polythene
Relative permittivity
1
1.0006
2.3
Waxed paper
Mica
Glycerin
2.7
5.4
43
Pure water
80
Strontium titanate
310
Capacitance of Metal Plates
+V
• Consider a metal plate A which
has a charge +Q as shown.
• If the plate is isolated, A will
+Q
then have some potential V
relative to earth and its
capacitance C = Q/V.
A
-q +q
• Now suppose that another metal B is brought
near to A.
•Induced charges –q and +q are then obtained
on B. This lowers the potential V to a value V’.
•So C’ = Q/V’ > C.
B
Combination of Capacitor (1)
• In series
Q  Q1  Q 2  Q 3
V  V1  V 2  V 3
1
C

1
C1
1

1

C2
V1 : V 2 : V 3 
1
C3
:
1
C1 C2
:
1
C3
The resultant capacitance is smaller than the smallest
Individual one.
Combination of Capacitors (2)
• In parallel
Q  Q1  Q 2  Q 3
V  V1  V 2  V 3
C  C1  C 2  C 3
Q1 : Q 2 : Q 3  C 1 : C 2 : C 3
The resultant capacitance is greater
Than the greatest individual one.
Measurement of Capacitance using
Reed Switch
• The capacitor is charged at a frequency f to
the p.d V across the supply, and each time
discharged through the microammeter.
V +
-
V
mA
During each time
interval 1/f, a
charge Q = CV is
passed through the
ammeter.
I 
Q
1
f
 fCV
Stray Capacitance
• The increased capacitance due to nearby
objects is called the stray capacitance Cs which
is defined by
• C = Co + Cs
– Where C is the measured capacitance.
• Stray capacitance exists in all circuits to some
extent. While usually to ground, it can occur
between any two points with different potentials.
• Sometimes stray capacitance can be used to
advantage, usually you take it into account but
often it's a monumental pain.
Measurement of Stray Capacitance
• In measuring capacitance of a capacitor,
the stray capacitance can be found as
follows:
C
C 
d
Cs
0
oA
1/d
 Cs
Time Constant ()
•  = CR
• The time constant is used to measure how long
it takes to charge a capacitor through a resistor.
• The time constant may also be defined as the
time taken for the charge to decay to 1/e times
its initial value.
• The greater the value of CR, the more slowly
the charge is stored.
• Half-life
– The half-life is the time taken for the charge in a
capacitor to decay to half of its initial value.
– T1/2 = CR ln 2
Energy Stored in a Charged Capacitor
• The area under
the graph gives
the energy stored
in the capacitor.
Q
E 
1
QV
2

0
V

1
CV
2
2
2
1 Q
2 C
Applications of Capacitors (1)
• The capacitance is varied by
altering the overlap between
a fixed set of metal plates
and a moving set. These are
• Press the key on a computer
keyboard reduce the capacitor
spacing thus increasing the
capacitance which can be
detected electronically.
Applications of Capacitors (2)
• Condenser microphone
– sound pressure changes the
spacing between a thin
metallic membrane and the
stationary back plate. The
plates are charged to a total
charge
– A change in plate spacing will
cause a change in charge Q
and force a current through
resistance R. This current
"images" the sound pressure,
making this a "pressure"
microphone.
Applications of Capacitors (3)
• Electronic flash on a camera
– The battery charges up the
flash’s capacitor over several
seconds, and then the capacitor
dumps the full charge into the
flash tube almost instantly.
– A high voltage pulse is generated
across the flash tube.
– The capacitor discharges
through gas in the the flash tube
and bright light is emitted.
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