Transcript Capacitance

Capacitance
Physics
Montwood High School
R. Casao
Capacitors and Dielectrics
• A capacitor is a device that stores electric
energy.
• A capacitor consists of two conductors
separated by air, a vacuum or an insulator.
• Capacitance: depends on the geometry of
the capacitor and on the material, called a
dielectric, that separates the conductors.
• A capacitor consists of two conductors
(known as plates) carrying charges of equal
magnitude but opposite sign.
• A potential difference
DV exists between the
conductors due to the
presence of the
charges.
Parallel Plate Capacitor
• When the switch is
closed, the battery
establishes an electric
field in the wire that
causes electrons to move
from the left plate into the
wire and into the right
plate from the wire.
• As a result, a separation
of charge exists on the
plates, which represents
an increase in electric
potential energy of the
system of the circuit.
• This energy in the system
has been transformed
from chemical energy in
the battery.
Definition of Capacitance
• Experiments show the quantity of electric charge
Q on a capacitor is linearly proportional to the
potential difference between the conductors, that
is Q ~ DV; mathematically: Q = C·DV
• The capacitance C of a capacitor is the ratio of
the magnitude of the charge on either conductor
to the magnitude of the potential difference DV
between them:
Q
C is always
C
positive.
DV
• SI Unit: farad (F), 1F = 1 C/V
• Commonly used unit is the microfarad, μF, where
1 μF = 1 x 10-6 F
Parallel - Plate Capacitors
A parallel-plate capacitor
consists of two parallel
conducting plates, each of
area A, separated by a
distance d. When the capacitor
is charged, the plates carry
equal amounts of charge. One
plate carries positive charge,
and the other carries negative
charge.
The plates are charged by connection to a
battery.
Parallel-Plate Capacitors
d
Two parallel metallic plates
of equal area A separated by
a distance d as shown.
One plate carries a charge Q
and the other carries a
charge –Q.
A
If the plates are large, charges can distribute
themselves over the large area and the
amount of charge that can be stored on a
plate for a given potential difference
increases as the area A is increased.
Parallel-Plate Capacitors
• The capacitance of a parallel plate capacitor is
directly proportional to the area of the plates and
inversely proportional to the distance between the
plates.
• When air or a vacuum is the dielectric material
between the plates:
εo  A
C
d
• When another material is present as the dielectric
material between the plates, the equation includes a
dielectric constant k:
k  εo  A
C
d
Parallel-Plate Capacitors
• εo is the permittivity of the
insulating material (called the
dielectric) between the plates.
ε o  8.854 x 10
12
C
2
N  m2
• For parallel plate capacitors, the
electric field between the plates is:
V
E
d
•V is the voltage across the
plates of the capacitor.
•d is the distance between the
capacitor plates.
Capacitors in Parallel
• When a potential difference V is applied
across several capacitors connected in
parallel, that potential difference V is
applied across each capacitor.
• The total charge q stored on the
capacitors is the sum of the charges
stored on all the capacitors.
• Capacitors connected in parallel can be
replaced with an equivalent capacitor that
has the same total charge q and the same
potential difference V as the actual
capacitors.
• Equivalent capacitor is a single capacitor
that has the same capacitance as a
combination of capacitors.
Ceq = C1 + C2 + C3 + …
Parallel Combination
The individual potential differences across capacitors
connected in parallel are all the same and are equal
to the potential difference of the voltage source.
Parallel Combination
When the uncharged capacitors are first connected, electrons
transfer between the wires and the plates leaving the left plates
positively charged and right plates negatively charged.
The energy source for this charge transfer is the internal
chemical energy stored in the battery.
Flow of charges onto the plates stops when the voltage across
the capacitors is equal to that across the battery terminals.
Capacitors reach their maximum charge when the flow of
charges stops.
Parallel Combination
The total charge Q stored by the two capacitors: Q = Q1 + Q2.
The voltage across each capacitor is the same:
Q1 = C1·DV,
Q2 = C2·DV
Total charge can also be found using: Q = CeqDV
Equivalent capacitance: Ceq ·DV = C1·DV + C2·DV
DV cancels and Ceq = C1 + C2
In general: Ceq = C1 + C2+ C3+ …
Capacitors in Parallel
C3
Q3
Q2
a

Qtotal  Q1  Q2  Q3
V  V1  V2  V3
C2
C1
Q1
Q total  Ceq  V
b
Q1  C1  V
Q 2  C2  V
Q3  C3  V
Ceq  V  C1  V  C2  V  C3  V
V  Vab
Ceq  C1  C2  C3
Capacitors in Series
• When a potential difference V is applied
across several capacitors connected in
series, the capacitors have identical charge
q.
• The sum of the potential differences across
all the capacitors is equal to the applied
potential difference V.
• Capacitors that are connected in series
can be replaced with an equivalent
capacitor that has the same charge q and
the same total potential difference V as
the actual series capacitors.
1
1
1
1




Ceq C1 C2 C3
Series Combination
When a battery is connected to the uncharged
capacitors, electrons transfer out of the left plate of C1
and into the right plate of C2.
As this charge accumulates on the right plate of C2, an
equivalent amount of negative charge is forced off the
left plate of C2 and this left plate therefore has an
excess positive charge.
Series Combination
The negative charge leaving the left plate of C2 travels
through the connecting wire and accumulates on the
right plate of C1.
As a result, all right plates end up with a charge –Q
and all the left plates end up with a charge +Q.
Thus the charges on capacitors connected in series
are the same.
Series Combination
Voltage ΔV across the battery terminals is shared
between the two capacitors; ΔV = ΔV1 + ΔV2
Equivalent capacitor: Ceq = Q/ Δ V
For each capacitor, we have Δ V1 = Q/C1 and Δ V2 = Q/C2
Q/Ceq = Q/C1 + Q/C2 ; Q’s cancel
1/Ceq = 1/C1 + 1/C2
In general:
1
1
1
1




Ceq C1 C2 C3
Capacitors in Series
C1
C2
C3
Q1
Q2
Q3
a
Q
Q  Ceq V  V 
Ceq
V  V1  V2  V3
Q

V1 
b
C1
V  Vab
Qtotal  Q1  Q2  Q3
Q
V2 
C2
Q
V3 
C3
Q
Q Q Q
 

Ceq C1 C2 C3
It is easier to use the reciprocal key (x-1 or 1/x) on your
calculator:
Ceq = (C1-1 + C2-1 + C3-1 + …)-1
Example: Equivalent Capacitance
In series use 1/C = 1/C1 + 1/C2
2.50 mF
20.00 mF
6.00 mF
In series use 1/C = 1/C1 + 1/C2
8.50 mF
In parallel use C = C1 + C2
20.00 mF
5.965 mF
Example: Equivalent Capacitance
In parallel use C = C1 + C2
In parallel use C = C1 + C2
In series use 1/C = 1/C1 + 1/C2
In parallel use
1
1 1


In series use
CA C C
Ceq  C 
C
C/2
In series use
1
1 1 1
  
CB C C C
C/3
C C

2 3
Capacitors with Dielectrics
A dielectric is a nonconducting (insulating)
material, such as rubber, glass, or waxed
paper.
When a dielectric is inserted between the
plates of a capacitor, the capacitance
increases.
If the dielectric completely fills the space
between the plates, the capacitance
increases by a dimensionless factor k ,
which is called the dielectric constant.
Dielectric constant is a property of a
material and varies from one material to
another.
A charged capacitor (a) before and (b) after insertion
of a dielectric between the plates. The charge on the
plates remains unchanged, but the potential
difference decreases from ΔVo to ΔV = ΔVo/k, thus
the capacitance increases from Co to k·Co.
Note no battery is involved in this example.
Capacitors with Dielectrics
Capacitance increases by the factor k when a
dielectric completely fills the region between the
plates.
If the dielectric is introduced while the potential
difference is being maintained constant by a
battery, the charge increases to a value
Q = k ·Qo . The additional charge is supplied by
the battery and the capacitance again increases
by the factor k.
For a parallel plate capacitor:
k  εo  A
C
d
• The surface charges on the dielectric
reduce the electric field inside the
dielectric.
• This reduction in the electric field is
described by the dielectric constant k,
which is the ratio of the field magnitude E0
without the dielectric to the field
Eo
magnitude Eind inside the dielectric:
k
E ind
• The dielectric constant is a measure of the
degree of dipole alignment in the material.
Properties of Dielectrics
Redistribution of charge is
called polarization.
C
K
C0
We assume that the
induced charge is directly
proportional to the E-field
in the material.
In dielectrics, induced
charges do not exactly
compensate charges on the
capacitance plates.
Effect of a Dielectric on Capacitance
E dielectric
Vdielectric
Eo

K
Vo

K
Potential difference with a
dielectric is less than the
potential difference across free
space.
Q
Q
C   K
 K  Co
V
Vo
Results in a
higher
capacitance.
Allows more charge to be stored before
breakdown voltage.
Dielectric breakdown — the maximum
potential difference between the capacitor
plates before sparking or discharge.
Dielectric strength — the maximum E
field between the capacitor plates before
the dielectric breaks down and acts as a
conductor between the plates (sparks or
discharges).
E max
Vmax

d
Capacitors with Dielectric Material
What are the advantages of dielectric material in a
capacitor?
• Increase the capacitance
• Increase the maximum operating voltage
• Possible mechanical support between the
plates, which allows the plates to be close
together without touching, thereby decreasing d
and increasing C.
Types of Capacitors
(a) A tubular capacitor, whose plates are separated by paper
and then rolled into a cylinder.
(b) A high-voltage capacitor consisting of many parallel
plates separated by insulating oil.
(c) An electrolytic capacitor.
Energy Stored in an Electric Field
The potential energy of a charged capacitor may be
viewed as being stored in the electric field between its
plates.
The work required to bring the total capacitor charge up
to a final value Q is W = 0.5·C·V2.
Work done in charging the capacitor = electric potential
energy U stored in the capacitor.
This result applies to any capacitors, regardless of its
geometry.
U 
Q2
and U  0.5  Q  V
2 C
Energy Stored in an Electric Field
Electric potential energy stored = amount of work
done to charge the capacitor (i.e. to separate charges
and place them onto the opposite plates).
Capacitor has the ability to hold both charge and
energy.
2
Q
U  0.5  C  V 2 and U 
and U  0.5  Q  V
2 C
Two Things to Remember About ParallelPlate Capacitors
• If the battery remains connected to the
parallel-plate capacitor:
 voltage across the plates remains constant;
 amount of charge on the plates can change.
• If the battery is disconnected from the
parallel-plate capacitor:
 voltage across the plates can change if the
plate separation changes;
 amount of charge on the plates remains
constant.
Example
Four capacitors are connected
as shown.
(a) Find the equivalent
capacitance between points
a and b.
(b) Calculate the charge on each
capacitor if ΔVab = 15.0 V.
1
1
1


Cs 15.0 3.00
Cs  2.50 m F
C p  2.50  6.00  8.50 m F
1
 1
1 
Ceq  

  5.96 m F
 8.50 m F 20.0 m F 
Q  C  DV   5.96 m F   15.0 V   89.5 mC
Q 89.5 m C

 4.47 V
C 20.0 m F
15.0  4.47  10.53 V
DV 
Q  C  DV   6.00 m F   10.53 V   63.2 m C on 6.00 m F
89.5  63.2  26.3 mC
Example
Find the equivalent capacitance
between points a and b for the group
of capacitors connected as shown.
Take C1 = 5.00 μF, C2 = 10.0 μF, and
C3 = 2.00 μF.
1 
 1

Cs  

 5.00 10.0 
1
 3.33 m F
C p1  2  3.33  2.00  8.66 m F
C p 2  2 10.0   20.0 m F
Ceq
1 
 1



20.0
8.66


1
 6.04 m F