Capacitance and the Storage of Electric Energy
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Transcript of Capacitance and the Storage of Electric Energy
CAPACITANCE and The Storage of Electric EnergyElectric Energy
PARALLEL PLATEPARALLEL PLATE CAPACITORS
Outline Objectives
1. Capacitors and Capacitance2 Th C bi i f
At the end of this chapter, you should be able to:
2. The Combination of Capacitors3. Energy Storage in
1. Explain how electrostatic energy is stored;2. Define capacitance;3. Energy Storage in
Capacitors4. Capacitors and
3. Define a dielectric and explain how dielectrics affect the energy stored in a capacitor; andDielectrics capacitor; and4. Solve problems involving capacitors.
Chapter TwoChapter TwoChapter TwoChapter Two
The First of ThreeThe First of ThreeThe First of ThreeThe First of ThreeThe Capacitor is one of the three simple circuit elements that can be connected with wires to form an electric circuit!
Capacitors have variety of uses, ranging y g gfrom: radio fine tuner circuits to camera flashes to defibrillatorsflashes to defibrillators
ceceCapacitor ◦ is a system composed
itanc
itanc
y pof two conductors (plates) with equal and opposite charges
apac
apac
pp gon them!
ΔV
of C
of C
ΔV ◦ exists between the
plates because of the f ld b h
tion
tion field between them
Question???:
efin
itef
init Question???:◦ What determines the
amount of charge Q th l t t
Experiment shows that the amount Q is
proportional to ΔV!
1. D
1. D
on the plates at a given V?
ee D fi iti S ta
nce
tanc
e Definition of C Unit Some
Notes
apac
itap
acit
Amount of Charge Stored in a given
PotentialThe Farad Always positive
of C
aof
Ca Potential
Does not actually
ion
oio
n o
C = Q/ΔV 1 F = 1C/1V
ydepend on the
charge or potential, but on the geometry.
efin
itef
init
The Farad is a large unit!
C is Proportionality
Constant
I. D
eI.
De
Initially there are no charges at the plates.
k?k?After making the connection, charging happens.
Wor
kW
ork
Let us focus on the negative plate◦ Charging Stops if the wire plate
tors
Wto
rs W Charging Stops if the wire, plate,
and terminal are all at the same potential!
paci
tpa
cit The positive plate also
experiences a similar phenomenon!
w C
aw
Ca
In the final configuration the potential difference between the plates is the same as the
How
How
the plates is the same as the battery!
A h Q l A capacitor stores charge Q at a potential difference ΔV. If the voltage applied by a battery to the capacitor is doubled to 2ΔV, battery to the capacitor is doubled to 2ΔV,
(a) the capacitance falls to half its initial value
2.1
2.1
( ) pand the charge remains the same (b) the capacitance and the charge both fall to half their initial values
oint
2oi
nt 2 to half their initial values
(c) the capacitance and the charge both double
eckp
oec
kpo (d) the capacitance remains the same and the charge doubles.
Che
Che
tors
tors Parallel Plate Capacitors
Th C i f PP i ap
acit
apac
it ◦ The Capacitance of PP is related only to the area A of the plates and the separation
of C
aof
Ca the plates and the separation
distance d between them!
ance
an
ce
paci
tapa
cita
he C
ahe
Ca
1. T
h1.
Th
Many computer keyboard buttons are constructed of capacitors, as shown in the Figure. When a key is pushed down, the soft insulator between the movable plate and the fixed plate is compressed. When the key
2.2
2.2
p yis pressed, the capacitance
(a) increases
oint
2oi
nt 2 (a) increases,
(b) decreases, or (c) changes in a way that we
eckp
oec
kpo ( ) g y
cannot determine because the complicated electric circuit connected to the
Che
Che keyboard button may cause
a change in ΔV.
A parallel-plate capacitor with air between h l h A 2 00 10 4 2 d the plates has an area A = 2.00 x10-4 m2 and
a plate separation d = 1.00 mm.Find its capacitance.
Ans: 1.77 pF
1 1m
ple
mpl
e Ex
amEx
am
ors
ors Cylindrical Capacitors (Coaxial)
acito
acito
Cap
a C
apa
L is length of the conductors, b is outer radius and a is inner radius
ce o
f ce
of
Spherical Capacitors (Concentric)
itanc
itanc
Spherical Capacitors (Concentric)
apac
apac a is inner radius, b is outer radius. If b→ ∞ then we
have an isolated conductor with a new capacitance
1. C
a1.
Ca
ss 1. An isolated spherical capacitor has a capacitance of 1F Calculate the radius of
cito
rci
tor capacitance of 1F. Calculate the radius of
the spherical capacitor.
Cap
acC
apac
2. You bought a 1-m coaxial cable for your TVR If it’s indicated that the outer radius is
her
Che
r C TVR. If it s indicated that the outer radius is
2mm and the capacitance is 2μF, find the inner radius of the coaxial cable.
: Oth
: Oth
inner radius of the coaxial cable.
3 Your laboratory instructor asked you to
mpl
em
ple 3. Your laboratory instructor asked you to create a spherical capacitor with capacitance 12 pF. The instructor gave you
Exam
Exam
p p g ya solid sphere of radius 2.4mm, what should be the diameter of the shell enclosure?
nts
nts As mentioned
li i h em
enem
en earlier in the chapter, capacitors
d i l i
uit
Ele
uit
Ele are used in electric
circuits.
Circ
uC
ircu
In circuit analysis, we
rs a
s rs
as study pictorial
representations of
acito
rac
itor
circuits known as circuit diagrams
Cap
aC
apa
2 Combinations of Capacitors2 Combinations of Capacitors2. Combinations of Capacitors2. Combinations of Capacitors
There are two types It’s series if the ypof circuit element combination:
elements are connected from end-
A. Series
to-end.
B. Parallel It’s parallel if the elements are connected at connected at common ends.
With parallel: the capacitors are at a common potential!
The equivalent capacitance is Ceq
ion
ion
common potential! capacitance is Ceqwhich is the sum of
the individual capacitances
bina
tibi
nati
Com
bC
omb
lel C
lel C
Para
lPa
ral
2.1
P2.
1 P
The individual charges can be found by QN = CNΔV
With series: the same charge Q is stored among all the capacitors!
ononis stored among all the capacitors!
The equivalent capacitance can be found by taking the reciprocal of the
natio
natio
y g psum of the reciprocal capacitances.
ombi
nom
bin
es C
oes
Co
The individual
Seri
eSe
rie voltage across each
capacitor can be found by
2.2
S2.
2 S
ΔVN = Q/CN
t ed
el!
el!
ival
ent
s st
ore
V = 18 Volts
Para
llePa
ralle
e eq
uiha
rges
ies
ies--
PPfin
d th
d th
e c
or!
: Ser
i: S
eri
rcui
t, f
ce a
ndap
acito
Vab = 15 Volts
mpl
em
ple
each
ci
acita
ncea
ch c
a ab
Exam
Exam In e
capa
in e
0)0)Find the equivalent capacitance. Note that
h i h h i10
/10
10/1
0 each capacitor has the same capacitance.ee
t, 1
eet,
12
she
2 sh
e5 5
(1/2
(1/2
IZ #
5IZ
#5
QU
IQ
UI
gygyEn
erg
Ener
gel
d E
eld
Etic
Fie
tic F
ieos
tat
osta
tec
tro
ectr
o3.
El
3. E
l
gygyEn
erg
Ener
gel
d E
eld
Etic
Fie
tic F
ieos
tat
osta
tec
tro
ectr
o3.
El
3. E
l
gygyFigure to the right
shows the linear En
erg
Ener
g shows the linear relationship between Q and ΔV.
eld
Eel
d E
The Energy U can be computed by taking
tic F
ietic
Fie computed by taking
the area under the curve!
osta
tos
tat
The energy density uE(Energy/Volume)
ectr
oec
tro (Energy/Volume)
◦ This is the energy that is stored in an electric
3. E
l3.
El field, regardless of the
configuration!
You have three capacitors and a battery. In hi h f h f ll i bi i f h which of the following combinations of the
three capacitors will the maximum possible b d h h bi i i energy be stored when the combination is
attached to the battery?
2.3
2.3
(a) series (b) parallel
oint
2oi
nt 2 (c) Both combinations will store the same
amount of energy.
eckp
oec
kpo gy
Ans: (b) for a given voltage capacitances add up
Che
Che
Ans: (b) for a given voltage, capacitances add up when in parallel and U = ½ C(ΔV)2
You charge a parallel-plate capacitor, remove it from the battery, and prevent the wires from the battery, and prevent the wires connected to the plates from touching each other. When you pull the plates apart to a larger separation do the following quantities increase separation, do the following quantities increase, decrease, or stay the same?
(a) C;
2.4
2.4
( ) ;(b) Q; (c) E between the plates;
oint
2oi
nt 2 (d)ΔV
(e) Energy stored (U) in the capacitor.
eckp
oec
kpo
Ans: C decreases, Q stays the same, E remains constant, V increases, U increases because U =
Che
Che
constant, V increases, U increases because U ½ QΔV
ors
ors Dielectrics are
insulators!pa
cito
paci
toinsulators!◦ Some examples of
dielectrics are air, paper, wax rubber and glass
Cap
Cap
wax, rubber, and glass.◦ Characterized by the
dielectric constant κ(>1) which modifies
and
a
nd (>1) which modifies
the permittivity of free space ε0!
ctri
csct
rics “When dielectrics
occupy the space
iele
cie
lec occupy the space
between the plates of a capacitor, the capacitance
4. D
4. D
capacitance increases!”
WARNING!er
y!er
y!◦ Before analyzing and solving for the effect of the dielectric on electrical
properties such as charge and potential… we have to ask…
Batt
eBa
tte
“IS THE CAPACITOR CONNECTED TO A BATTERY?”
f the
f t
he
CASE 1:If it was connected then removed before dielectric was inserted!
ct o
f ct
of If it was connected then removed before dielectric was inserted!
• The charge on the capacitor remain the same
CASE 2:
Effe
c E
ffec CASE 2:
If it remains connected when the dielectric was connected• The voltage across the capacitor remains the same
The
T
he
o a
o a
c in
toc
into
lect
rile
ctri
itor!
itor!
a di
ea
die
capa
cica
paci
rtin
g rt
ing
ged
cge
d c
Inse
rIn
ser
char
gch
arg
When the dielectric is
o a
o a
inserted, the charge remains the same, voltage
drops by:
c in
toc
into
lect
rile
ctri
itor!
itor!
If voltage drops, then capacitance increases
a di
ea
die
capa
cica
paci
pby:
rtin
g rt
ing
ged
cge
d c
Inse
rIn
ser
char
gch
arg
or is
or
is
When the dielectric is inserted, the voltage
capa
cito
capa
cito remains the same, to
accomplish this battery must supply additional
h h
le t
he c
le t
he c
ery
ery
charge so charge increases by:
tric
whi
tric
whi
he b
atte
he b
atte
If charge increases on
die
lect
die
lect
ed t
o th
ed t
o th
gthe plates,
capacitance also increases by:
ing
the
ing
the
onne
cte
onne
cte
Inse
rti
Inse
rti
still
co
still
co
1. A capacitor is to be constructed by making circular parallel plates (radius 1cm) and with separation of parallel plates (radius 1cm) and with separation of 2.5mm with a paper dielectric between the plates (κ= 3.7). Find the capacitance of this capacitor.
2. A 10nF capacitor is charged with a 12V battery. After fully charging the capacitor the battery was After fully charging the capacitor, the battery was removed and a dielectric (κ = 2) was inserted between the plates of the capacitor. Find the f ll i ( ) i i i l d fi l h (b) fi l l
s:s:
following: (a) initial and final charges, (b) final voltage, (c) final capacitance
mpl
em
ple
3. A 4.7nF capacitor is charged with a 9V battery. Then a glass dielectric was inserted between the
Exam
Exam plates of the capacitor. Find the following: (a) initial
and final charges, (b) final voltage, (c) final capacitance
CHAPT
This Chapter is divided into two parts:TER TH
parts:
CIRCUIT ELEMENTSH
REE: D
1.Resistors2. Batteries3. Combination of Resistors
DIRECT C
Resistors
DC CIRCUITS1 AnalysisCU
RREN
1. Analysis2. RC Circuits
NT CIRCCU
ITS
CHAPTER OBJECTIVES
1. Define steady state currents and its relation to ya material’s resistance.2. Relate voltage, current and resistance of
i tresistors.3. Differentiate a real from an ideal battery and monitor the energies of the circuital parametersmonitor the energies of the circuital parameters4. Compute for the equivalent resistance of a network of resistors5. Analyze Direct Current Circuits using Kirchhoff’s Rules.6 Analyze the behavior of RC Circuits6. Analyze the behavior of RC Circuits.
1. Electric Current
CI
2. Resistance and Resistors: Ohm’s LawRCU
IT3. EMF Sources:
Real and Ideal B tt i
T ELEMBatteries
4. Combination of ResistorsM
ENT
5. Energy in Electric CircuitsTS
ELECTRIC CURRENTI h f fl f h Is the rate of flow of charges per unit time
SI Unit: Ampere (A) after Andre Marie Amperep
1 A = 1C/1s
The direction of current is the direction of flow of positive charges
Many types:1. Electron in Hydrogen
Atom2. Electron Beam in Cathode
Ray Tubes in TV’s3. Electricity in Wires
ELECTRIC CURRENT AND THE ELECTRICFIELD
Since the I
direction of current is the direction of Eflow of positive charges: OHM’s LAW:
Current is related to the Electric The direction of the electric field is in the
Current is related to the Electric Field via the conductivity σ of the and cross sectional area A of the material:field is in the
same direction as the electric current!current!
ELECTRICAL RESISTANCE
I
High V Low VWhen a current passes through a material it
E
material it encounters a potential drop!
This potential drop is related to the current via the current via the Ohm’s Relation!
R is the
OhmicNon
Oh iR is the proportionality constant called “RESISTANCE OF
Ohmic
RESISTANCE OF THE MATERIAL”
ELECTRICAL RESISTANCE ANDRESISTIVITY
The Unit for Electrical R i i h Oh ( ) Resistance is the Ohm (Ω) after Georg Ohm
J lik i ρ is the resistivity of the Just like capacitance, resistance is not dependent on V or I, it’s dependent on the geometry and the kind
ρ s t e es st v ty o t e material in (Ω -m), L is the
length (m) and A(m2) is the cross sectional area of the material!the geometry and the kind
of material we have.
Examples:Examples:
A Nichrome wire ( ρ=10-6 Ω-m) has a radius of 0.65mm. What length of wire is needed to obtain a resistance of 2.0 Ω)?
RESISTANCE AND RESISTORS
Resistors are devices Schematic Symbol for Resistors
that provide resistance in a circuit.
Resistors and the resistance they carry resistance they carry have many purposes and applications in a variety circuitsvariety circuits.
(ELECTROMOTIVE FORCE)EMF SOURCES
An emf source is a device th t l t th that elevates the potential of a charge across its terminal. Positive Terminal
The potential gain is the emf of the battery!
It serves as the source or the pump of current in the circuit!
A very good example of emf source is the b !battery! +
Negative Terminal
REAL AND IDEAL BATTERIESTh di i i The distinction between an ideal and real battery is in their yterminal voltages (TV).
Ideal Battery:
Real Battery: (Ideal Battery plus a small y pinternal resistance r)
IDEAL REAL
BATTERIES…
Thus, the TV of
Rate and Energy Stored:
,a real battery is always less than an EMFan EMF.
Malfunctioning
Rate:1 Ah = 3600 C
Malfunctioning batteries have very large internal T l E internal resistances.
Total Energy StoredW = Qξξ
ANALYZING A BASIC CIRCUIT
ENERGY IN ELECTRIC CIRCUITS
The Unit of Power is Watts.
Power delivered by Power delivered by a battery ideal real
Powered dissipated i t
If V and R are givenacross a resistor If I and R are given
g
Example: Example: 1. A 12-Ω resistor carries a current of 3 A. Find the power dissipated in this resistor.
2. A wire of resistance 5 Ω carries a current of 3A for 6s.(a) How much power is put into the wire? (45W)
(b) How much thermal energy is produced? (270 J)
EXAMPLE:An 11-Ω resistor is connected across a battery of yemf 6V and internal resistance 1 Ω.
Find the followingFind the following(a) The current(b) The terminal voltage of the batteryg y(c) The power delivered by the emf source(d) The power delivered to the external resistor
Th di i t d b th b tt ’ i t l (e) The power dissipated by the battery’s internal resistance
(f) If the battery is rated at 150 A•h, how much energy does it store?
COMBINATIONS OF RESISTORS
Resistors are also known as “Loads”
SERIES PARALLEL
RESISTORS IN SERIES oYou can replace R1and R2 with a single 2 gresistor with a resistance Req.
oFor series connection, the
i h Req
current is the same across each capacitor but there is a potential drop across each resistor!
RESISTORS IN PARALLEL You can replace R1 and R2 with a resistor with R2 with a resistor with resistance Req.
For parallel connection the voltage across each resistor is the same R resistor is the same but the current splits along the junctions.
Req
EXAMPLE:
For the circuit that appears below find the following:a) I, I1 and I2b) Rb) Reqc) Voltage drop across each resistor
1. Kirchhoff’s Rules
C
Rules
2 RC IRCUI
2. RC Circuits
IT ANAALYSIIS
KIRCHHOFF’S RULES I2
Junction Rule:I3I1
All currents in and all currents out the junction are equal I4
I4 = I1+I2+I3
Iin = Iout
Loop Rule:
4
Current I Loop directionLoop Rule:In a single loop, all voltage gain is equal to all voltage drop
Current, I Loop direction
DROP GAIN
all voltage dropVgain = VdropIt is important to take note of the loop
Loop Directionnote of the loop direction GAIN DROP
ANALYSIS OF CIRCUITS1 Si l L 2 M l il Ci i1. Single Loop
Find the current in this
2. Multiloop Circuits
Find the currents I I Find the current in this circuit
Find the currents I1, I2, and I3.
MORE KIRCHHOFF’SFind all the currents through junction bg j
RC CIRCUITSC i i d Contains a resistor and a capacitor.I flows in a single direction but its direction but its magnitude varies with time.RC Circuit “charges” and g“discharges”
For charging: we put in g g pthe maximum amount of charge possible in the capacitor over a time constantconstant
For discharging: we drain the charge until it’s value the charge until it s value is negligible!
CHARGING RCW h h We assume that the capacitor is initially uncharged.g
Charge will increase in the capacitor, however, current decreases.
Charge in the capacitor at some time later, will reach its maximum value of Q = Cξ when the current I equals
Qf is the maximum charge that can be stored in a capacitor
the current I equals zero. I0 is the initial current in
the circuit
DISCHARGING RCDischarge happens Discharge happens because when the switch is closed at t = 0, there is a potential drop across
the resistor, meaning there is current in it.
Af i h After some time, the charge on the capacitor is reduced, hence the current is also reduced! current is also reduced! (Why is this happening?)
This happens again and pp gagain, until at some time, the charge and the current are both negligible hence
Qo is the initial charge that is stored in a capacitor
I0 is the initial current in negligible hence “discharged”
I0 is the initial current in the circuit
EXAMPLES:1. An uncharged capacitor and a resistor are connected in series to a battery. If ξ =12 0 V C = 5 00 μF and R = 12.0 V, C 5.00 μF, and R 8.00 x 105 Ω, find the time constant of the circuit, the maximum charge on the
it d th capacitor, and the maximum current in the circuit.
2. Consider a capacitor of capacitance C that is being discharged through a discharged through a resistor of resistance R, as shown in the figure. After how many time constants is th h g th it the charge on the capacitor one-fourth its initial value?