Download Lecture 14: Capacitors in Series and Parallel

Electricity and Magnetism
Circuits
Capacitors in Series and Parallel
Lana Sheridan
De Anza College
Oct 13, 2015
Last time
• Capacitance
• capacitors of different shapes
Warm Up Question
True or false: A component (fixed) capacitor has a capacitance,
even when storing no charge.
(A) true
(B) false
Warm Up Question
True or false: A component (fixed) capacitor has a capacitance,
even when storing no charge.
(A) true
(B) false
←
Overview
• Parallel plate capacitors
• Circuits and circuit diagrams
• Capacitors in series and parallel
• Energy stored in a capacitor
Capacitance
Q = C (∆V ) ⇒ C =
Q
∆V
C is a property of the geometry of the capacitor.
A particular capacitor will have a particular fixed value of C , just
like a given resistor will have a constant value of resistance R.
For a parallel plate capacitor:
C=
0 A
d
where d is the separation distance of the plates and A is the area
of each plate
Circuits
Circuits consist of electrical components connected by wires.
Some types of components: batteries, resistors, capacitors,
lightbulbs, LEDs, diodes, inductors, transistors, chips, etc.
The wires in circuits can be thought of as channels for an electric
field that distributes charge to (or charge flow through) the
components.
Circuit
substitution problem.
of is
superconducting
magne
Notice
that
this
the
as Equation
26.2
Notice
that
thisexpression
expression is the
samesame
asthis
Equation
26.2, the capa
Capacitor
section.
Th
approximately
ten
times
g
component
symbols
symbol
initially
unchar
Use Equation 30.17
to express the magnetic
field
in the
tromagnets.
Such
superco
In Supercond
studying e
storing energy.
interior of the base solenoid:
26.3circuit
Combinations
resonance
imaging,
or MR
diagram
"
Battery
26.3
Combin
Find the mutual inductance, noting
that
magnetic
Twothe
or more
capacitors
oftenfo
a
organs
without
the
need
elements.
The
symbol
battery
∆V
the equivalent capacitance of c
! coil caused
flux F BH through the
handle’s
by
the
magful radiation.
orThroughout
more
capacito
Capacitor
this Two
section.
this se
wires
between
symbol
initially
theuncharged.
equivalent capac
netic field
the base
coil is BA:
The of
direction
of the
effective flowCapacitor
of positive
and switches
asw
In
studying
electric
circuits,
this
section.
Through
diagram.
Such The
a diagra
"a number circuit
Switch
Wireless
charging
is used
ofinitially
other
“cordless”
ds
Battery in
ure
26.6.
Open
charge
is clockwise.
capacitor
Csymbol
27.6
Electrical
elements.
Theuncharged.
circuit symbolsP
symbol
!
symbol
used by some manufacturers of electricwires
cars
that
avoids
model
for
a dire
cap
In
studying
electric
between
the
circuit
elem
I
andcircuit
switches
as
wellcircuits,
as theSuch
colo
diagram.
In typical
electric
ing apparatus.
Closed
is
at
the
higher
"
Battery
Switch
ure 26.6. The symbol for the ca
Open
elements.
The
circuit
a source
such
as a battery
symbol
model for
a capacitor,
a pair of
!
switch Figure 26.6
Ssymbol
CircuitClosed
symbols
forisdetermine
between
the
atwires
the higher
potential
and ci
is r
Let’s
an express
batteries,
and
switches.
Parallel
Com
and
switches
as
well
b capacitors,
c symbols
Figure
26.6 Circuit
for
transfer. First, consider thea
capacitors,
batteries, and
Parallel
Combination
Notice that
capacitors
areswitches.
in
Switch
ureTwo
26.6.
The symbol
#
Open
capacitors
to a resistor.
(Resistors
are
Notice that capacitors
are in
Rin green,
Two
capacitors
as sho
!V batteries
blue,
are
and
symbol
model
forconnected
a capacitor
blue,
resistor
R batteries are in green, andconnecting
"
wires
also
have
nation of
capacitors.
Figure
26.
nation
of
capac
switches
are
in
red.
The
closed
switches
are
in
red.
The
closed
Closed
is
at
the
higher
poten
a
d whereas
capacitors.
The
left plates
of the
switch can carry current,
some
to
the
resistor.
Unles
capacitors.
The
switch can
whereas the battery by a conducting wire
thecarry
open onecurrent,
cannot.
!
Figure 26.6 Circuit symbols
for ais capacitor
When
is conn
wires
small
the open
one batteries,
cannot.
the compared
battery
bywit
a
C
L
capacitors,
and
switches.
Parallel
Combinat
" inductor
L
delivered
to
the
wires
is
ne
combination
is
an
LC
circu
Q max
Notice that capacitors are in
Two capacitors
following
acurr
pos
blue, batteries are in green,then
and Imagine
closed,
both theconne
Figure 27.11
A circuit consistnation
of capacitors.
switches are in red. The closedcircuit in
Figure
27.11
from
late
between
maximum
p
ing of a resistor
of can
resistance
R
capacitors. The left pl
switch
carry current,
whereas
We
identify
the
entire
circu
S
and a battery
having
a
potential
cuit is zero, no energy is tr
the open one cannot.
32.5 Oscillations
the battery by a condu
this sectio
initially u
In stud
circuit di
"
Battery
elements.
symbol
!
wires bet
and
Batteries cause a potential difference between two parts
of the switc
circuit.
Switch
ure 26.6.
Open
This drives a charge flow.
symbol
model for
Closed
is at the h
Circuits: Batteries
Capacitor
symbol
8
Circuits
HALLIDAY REVISED
The different elements can be combined together in various ways
to make complete circuits: paths for current to flow from one
terminal of a battery or power supply to the other.
l
h
+
B
C
Terminal
C
l
h
+
B–
V
–
S
(a )
S
Terminal
(b)
This circuit is said to be incomplete while the switch is open.
25-4 (a) Battery B, switch S, and plates h and l of capacitor C, connected in a cir-
(b) A schematic diagram with the circuit elements represented by their symbols.
negative
charges moving horizontally
Flow of charge in a circuit
gureConventional
27.4. Rank
the current in these four
current is said to flow from the positive terminal to
the negative terminal.
However, actually it is negatively charged electrons that flow
through metal wires:
#
"
"
#
#
#
−
c
1
←
d current, i ←
Figure from Serway and Jewett, 9th ed.
+
Series and Parallel
Series
Parallel
When components are
connected one after the other
along a single path, they are
connected in series.
When components are
connected side-by-side on
different paths, they are
connected in parallel.
R1
V
V
R2
R1
R2
+
B–
es stored on all the capacitors.
Capacitors in Parallel
V
–q 3 C 3
V
–q 2 C
Capacitors
parallel
all haveitthe
same potential
difference
T 3
(a) PA R
Terminal
in parallel,
weincan
simplify
with
across them.
N PARALLEL AND IN SERIES
Parallel c
their equi
Equivalent circuit:
the same
663
Three capacitors in parallel:
Terminal capacitor that
d with an equivalent
l difference V as the actual
+
B–
+q 3
V
–q 3 C 3
V
+q 2
–q 2 C 2
V
+q 1
V
–q 1 C 1
+
B–
+q
V
–q Ceq
(b)
nsense
word
“par-V,” which is close
(a)
Terminal
e the same V.”) Figure 25-8b shows
Fig. 25-8 (a) Three capa
Parallel capacitors and
We could
all three
capacitors
with one
acitance
Ceq)replace
that has
replaced
thein theincircuit
parallel
to battery B. The
their equivalent
have
equivalent
capacitance.
The
current
and
potential
difference
in the
C2, and C3) of Fig. 25-8a.
tains potential difference V
same V (“par-V”).
rest of the circuit the
is unchanged
by this.
-8b, we first use Eq. 25-1 to find the
nals and thus across each ca
+
+q
equivalent capacitor, with c
+
B–
es stored on all the capacitors.
Capacitors in Parallel
V
–q 3 C 3
V
–q 2 C
Capacitors
parallel
all haveitthe
same potential
difference
T 3
(a) PA R
Terminal
in parallel,
weincan
simplify
with
across them.
N PARALLEL AND IN SERIES
Parallel c
their equi
Equivalent circuit:
the same
663
Three capacitors in parallel:
Terminal capacitor that
d with an equivalent
l difference V as the actual
+
B–
+q 3
V
–q 3 C 3
V
+q 2
–q 2 C 2
V
+q 1
V
–q 1 C 1
+
B–
+q
V
–q Ceq
(b)
nsense
word
“par-V,” which is close
(a)
Terminal
e the same V.”) Figure 25-8b shows
Fig. 25-8 (a) Three capa
Parallel capacitors and
We could
all three
capacitors
with one
acitance
Ceq)replace
that has
replaced
thein theincircuit
parallel
to battery B. The
their equivalent
have
equivalent
capacitance.
The
current
and
potential
difference
in the
C2, and C3) of Fig. 25-8a.
tains potential difference V
same V (“par-V”).
rest of the circuit the
is unchanged
by this.
-8b, we first use Eq. 25-1 to find the
nals and thus across each ca
What would be the capacitance of this equivalent
capacitor?
equivalent
capacitor, with c
+q
+
Capacitors in Parallel
Capacitors in parallel all have the same potential difference
across them.
∆V1 = ∆V2 = ∆V3 = ∆V
The total charge on the three capacitors is the sum of the charge
on each.
qnet = q1 + q2 + q3
where q1 = C1 ∆V .
Capacitance is C = q/(∆V ):
Ceq =
qnet
∆V
Capacitors in Parallel
Equivalent capacitance:
Ceq =
=
qnet
∆V
q2
q3
q1
+
+
∆V
∆V
∆V
= C1 + C2 + C3
Capacitors in Parallel
Equivalent capacitance:
Ceq =
=
qnet
∆V
q2
q3
q1
+
+
∆V
∆V
∆V
= C1 + C2 + C3
So in general, for any number n of capacitors in parallel, the
effective capacitance of them all together is:
Ceq = C1 + C2 + ... + Cn =
n
X
i=1
Ci
+
B–
Capacitors in Series
+q
V2
V
–q C 2
Capacitors in series all store the same charge.
664
CHAPTER 25 CAPACITANCE
Three capacitors in series:
Terminal
+q
V1
+
B–
–q C 1
+q
V
V2
–q C 2
+q
V3
Terminal
(a)
+q
V3
–q C 3
produces charge !
negative charge fr
Terminal
Series c
repelled negative
Equivalent circuit:
(a)
their
equ
charge !q).
That
the the
same
charge from
to
+q
plate
of
capacitor
+
B – V of capacitor 1 help
–q C
battery,eqleaving th
Here are two
(b)
1. When charge is
(a)
Three
capac
it can
move
alo
nected in series
to
battery
B.
Fig. 25-9a. If the
Series capacitors and
maintains 2.
potential
differenc
The battery
di
their equivalent have
–q C 3
Fig. 25-9
Capacitors in Series
Again, we could replace all three capacitors in the circuit with one
equivalent capacitance and we can find the capacitance of this
equivalent capacitor.
The sum of the potential differences across capacitors in series
is V , the battery’s supplied potential difference.
∆V = ∆V1 + ∆V2 + ∆V3
where ∆V1 = q/C1 , etc.
Then,
Ceq =
q
∆V
Capacitors in Series
Equivalent capacitance:
Ceq =
q
∆V
q
∆V1 + ∆V2 + ∆V3
V1 + V2 + V3 −1
=
q
∆V1 ∆V2 ∆V3 −1
=
+
+
q
q
q
1
1 −1
1
=
+
+
C1 C2 C3
=
Capacitors in Series
In general, for any number n of capacitors in series, we can always
relate the effective capacitance of them all together to the
individual capacitances by:
n
X
1
1
1
1
1
=
+
+ ... +
=
Ceq
C1 C2
Cn
Ci
i=1
The equivalent capacitance of capacitors in series is always less
than the smallest capacitance in the series.
Practice
A 5.0 µF capacitor is connected in parallel with a 10 µF capacitor.
What is the equivalent capacitance of this arrangement?
Practice
A 5.0 µF capacitor is connected in parallel with a 10 µF capacitor.
What is the equivalent capacitance of this arrangement?
Ceq = 15 µF
Practice
A 5.0 µF capacitor is connected in parallel with a 10 µF capacitor.
What is the equivalent capacitance of this arrangement?
Ceq = 15 µF
A 5.0 µF capacitor is connected in series with a 10 µF capacitor.
What is the equivalent capacitance of this arrangement?
Practice
A 5.0 µF capacitor is connected in parallel with a 10 µF capacitor.
What is the equivalent capacitance of this arrangement?
Ceq = 15 µF
A 5.0 µF capacitor is connected in series with a 10 µF capacitor.
What is the equivalent capacitance of this arrangement?
Ceq = 3.3 µF
We first reduce the
The equival
circuit to a single
parallel cap
What is the equivalent
capacitance of this arrangement?
capacitor.
is larger.
More Practice
A
A
C1 =
12.0 µ F
V
C 12
17.3
C2 =
5.30 µ F
B
C3 =
4.50 µ F
(a)
V
C3
4.50
(b)
citance
More Practice
When solving this type of problem, take an iterative approach.
Identify sets of capacitors that are in parallel, then series, then
parallel, etc. and at each step replace with the equivalent
capacitance:
citance
and b for the
ure 26.9a. All
4.0
1.0
4.0
4.0
3.0
6.0
a
lly and make
re connected.
allel connec-
2.0
a
b a
b
8.0
8.0
b
8.0
heyMore
can be Practice
connected in series or
apacitance for the combination,
n parallel
waytype
because
When(c) either
solving this
of problem, take an iterative approach.
Identify sets of capacitors that are in parallel, then series, then
parallel, etc. and at each step replace with the equivalent
capacitance:
r the
. All
4.0
4.0
4.0
2.0
3.0
6.0
a
make
cted.
nec-
ains
1.0
2.0
a
b a
b
8.0
8.0
b
b a 6.0 b
a
8.0
4.0
c
d
Figure 26.9 (Example 26.3) To find the equivalent capacitance
We first reduce the
The equival
More Practice circuit to a single
parallel cap
What is the equivalent
capacitance of this arrangement:is larger.
capacitor.
A
A
C1 =
12.0 µ F
V
C 12
17.3
C2 =
5.30 µ F
B
C3 =
4.50 µ F
(a)
V
C3
4.50
(b)
We first reduce the
The equival
More Practice circuit to a single
parallel cap
What is the equivalent
capacitance of this arrangement:is larger.
capacitor.
A
A
C1 =
12.0 µ F
V
B
C3 =
4.50 µ F
Ceq = 3.57 µF.
C 12
17.3
C2 =
5.30 µ F
(a)
V
C3
4.50
(b)
Energy Stored in a Capacitor
A charged capacitor has an electric field between the plates. This
field can be thought of as storing potential energy.
As you might expect, the energy stored is equal to the work done
charging the capacitor. (Energy Conservation!)
26.4 Energy Stored in a Charged Capacitor
Energy
Stored in a Capacitor
787
charged
capacitor has an electric field between the plates.
at occurs,Athere
is a transformation
s closed,field
energycan
is stored
as chemi- of as storing potential energy.
be thought
is transformed during the chemical
is operating in an electric circuit.
al potential energy in the battery is
mightof expect,
ated withAs
theyou
separation
positive the energy stored is equal to the work
charging the capacitor. (Energy Conservation!)
This
done
or, we shall assume a charging proThe work required to move charge
cribed in Section 26.1 but that gives dq through the potential
ed because the energy in the final difference !V across the capacitor
But process.
how much
work
done?
Wapproximately
3 Imagine
app = q ∆V
is given
by , yet the potential
arge-transfer
the is plates
the areacharge
of the shaded
rectangle. on the capacitor plates.
u transfer
the charge changes
mechanically
difference
as more
is placed
s. You grab a small amount of posi!V
causes this positive charge to move
k on the charge as it is transferred
required to transfer a small amount
t once this charge has been transen the plates. Therefore, work must
s potential difference. As more and
the other, the potential difference
ed. The overall process is described
q
uation 8.2 reduces to W 5 DU E ; the
Q
t appears as an increase in electric
me instant during the charging pro-
dq
dq through the potential
gy in the final difference !V across the capacitor
Energy Stored in a Capacitor
.3 Imagine the plates is given approximately by
area of the
shaded
rectangle.
How much workthe
is done?
dWapp
= (∆V
) dq
e mechanically
mount of posi!V
harge to move
is transferred
small amount
as been transore, work must
e. As more and
tial difference
ess is described
q
W 5 DU E ; the
Q
ase in electric
dq
charging
pro→ Need
to integrate!
Figure 26.11 A plot of potential
or is DV 5 q/C.
Energy Stored in a Capacitor
q
C
For a fixed capacitor (plates are not changing configuration or
shape), C is a constant.
∆V =
ZQ
UE = Wapp =
=
q
dq
0 C
1 Q2
2 C
The energy stored in a capacitor with charge Q and capacitance C :
1 Q2
U=
2 C
Energy Stored in a Capacitor
The energy stored in a capacitor with charge Q and capacitance C :
1
U=
2
Q2
C
Since Q = C (∆V ) we can also write this as:
1
U = C (∆V )2
2
And:
1
U = Q ∆V
2
Stored Energy Example
Suppose a capacitor with a capacitance 12 pF is connected to a
9.0 V battery.
What is the energy stored in the capacitor’s electric field once the
capacitor is fully charged?
Stored Energy Example
Suppose a capacitor with a capacitance 12 pF is connected to a
9.0 V battery.
What is the energy stored in the capacitor’s electric field once the
capacitor is fully charged?
UE = 4.9 × 10−10 J
Energy Density
It is sometimes useful to be able to compare the energy stored in
different charged capacitors by their stored energy per unit volume.
We can link energy density to electric field strength.
This will make concrete the assertion that energy is stored in the
field.
Energy Density
It is sometimes useful to be able to compare the energy stored in
different charged capacitors by their stored energy per unit volume.
We can link energy density to electric field strength.
This will make concrete the assertion that energy is stored in the
field.
For a parallel plate capacitor, energy density u is:
uE =
UE
Ad
(Ad is the volume between the capacitor plates.)
Energy Density and Electric Field
uE
=
=
UE
Ad
C (∆V )2
2Ad
Energy Density and Electric Field
uE
=
=
Replace C =
UE
Ad
C (∆V )2
2Ad
0 A
d :
uE
=
=
0 A ∆V 2
d 2Ad
0 ∆V 2
2
d
Energy Density and Electric Field
uE
=
=
Replace C =
UE
Ad
C (∆V )2
2Ad
0 A
d :
uE
=
=
0 A ∆V 2
d 2Ad
0 ∆V 2
2
d
Lastly, remember ∆V = Ed in a parallel plate capacitor, so:
1
uE = 0 E 2
2
Energy Density and Electric Field
Energy density in a capacitor:
1
uE = 0 E 2
2
The derivation of this expression assumed a parallel plate
capacitor. However, it is true more generally. (General proof
requires vector calculus.)
It is also true for varying electric fields, in which case the energy
density varies.
Energy density of an electric field ∝ E 2
Summary
• parallel plate capacitors
• circuits, circuit diagrams
• capacitors in series and parallel
• practice with capacitors in circuits
• energy stored in a capacitor
Homework
Serway & Jewett:
• PREVIOUS: Ch 26, onward from page 799. Problems: 1, 5, 7,
11, 51
• NEW: Ch 26. Problems: 13, 17, 21, 25, 31, 33, 35