ch 17 student_201516
TRANSCRIPT
Topic --- Capacitor & Dielectrics
17.1 Capacitance & Capacitors In Series &
Parallel (1 Hour)
17.2 Charging & Discharging Of Capacitors
(1 Hour)
17.3 Capacitors With Dielectrics (1 Hour)
Topic --- Capacitor & Dielectrics
Ca
pa
cito
r &
Die
lectr
ics
Capacitance
Capacitors
In series
In parallel
Charging & discharging of capacitor
Dielectric
Topic --- Capacitor & Dielectrics
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
(a) Define and use capacitance,
(b) Derive and determine the effective
capacitance of capacitors in series and
parallel
(c) Derive and use energy stored in a capacitor,
V
QC
C
QQVCVU
22
2
1
2
1
2
1
Topic --- Capacitor & Dielectrics
Capacitor (Condenser)
A device that is capable of storing electric charges orelectric potential energy
Consist of two conducting plates separated by a small air gap or a thin insulator(dielectric) such as mica, ceramics or even oil
Uses: Photo-flash, on-off switches, smoothen direct current (d.c) voltages
OR
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
Topic --- Capacitor & Dielectrics
CVQ V
QC
VQ
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
Topic --- Capacitor & Dielectrics
V
+Q1QIC1,V1
+Q2
Q2C2,V
2+Q3Q3
C3,V3
+Q
QCeff,
V
V321
111
CCCQ
V
eff
1
CQ
V
capacitors in series
Where and
11
11
C
Q
C
QV
22
22
C
Q
C
QV
The magnitude of the charge Q on each plate is the same
The potential difference across each capacitor C1, C2 and C3 are V1,V2 and V3 respectively
The total potential difference V is given by
321 QQQQ
33
33
C
Q
C
QV
321 VVVV
321 C
Q
C
Q
C
Q
nC...
CCCC
11111
321eff
Topic --- Capacitor & Dielectrics
V +Q1QI
C1,V1
+Q2
Q2C2,V2
+Q3Q3
C3,V3
+Q
QCeff,
V
V
The potential difference across each capacitoris the same as the supply voltage (V)
321 VVVV
The charges stored by each capacitor C1,C2 and C3 are Q1,Q2 and Q3 respectively
The total charge Q on the effective capacitor is given by
where
;1111 VCVCQ ;2222 VCVCQ VCVCQ 3333
321 QQQQ VCVCVC 321
and321 CCC
V
Q effC
V
Q
nC...CCCC 321eff
capacitors in parallel
Topic --- Capacitor & Dielectrics
Calculate the total capacitance in (a), (b) and (c).
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
Topic --- Capacitor & Dielectrics
Determine the potential difference across and charges on capacitors X, Y and Z.
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
Topic --- Capacitor & Dielectrics
Determine the effective capacitance of the configuration shown in Figure 17.1. All the capacitors are identical and each has a capacitance of 2 F.
Figure 17.1
μF 11
30eff C OR μF 73.2
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
Topic --- Capacitor & Dielectrics
In Figure 17.2, C1= 100 F, C2
= 200 F and C3 = 300 F. The applied potential difference between points A and B is VAB = 8.0 V. Calculate
(a) the charge on each capacitor
(b) the potential difference across each capacitor
(c) the potential difference between points A and D.
A
B
D
1C
2C
3C
Figure 17.2
μC 1200123 QQQ
V 0.4321 VVV
V 0.412AD VV
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
Topic --- Capacitor & Dielectrics
• When a switch in circuit closed, charges begin to accumulate on the plates
• A small amount of work (dW) is done in bringing a small amount of charge (dQ) from the battery to the capacitor
• This is given by
• The total work W required to increase the accumulated charge from zero to Q is given by
dQ
VandVdQdW C
QV
dQC
QdW
QW
dQC
QdW
00
C
QWU
2
2
1
2
2
1CVU
QVU2
1
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
Topic --- Capacitor & Dielectrics
Figure 17.3 shows a combination of three capacitors where C1= 100 F, C2 = 22 F and C3 = 47 F. A 20 Vsupply is connected to the combination.Determine(a) the effective capacitance in the circuit,(b) the charge stored in the capacitor C1,(c) the potential difference across the
capacitor C2,(d) the energy stored in the capacitor C3,(e) the area of the each plate in capacitor C1 if the distance
between two plates is 0.02 m and the region between plates is vacuum.
(Permittivity of free space, 0 = 8.85 1012 C2 N1 m2)
1C
2C 3C
V 02
Figure 17.3
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
Topic --- Capacitor & Dielectrics
14
Consider the circuit shown in Figure 17.4, where C1= 50 F, C2 = 25 F and V = 25.0 V.
Capacitor C1 is first charged by
closing a switch S1. Switch S1 is
then opened, and then the
Charged capacitor is connected
to the uncharged capacitor C2 by
closing a switch S2.
Calculate the initial charge acquired by C1 and the final charge on each capacitor.
1C 2CV
1S 2S
Figure 17.4
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
Topic --- Capacitor & Dielectrics
Given 0 = 8.85 1012 C2 N1 m2
1. Four capacitors are connected as shown in Figure 17.5.
Figure 17.5
Calculate
(a) the equivalent capacitance between points a and b,
(b) the charge on each capacitor if Vab=15.0 V.
(Physics for scientists and engineers,6th
edition,Serway&Jewett, Q21, p.823)
ANS: 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
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
Topic --- Capacitor & Dielectrics
2. Determine the equivalent capacitance between points a and b for the group of capacitors connected as shown in Figure 17.6.
Figure 17.6
Take C1 = 5.00 F, C2 = 10.0 F and C3 = 2.00 F.
(Physics for scientists and engineers,6th
edition,Serway&Jewett, Q27, p.824)
ANS: 6.04 F
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
Topic --- Capacitor & Dielectrics
17
3. An electronic flash unit for a camera contains a capacitor of capacitance 850 F. When the unit is fully charged and ready for operation, the potential difference across the plates is 330 V.
(a) What is the magnitude of the charge on each plate of the fully charged capacitor?
(b) Calculate the energy stored in the “charged-up” flash unit.(Physics,3rd edition, J.S.Walker, Q59, p.692)
ANS: 0.28 C; 46 J4. A parallel-plate capacitor has plates with an area of 405 cm2 and an
air-filled gap between the plates that is 2.25 mm thick. The capacitor is charged by a battery to 575 V and then is disconnected from the battery.
(a) How much energy is stored in the capacitor?(b) The separation between the plates is now increased to 4.50 mm.
How much energy is stored in the capacitor now?(c) How much work is required to increase the separation of the
plates from 2.25 mm to 4.50 mm?(Physics,3rd edition, J.S.Walker, Q60, p.692)
ANS: 2.63 105 J; 5.27 105 J; 2.63 105 J
17.1 CAPACITANCE & CAPACITORS IN SERIES & PARALLEL
Topic --- Capacitor & Dielectrics
17.2 CHARGING & DISCHARGING OF CAPACITORS
18
(a) Define and use time constant, = RC
(b) Sketch and explain the characteristics of Q-t and I-t
graph for charging and discharging of a capacitor
(c) Use
for discharging and
for charging
RC
t
eQQ
0
RC
t
eQQ 10
Topic --- Capacitor & Dielectrics
Charging Discharging
17.2 CHARGING & DISCHARGING OF CAPACITORS
Topic --- Capacitor & Dielectrics
20
Time constant,
• Scalar quantity
• Unit: s
• A measurementof how quickly the capacitor charges or discharges
RC
17.2 CHARGING & DISCHARGING OF CAPACITORS
Topic --- Capacitor & Dielectrics
• The voltage V across the capacitor, increase from zero at t = 0 to maximum values V0 after a very long time
• is defined as the time required for the capacitor to reach (1 e1) = 0.63 or 63% of its maximum voltage
Charging
0 t,time
V
0V
063.0 V
RCτ
RC
t
eVV 10
00 V
C
Q
RC
t
eC
Q
C
QV 10
and
17.2 CHARGING & DISCHARGING OF CAPACITORS
Topic --- Capacitor & Dielectrics
• The charge Q across the capacitor, increase from zero at t = 0 to maximum values Q0 after a very long time
• is defined as the time required for the capacitor to reach (1 e1) = 0.63 or 63% of its maximum charge
Charging
0 t,time
Q
0Q
063.0 Q
RCτ
RC
t
eQQ 10
00 CVQ and
17.2 CHARGING & DISCHARGING OF CAPACITORS
Topic --- Capacitor & Dielectrics
• the current drops exponentially in time constant
• is defined as the time required for the current drops to 1/e = 0.37 or 37% of its initial value(I0)
Q0: maximum chargeV0: maximum (supply) voltage
I0: maximum currentR: resistance of the resistor
C: capacitance of the capacitor
23
Charging
23
0I
0 t,time
I
037.0 I
RCτ
and
RC
t
eII
0
R
VI 0
0
17.2 CHARGING & DISCHARGING OF CAPACITORS
Topic --- Capacitor & Dielectrics
24
0V
0 t,time
V
037.0 V
RCτ
RC
t
eVV
0
• the charge Q, the voltage Vand the current I is seen to decrease exponentially in time constant
• is defined as the time required for the charge on the capacitor (or voltage across it or current in the resistor) decreases to 1/e = 0.37 or 37% of its initial value
Discharging: V-t, Q-t & I-t graph
17.2 CHARGING & DISCHARGING OF CAPACITORS
Discharging
Topic --- Capacitor & Dielectrics
25
Charge on capacitor Current through resistor
0I
0 t,time
I
RCτ
037.0 I
The charge on the capacitor decreases exponentially with time
The current through the resistor decreases exponentially with time
0Q
0 t,time
Q
037.0 Q
RCτ
RC
t
eQQ
0
RC
t
eII
0
17.2 CHARGING & DISCHARGING OF CAPACITORS
Discharging
Topic --- Capacitor & Dielectrics
In the RC circuit shown in Figure 17.15, the battery has fully charged the capacitor. At time t = 0 s, a switch S is thrown from position a to b. The battery voltage V0 is 12.0 V and the capacitance C = 3.00 F. The current I is observed to decrease to 0.45 of its initial value in 60 s.Determine(a) the value of R.(b) the time constant, (c) the value of Q, the charge on the capacitor at t = 0.(d) the value of Q at t = 100 s
C
R
0V
S
b
a
Figure 17.7
Topic --- Capacitor & Dielectrics
17.3 CAPACITORS WITH DIELECTRICS
(a) Calculate capacitance of air-filled parallel plate
capacitor,
(b) Define and use dielectric constant
(c) Describe the effect of dielectric on a parallel plate
capacitor.
(d) Use capacitance with dielectric, 0CC r
0
rε
εε
d
AεC 0
Topic --- Capacitor & Dielectrics
• Consider a two parallel metallic plates capacitor of equal area A are separated by a distance d and the space between plates is vacuum or air as shown in Figure 17.8.
• When the capacitor is charged, its plates have charges of equal magnitudes but opposite signs: + Q and Q then the potential difference Vacross the plates is produced.
dQ
QV
A
positive
terminal
negative
terminal
E
Figure 17.8
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
• Since d << A so that the electric field strength E is uniformbetween the plates.
• The magnitude of the electric field strength within the plates is given by
where, : surface charge density on either side (Unit: C m-2 and a scalar quantity)
• Since Q=CV and V=Ed (Uniform E) then equation (17.1) can be written as
0ε
σE
A
Qσ and
0Aε
QE (17.1)
0Aε
CVE
0Aε
CEdE
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
0 = permittivity of free space = 8.85 x 10-12 C2 N-1 m-2
= permittivity of dielectric materiald = distance between the two plates
A = area of each plate
• From equations (17.2) and (17.3), The capacitance, C of a
parallel-plate capacitor is proportional to the area, A of its plates
and inversely proportional to the plate separation, d.
(17.2)
d
AεC 0
and
(17.3)
d
εAC
Parallel-plate capacitor
separated by a vacuum
Parallel-plate capacitor
separated by a dielectric
material
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
A parallel-plate capacitor has plates of area 280 cm2 are separated by
a distance 0.550 mm. The plates are in vacuum. If a potential
difference of 20.1 V is supplied to the capacitor, determine
a. the capacitance of the capacitor.
b. the amount of charge on each plate.
c. the electric field strength was produced.
d. the surface charge density on each plate.
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
A circular parallel-plate capacitor with radius of 1.2 cm is connected to
a 6.0 V battery. After the capacitor is fully charged, the battery is
disconnected without loss of any of the charge on the plates. If the
separation between plates is 2.5 mm and the medium between plates
is air.
a. Calculate the amount of charge on each plate.
If their separation is increase to 8.0 mm after the battery is
disconnected, determine
b. the amount of charge on each plate.
c. the potential difference between the plates.
d. the capacitance of the capacitor.
(Given permittivity of free space, 0 = 8.85 1012 C2 N1 m2)
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
1. a. A parallel-plate, air-filled capacitor has circular plates separated by 1.80 mm. The charge per unit area on each plate has magnitude of 5.60 pC m2. Calculate the potential difference between the plates of the capacitor.
(University physics,11th edition, Young&Freedman, Q24.4, p.934)
b. An electric field of 2.80 105 V m1 is desired between two parallel plates each of area 21.0 cm2 and separated by 0.250 cm of air. Determine the charge on each plate. (Physics for scientist & engineers ,3rd
edition, Giancoli, Q14, p.628) ANS: 1.14 mV; 5.20 109 C
2. When the potential difference between the plates of a capacitor is increased by 3.25 V, the magnitude of the charge on each plate increases by 13.5 C. What is the capacitance of this capacitor?
(Physics,3rd edition, J.S.Walker, Q86, p.694) ANS: 4.15 F
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
3. A 10.0 F parallel-plate, air-filled capacitor with circular plates is connected to a 12.0 V battery. Calculatea. the charge on each plate.b. the charge on each plate if their separation were twice while the capacitor remained connected to the battery.c. the charge on each plate if the capacitor were connected to the 12.0 V battery after the radius of each plate was twice without changing their separation.
(University physics,11th edition, Young&Freedman, Q24.5, p.934) ANS: 120 C; 60 C; 480 C
4. A capacitor stores 100 pC of charge when it is connected across a potential difference of 20 V. Calculatea. the capacitance of the capacitor,b. the amount of charge to be removed from the capacitor to reduce its potential difference to 15 V.
ANS: 5.0 pF; 25 pC
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
• Consider a parallel-plate capacitor as shown in Figure 17.19
Figure 17.9
Q Qd
0E
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
• Initially the plates are separated by a vacuum and connected to a battery, giving the charge on the plates +Q and –Q
• The battery is now removed and the charge on the plates remains constant
• The electric field between the plates is uniform and has a magnitude of E0
• Meanwhile the separation between plates is d
• When a dielectric is placed in the electric field between the plates, the molecules of the dielectric tend to become orientedwith their positive ends pointing toward the negatively charged plate and their negative ends pointing toward the positively charged plated as shown in Figure 17.10
Q Q
Figure 17.10
0E
E
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
• The result is a buildup of positive charge on one surface of the dielectric and of negative charge on the other as shown in Figure 17.11 Q Q
Figure 17.11
E
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
38
• From Figure 17.11, the number of field lines within the dielectric is reduced thus the applied electric field E0 is partially canceled.
• Because the new electric field strength (E < E0) is less then the potential difference, V across the plates is less as well.
• Since V is smaller while Q remains the same the capacitance,
is increased by the dielectric.
EdV
V
QC
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
• is defined as a ratio between the capacitance of a given capacitor with space between plates filled with dielectric, Cand the capacitance of the same capacitor with plates in a vacuum, C0
Mathematically,
No unit for dielectric constant.• For parallel-plates capacitor:
0
rC
Cε (17.10)
d
εAC
d
AεC 0
0
and
d
Aε
d
εA
ε0
r
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
then the equation (17.10) can be written as
: permittivity of dielectric material• From the definition of the capacitance,
hence the equation (17.10) can be written as
V: potential difference across capacitor with dielectricV0: potential difference across capacitor in vacuum
0
rε
εε 0rεεε OR (17.11)
V
QC
0
0V
QC and where Q is constant
V
Vε 0
r (17.12)
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
• From the relationship between E and V for uniform electric field,thus the equation (17.12) can be written as
E0: electric field strength of the capacitor in vacuumE0: electric field strength of the capacitor with dielectric
• The dielectric constant depends on the insulating material used.
• Table 17.1 shows the value of dielectric constant and the dielectric strength for several materials.
EdV dEV 00 and
E
Eε 0
r (17.13)
Ed
dE
V
Vε 00
r
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
• The dielectric strength is defined as the electric field strength at which dielectric breakdown occurs and the material becomes a conductor.
Material Dielectric constant, r
Dielectric Strength
(106 V m1)
Air 1.00059 3
Mylar 3.2 7
Paper 3.7 16
Silicone oil 2.5 15
Water 80 -
Teflon 2.1 60
Table 17.1
Note:
E
E
V
V
ε
ε
C
Cε 00
00
r
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
A vacuum parallel-plate capacitor has plates of area A = 150 cm2 and
separation d = 2 mm. The capacitor is charged to a potential difference
V0 = 2000 V. Then the battery is disconnected and a dielectric sheet of
the same area A is placed between the plates as shown in Figure
17.12. In the presence of the dielectric, the potential difference across
the plates is reduced to 500 V. Determine
(a) the initial capacitance of the capacitor,
(b) the charge on each plate before the dielectric is inserted,
(c) the capacitance after the dielectric is in place,
(d) the relative permittivity,
(e) the permittivity of dielectric material,
(f) the initial electric field,
(g) the electric field after the dielectric is inserted.
(0 = 8.85 1012 C2 N1 m2)
Figure 17.12
dielectric
d
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
A parallel-plate capacitor has the space between the plates filled with two
slabs of dielectric constants 1 and 2 as shown in Figure 17.13.
Each slab has thickness d/2, where d is the plate separation. Show that
the capacitance of the capacitor is
Figure 17.13
21
2102
d
AC
d/2
d/22
1
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
Given 0 = 8.85 1012 C2 N1 m2
1. What are the maximum and minimum equivalent capacitances that
can be obtained by combinations of three capacitors of 1.5 F, 2.0
F and 3.0 F?
(College Physics,6th edition, Wilson, Buffa & Lou, Q97, p.566)
ANS: 6.5 F; 0.67 F
2. The dielectric of a parallel-plate capacitor is to be constructed from
teflon that completely fills the volume between the plates. The area
of each plate is 0.50 m2.
a. What is the thickness of the teflon if the capacitance is to
be 0.10 F?
b. Calculate the charge on the capacitor if it is connected to a
12 V battery.
(Dielectric constant for teflon is 2.1)
ANS: 92.9 m; 1.2 C
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
3. Explain clearly why the electric field between two parallel plates of
a capacitor decreases when a dielectric is inserted if the capacitor
is not connected to a power supply, but remains the same when it
is connected to a power supply.
(College Physics,6th edition, Wilson, Buffa & Lou, Q79, p.566)
4. An air-filled parallel-plate capacitor has rectangular plates with
dimensions of 6.0 cm 8.0 cm. It is connected to a 12 V battery.
While the battery remains connected, a sheet of 1.5 mm thick
paper is inserted and completely fills the space between the
plates.
a. Explain briefly what is happen to the charge on the plates
of the capacitor while the dielectric was being inserted.
b. Determine the change in the charge storage of the
capacitor because of the dielectric insertion.
(Dielectric constant for paper is 3.7)
ANS: 0.92 nC
17.3 CAPACITORS WITH DIELECTRICS
Topic --- Capacitor & Dielectrics
47
Next Chapter…CHAPTER 18 :
Electric current
&
direct-current circuits