(Prof. Dr. I. M. A. Nasser)
CHAPTER 26
CAPACITORS and DIELECTRIC
A capacitor is an electronic device, which is used to store electric charge, and
hence electrical energy. A capacitor stores electric charge on its plates. There are
many types of capacitors available; one of them is the parallel plate capacitor. A
parallel-plate capacitor consists of two metal plates separated by an insulator (e.g.
air, waxed paper, etc.). When it is connected to a battery (charging), the two plates
are charged with equal and opposite charges.
Metallic plate
I
+
+
+
+
+
+
+
-
Figure 1 – Air capacitor, ceramic or paper capacitor, and schematic symbol.
Capacitance: is the ability to store charge- and measured in farads (F). “A
capacitance of one farad can store charge of one coulomb when the potential
difference across it is one volt”. It could be written as:
Capacitance (C ) =
Charge stored (Q)
Potential difference (V )
In symbols
C=
Q
V
If charge is measured in coulombs and potential in volts, capacitance is measured
in
Coulombs
≡ Farad [F] . Capacitance of one Farad is enormous. Most small
Volts
electronic components are measured in microfarads (µF), 1 µF = 10-6 F.
(Prof. Dr. I. M. A. Nasser)
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(Prof. Dr. I. M. A. Nasser)
EX.: A capacitor of capacitance 10 µF is charged to a P.D. of 5 volt. How much
charge is stored on each plate?
⇒
Q = CV = 10 × 10-6 × 5 = 50 × 10-6 C = 50 µ C
Ex: find the capacitance of the parallel-plate capacitor, with the separation d, area
A and carries a charge Q, in vacuum.
⇒
C =
Q
Q
Q
Q
A
=
=
=
= εo
V
Ed
d
(σ / ε o ) d
(Q / Aε o ) d
H.W. A parallel-plate capacitor has a capacitance of 2.0 µF. If the plates are 2.0
mm apart, what is the area of the plates? [Ans: 4.4 x 108 m2].
Ex: find the capacitance of the spherical shell capacitor of radius R.
⇒
C =
Q
Q
=
= 4 πε o R
( kQ / R )
V
Note: For one Farad, the radius of the sphere will be R = k = 9x109 m !!!
H.W. Find the capacitance of two concentric spherical conducting shells (of radii
a and b , b b > a ) and they are separated by vacuum. [Ans: C = 4πε o ab ]
b −a
The capacitance can be increased by:
1. Increasing the area of the plates. By increasing the area, more charge is
stored, so the capacitance is increased.
2. Decreasing the separation of the plates
3. Adding an insulator material (a dielectric) between the plates. The electric
field between the plates distorts the atoms of dielectric material, and the
displaced electrons and nuclei lower the potential, so the capacitance
increases.
Laws for Series Circuits:
V ab =V 1 +V 2 +V 3 + "
Q = Q1 = Q 2 = Q 3 = "
1
C eq
=
1
C1
+
1
C2
+
1
C3
C1, V1
a
C2, V2
C3, V3
b
+"
(Prof. Dr. I. M. A. Nasser)
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(Prof. Dr. I. M. A. Nasser)
Laws for Parallel Circuits:
C3, Q3
Q = Q1 + Q 2 + Q 3 + "
C2, Q2
C eq = C 1 + C 2 + C 3 + "
C1, Q1
V ab =V 1 =V 2 =V 3 = "
b
a
Energy Stored in a Capacitor
1Q2 1
1
U =
= QV = CV
2 C
2
2
2
Energy Denisty
u =
U
1
= εo E 2
Volume
2
Capacitor with Dielectric
C = κC o ;
κ = dielectric constant >1
where C o is the capacitance in the absence of the dielectric. Following that
V =V o / κ ; E = E o / κ , U = κU o
(Prof. Dr. I. M. A. Nasser)
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(Prof. Dr. I. M. A. Nasser)
(Prof. Dr. I. M. A. Nasser)
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(Prof. Dr. I. M. A. Nasser)
(Prof. Dr. I. M. A. Nasser)
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(Prof. Dr. I. M. A. Nasser)
EXTRA POBLEMS
C2
V
C1
T-031
C3
17 Q0
Q0
26 Q0
Q0
Q0
A1
A2
A3
A4
A5
Q0
18 Q0
Q0
26 Q0
Q0
Q0
A1
A2
A3
A4
A5
Q0
19 Q0
26 Q0
Q0
Q0
Q0
Q0
A1
A2
A3
A4
A5
Q0
20 Q0
26Q0
Q0
Q0
Q0
Q0
A1
A2
A3
A4
A5
Figure (8)
Find the equivalent capacitance of three capacitors
connected in series. Assume the three capacitors are:
C1 = 2.00 micro-F, C2 = 4.00 micro-F and
C3 = 8.00 micro-F.
1.14
0.88
3.01
26.1
15.4
micro-F.
micro-F.
micro-F.
micro-F.
micro-F.
In figure (8), find the total charge stored by the three
capacitors if the potential difference “V” is 10.0 volts.
Assume C1 = 10.0 micro-F, C2 = 5.00 micro-F and
C3 = 4.00 micro-F.
31.6
22.1
61.3
26.1
63.4
micro-C.
micro-C.
micro-C.
micro-C.
micro-C.
An air filled parallel-plate capacitor has a capacitance of
1.00*10**(-12) F. The plate separation is then doubled and a
wax dielectric is inserted, completely filling the space
between the plates. As a result the, capacitance becomes
2.00*10**(-12) F. The dielectric constant of the wax is:
4.00.
0.25.
2.00.
0.50.
8.00.
Two capacitors, C1 and C2, are connected in series and a
potential difference is applied to the combination. If the
capacitor that is equivalent to the combination has the same
potential difference, then the charge on the equivalent
capacitors is the same as:
The
The
The
The
The
charge on C1 or C2.
sum of the charges on C1 and C2.
difference of the charges on C1 and C2.
product of the charges on C1 and C2.
ratio of the charges on C1 and C2.
(Prof. Dr. I. M. A. Nasser)
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(Prof. Dr. I. M. A. Nasser)
T-012
Q0
Q15Q0
012Q0
26Q0
Q0
Q0
A1
A2
A3
A4
A5
Q0
Q16Q0
012Q0
26Q0
Q0
A1
A2
A3
A4
A5
Q0
Q17Q0
012Q0
26 Q0
Q0
Q0
A1
A2
A3
A4
A5
Q0
A parallel-plate capacitor has a plate area of 0.2 m**2 and
a plate separation of 0.1 mm. If the charge on each plate has
a magnitude of 4.0*10**(-6) C the electric field between the
plates is approximately:
2.3*10**6
4.2*10**6
1.4*10**4
9.2*10**3
Zero.
V/m.
V/m.
V/m.
V/m.
A 2 micro-F and a 1 micro-F capacitor are connected in series
and a potential difference is applied across the combination.
The 2 micro-F capacitor has:
half the potential difference of the 1 micro-F capacitor.
twice the charge of the 1 micro-F capacitor.
half the charge of the 1 micro-F capacitor.
twice the potential difference of the 1 micro-F capacitor.
zero of stored energy.
Capacitors A and B are identical. Capacitor A is charged so
it stores 4 J of energy and capacitor B is uncharged. The
capacitors are then connected in parallel. The total stored
energy in the capacitors is now:
2
16
8
1
4
Joules.
Joules.
Joules.
Joules.
Joules.
T-011
1. Consider the combination of capacitors in Fig. (3). The energy
stored in the 5 micro-F capacitor is 0.20 J. The energy stored in 4
micro-F capacitor is:[0.16 J].
2. An isolated capacitor, C1 = 20.0 micro-F has a potential difference
of 26.0 V. When an uncharged capacitor C2, of unknown value, is
connected across C1, the potential difference becomes 16.0 V for
both. What is the value of C2? [12.5*10**(-6) F].
a
4µF
5µF
3µF
6µF
b
(Prof. Dr. I. M. A. Nasser)
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(Prof. Dr. I. M. A. Nasser)
T-002
Q0
16 Q0
002Q0
26 Q0
Q0
A1
A2
A3
A4
A5
Q0
17 Q0
002Q0
26 Q0
Q0
Q0
Q0
Q0
A1
A2
A3
A4
A5
Q0
18 Q0
26 Q0
991Q0
002Q0
A1
A2
A3
A4
A5
Q0
The equivalent capacitance between points a and b in
the combination of capacitors in figure 6 is:
1.0*10**(-6)
2.0*10**(-6)
1.5*10**(-6)
0.5*10**(-6)
3.0*10**(-6)
F.
F.
F.
F.
F.
A parallel-plate capacitor, of capacitance 1.0*10**(-9) F,
is charged by a battery to a potential difference of 12.0 volts.
The charging battery is then disconnected and oil with
dielectric constant = 4.0 fills the inside space between the
plates. The resulting potential difference, in volts, between
the plates is:
3.
12.
48.
1.0*10**(-9).
3.0*10**(-9).
If Vab is equal to 50 V, find the charge stored and the
potential difference across the 25 micro-F capacitor shown
in Figure 5.
250
300
600
600
250
micro-C
micro-C
micro-C
micro-C
micro-C
and
and
and
and
and
10
20
10
20
40
V.
V.
V.
V.
V.
2µF
35µF
2µF
2µF
a
a
1µF
1µF
2µF
25µF
b
b
FIGURE 6
15µF
FIGURE 5
(Prof. Dr. I. M. A. Nasser)
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(Prof. Dr. I. M. A. Nasser)
T-001
Q0
15 Q0 Find the WRONG statement:
26 Q0 When a dielectric materials is inserted between the plates of
Q0 an isolated capacitor, it will provide the following
Q0 advantages:
Q0
A1 Increase the original charge on the conducting plates.
A2 Increase the maximum energy that can be stored in the
A2 capacitor.
A3 Increase the capacitance of the capacitor.
A4 Increase the maximum operating voltage of the capacitor.
A5 Mechanical support between the conducting plates.
Q0
17 Q0 An isolated capacitor, C1 = 20.0 micro-F has a potential
26 Q0 difference of 26.0 V. When an uncharged capacitor C2, of
Q0 unknown value, is connected across C1, the potential
Q0 difference becomes 16.0 V for both. What is the value of C2?
Q0
A1
12.5*10**(-6) F.
A2
25.0*10**(-6) F.
A3
20.0*10**(-6) F.
A4
1.00*10**(-6) F.
A5
10.2*10**(-6) F.
Q0
18 Q0 A parallel plate capacitor of capacitance C has a charge
26 Q0 of magnitude q when connected to a battery of potential
Q0 difference V. After being fully charged, the capacitor is
Q0 disconnected from the battery and the separation between
Q0 the plates is doubled.
Q0 Which one of the following statements is TRUE?
Q0
A1 The voltage across the plates doubles.
A2 The voltage across the plates is halved.
A3 The capacitor's capacitance doubles.
A4 The magnitude of the charge on the plates doubles.
A5 The magnitude of the charge on the plates is halved.
Q0
T-099
Q0
16 Q0 A 72-V battery is connected across a 0.50 micro-F, air
26 Q0 filled, parallel-plate capacitor. With the battery still
Q0 connected, the space between the plates is filled with a
Q0 dielectric, whereupon the charge on the capacitor is
Q0 increased by 90 micro-C. What is the dielectric constant
Q0 of the dielectric?
Q0
A1 3.5
A2 2.5
A3 4.5
A4 1.5
A5 5.0
(Prof. Dr. I. M. A. Nasser)
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(Prof. Dr. I. M. A. Nasser)
18 Q0 A 9.0-V battery is connected to three capacitors as shown
26 Q0 in Figure 6. What is the energy stored in the 6 micro-F
Q0 capacitor?
Q0
A1 27 micro-J
3µF
6µF
A2 12 micro-J
A3 35 micro-J
A4 53 micro-J
2µF
A5 62 micro-J
Q0
9V
Figure 6
(Prof. Dr. I. M. A. Nasser)
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