Post on 11-Feb-2018
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Chapter 32 Inductance and Magnetic Materials
The appearance of an induced emf in a circuit associated with changes in its own magnet field is called self-induction. The corresponding property is called self-inductance.
A circuit element, such as a coil, that is designed specificallyto have self-inductance is called an inductor.
The appearance of an induced emf in one circuit due to changes in the magnetic field produced by a nearby circuit is called mutual induction. The response of the circuit is characterized by their mutual inductance.
Can we find an induced emf due to its own magnetic field changes? Yes!
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32.1 Inductance
Open: When the switch is opened, the flux rapidly decreases. This time the self-induced emf tries to maintain the flux.
Close: As the flux through the coil changes, there is an induced emf that opposites this change. The self induced emf try to prevent the rise in the current. As a result, the current does not reach its final value instantly, but instead rises gradually as in right figure.
When the current in the windings of an electromagnet is shut off, the self-induced emf can be large enough to produce a spark across the switch contacts.
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32.1 Inductance (II)
)( 1211111 Φ+Φ=Φ NNThe net emf induced in coil 1 due to changes in I1 and I2 is
Since the self-induction and mutual induction occur simultaneously, both contribute to the flux and to the induced emf in each coil.
)( 12111emf Φ+Φ−=dtdNV
The flux through coil 1 is the sum of two terms:
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Self-Inductance
dtdILV 1
111emf_ −=
It is convenient to express the induced emf in terms of a current rather than the magnetic flux through it.
The magnetic flux is directly proportional to the current flowing through it.
where L1 is a constant of proportionality called the self-inductance of coil 1. The SI unit of self-inductance is the henry (H). The self-inductance of a circuit depends on its size and its shape.
The self-induced emf in coil 1 due to changes in I1 takes the form
11111 ILN =Φ
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Mutual Inductance
dtdIMV 2
12emf_ −=
The magnetic flux produced by coil 2 is proportional to I2. Thus the flux contributed from coil 2 may be written as
where the constant of proportionality is M is called the mutual inductance of the two coils. The SI unit of mutual inductance is the henry (H). The mutual inductance of the two coils depends on its separation and orientation.
The emf induced in coil 1 due to changes in I2 takes the form
2121 MIN =Φ
Are the mutual inductances M12 and M21 the same? Why?
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Example 32.1
ALnLLIN
IALNBA
20
0
µ
µ
=
=Φ
==Φ
A long solenoid of length L and cross-sectional area A has Nturns. Find its self-inductance. Assume that the field is uniform throughout the solenoid.
Solution:
Notice that the self-inductance is proportional to the square of the number density.
This result may be compared to the capacitance of two parallel plates (C=ε0A/d), which depends on the geometry of the capacitor.
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abL
LIabIdx
xI
dxxIBdAd
xIB
b
a
ln2
ln22
2 ,
2
0
00
00
πµ
πµ
πµ
πµ
πµ
l
ll
l
=
===Φ
==Φ=
∫
Example 32.2A coaxial cable consists of an inner wire of radius a that carries a current I upward, and an outer cylindrical conductor of radius b that carries the same current downward. Find the self-inductance of a coaxial cable of length L. Ignore the magnetic flux within the inner wire.
Solution:
Hint1: The direction of the magnetic field.
Hint2: What happens when considers the inner flux?
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Example 32.3
F 54.7
)0004.0)(10)(1500)(104( 7
11202
121
12201212
u
ANnI
NM
AInAB
=×=
=Φ
=
==Φ
−π
µ
µ
A circular coil with a cross-sectional area of 4 cm2 has 10 turns. It is placed at the center of a long solenoid that has 15turns/cm and a cross-sectional area of 10 cm2, as shown below. The axis of the coil coincides with the axis of the solenoid. What is their mutual inductance?Solution:
Notice that although M12=M21, it would have been much difficult to find Φ21 because the field due to the coil is quite nonuniform.
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32.2 LR Circuits
)1(
:0
0 : 0
:
Let
0
emf
emf0
emfemf
00
emf
tLR
t
tt
eRVI
RVIt
RVRV
LRe
eIdtdIeII
dtdILIRV
−
−
−−
−=∴
−=−==
=⇒=−
=
−=⇒+=
=−−
β
ββ
α
αβ
α
αα
How does the current rise and fall as a function of time in a circuit containing an inductor and a resistor in series?
Rise
The quantity τ=L/R is called the time constant.
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32.2 LR Circuits
τ
α
αα
α
α
ttLR
t
tt
eRVe
RVI
RVIt
LRe
eIdtdIeII
dtdILIR
−−
−
−−
==∴
==
=
−=⇒=
=−−
emfemf
emf0
00
:0
:
Let
0
Decay
The quantity τ=L/R is called the time constant.
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31.3 Energy Stored in an InductorThe battery that establishes the current in an inductor has to do work against the opposing induced emf. The energy supplied by the battery is stored in the inductor.
In Kirchhoff’s loop rule, we obtain
22emf
2emf
emf
2
1 where, LIU
dtdURiiV
dtdiLiRiiV
dtdiLiRV
LL =+=
+=
+=
power supplied by the battery
power dissipatedin the resistor
energy change ratein the inductor
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Energy Density of the Magnetic FieldWe have expressed the total energy stored in the inductor in terms of the current and we know the magnetic field is proportional to the current. Can we express the total magnetic energy in terms of the B-field? Yes.
Let’s consider the case of solenoid.
space) freein field magnetic a ofdensity energy (The2
2)(
2
1
2
1
0
2
0
22
00
2
20
µ
µµ
µ
µ
Bu
ABAnILIU
AnL
B
L
=
===
=
ll
l
Although this relation has been obtained from a special case, the expression is valid for any magnetic field.
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Example 31.4
)21( d)(
)(
)1( , c)(
ms 5.111.0ln)1(90 (b)
ms 5 isconstant timeThe (a)
/2/2
emf2
/2/2
emf
/emf
/00
ττ
ττ
τ
τ τ
ttR
ttL
tL
t
eeRVRRIP
eeRVR
dtdU
eRVI
dtdILI
dtdU
teII.L/Rτ
−−
−−
−
−
+−
==
−
=
−==
=−=⇒−=
==
A 50-mH inductor is in series with a 10-Ω resistor and a battery with an emf of 25 V. At t=0 the switch is closed. Find: (a) the time constant of the circuit; (b) how long it takes the current to rise to 90% of its final value; (c) the rate at whichenergy is stored in the inductor; (d) the power dissipated in the resistor.Solution:
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Example 31.5
38
7-
22
0
3
2612-20
J/m 102.3
1042
20
2
1
J/m 40
)10)(3100.5)(8.85(2
1
×=
××==
=
××==
πµ
ε
BU
EU
B
E
The breakdown electric field strength of air is 3x106 V/m. A very large magnetic field strength is 20 T. compare the energy densities of the field.
Solution:
Magnetic fields are an effective means of storing energy without breakdown of the air. However, it is difficult to produce such large fields over large regions.
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Example 31.6
)ln(2
2
1)ln(
44
)(
4
)()2(
22
2
20
222
02
0
20
0
2
0
2
0
abhNL
LIabIhNdr
rNIhU
drrNIhrdrh
µBdV
µBdU
rNIB
b
aB
B
πµ
πµ
πµ
πµπ
πµ
=
===
===
=
∫
Use the expression for the energy density of the magnetic field to calculate the self-inductance of a toroid with a rectangular cross section.Solution:
Can we use the concept of magnetic flux to derive the self-inductance?
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31.4 LC OscillationsThe ability of an inductor and a capacitor to store energy leads to the important phenomena of electrical oscillations.
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31.4 LC Oscillations (II)Consider the situation shown in Fig. 32.11. A circuit consists of a inductor and a capacitor in series. According to Kirchhoff’sloop rule,
condition. initial from detremined becan and
,frequency)angular (natural 1
where
)sin(
0
sign) minus a is there(note
0
0
0
00
2
2
φ
ω
φω
QLC
tQQdtQdL
CQ
dtdQI
dtdIL
CQ
=
+=
=+
−=
=−
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31.4 LC Oscillations (III)
)cos(
)sin(
000
00
φωωφω+−=
+=tII
tQQThe current and charge relation
The energy stored in the system is
(constant) 2
1
)(cos2
1
)(cos2
1
2
1
)(sin2
1
2
1
20
022
0
022
020
2
022
02
QC
UUU
tQC
tLQLIU
tQC
QC
U
BE
B
E
=+=
+=
+==
+==
φω
φωω
φω
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The Analogies Between LC Oscillations and the Mechanical Oscillations
0001
2
2
2
2
2
2
=+=+=+ θθl
gdtdx
mk
dtxdQ
LCdtQd
LC Osc. Spring Pendulum.
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Example 32.7In an LC circuit, as in Fig. 32.10, L=40 mH, C=20 uF, and the maximum potential difference across the capacitor is 80 V. Find: (a) the maximum charge on C; (b) the angular frequency of the oscillation; (c) the maximum current; (d) the total energy.
Solution:
J 104.62
1 (d)
A 79.1 c)(
rad/s 112010201040
11 (b)
mC 4.2)80)(1020( (a)
220
000
630
60
−
−−
−
×==
==
=×××
==
=×==
QC
U
QILC
CVQ
ω
ω
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31.5 Damped LC Oscillations
.detremined be toparameters are and
,frequency)angular (damped )2
(1
where
)cos(
01
0
since 0
0
2
/0
2
2
φ
ω
φω
QLR
LC
teQQ
QLC
QLRQ
LCQ
dtdQ
LR
dtQd
dtdQI
dtdILRI
CQ
LRt
−=′
+′=⇒
=+′+′′∴=++
−==−−
−
Consider the situation shown in Fig. 32.14. A circuit consists of a inductor, a capacitor, and a resistor in series. According to Kirchhoff’s loop rule,
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31.5 Damped LC Oscillations (II)The behavior of the system depends on the relative value of ω0 and R/2L.
under damping critical and over damping
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32.6 Magnetic Properties of Matter
lity)susceptibi magnetic:( m0mM χχ BB =
When a material is placed in an external magnetic field B0, the resultant field within the material is different from B0. The field due to the material itself, BM, is directly proportional to B0:
The total field within the material is
ty)permeabili relative:(
)1(
m0m
0mM0
κκχ
BBBBB
=+=+=
The magnetic properties of matter can be classified according to their magnetic susceptibility.
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32.6 Magnetic Properties of Matter (II)
Ferromagnetic (χm=103~105): Fe, Ni, Co, Gd, and Dy.
Depends on temperature
Paramagnetic (χm=10-5): Al, Cr, K, Mg, Mn, and Na.
Depends on temperature
Diamagnetic (χm=-10-5): Cu, Bi, C, Ag, Au, Pb, and Zn.
Not depends on temperature
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32.6 Magnetic Properties of Matter (III)
When the surrounding temperature is greater than the Curie temperature, the ferromagnetic material becomesparamagnetic.
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32.6 Magnetic Properties of Matter (IV)
When the induced magnetic field “lags” behind the change in applied field B0, this lagging effect is called magnetic hysteresis.
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Exercises and Problems
Ch.32:
Ex. 13,16, 24, 29
Prob. 3, 4, 5, 9, 10