10 電磁感應
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From acoustic to electric guitars
10-1 Two symmetric Situations
current loop + magnetic field → torque
torque + magnetic field → current
10-2 Two Experiments
First Experiment
A Magnet is moving with respect to the loop.
Second Experiment
The switch S is closed or opend.
10-3 Faraday ’s Law of Induction
An emf is induced when the number of
magnetic field lines that pass through the loop is changing.
– A Quantitative Treatment
– The magnetic flux
m2
1T
= 1Wb
= 1Weber
A
B
B d Electromotive force
Faraday’s law
ε
ε (N turns)
ε
B
B
d dt N d
dt Ε ds
m2
1T
= 1Wb
= 1Weber
A
B
B d emf magnetic flux
induced emf
induced electric field
Ex.1 A solenoid contains a coil
i =1.5 A ns = 2.2×104 turns / m
A = 3.46 ×10-4m2 Nc = 130 turns
電流在25ms內穩定降至0
計算
mV 75
/
Wb 4
. 14
T 10
15 .
4
,
2 0
t N
BA in B
B i
B i
E
i =1.5 A n = 2.2×104 turns / m
A = 3.46 ×10-4m2 N = 130 turns
電流在25ms內穩定降至0
e
感應電流所生的磁場與感應出該電流的磁場反向
10-4 Lenz’s Law
Opposition to pole Movement
Opposition to flux change
Electric Guitars
A 6 . 1
R R
i Enet Eind Ebat
2
(8.0 2.0)
2
5.2 ( 10)
d B dB
dt A dt
r t
V t
E
Ex.2 B = 4.0 t
2+2.0 t + 3.0
Ebat = 2.0V r = 0.20m R = 2.0
increasing
clockwise
e
V 14 144
t
dt d B E
0 . 3 0
2 2
2
2 2 B
72 4
4
t dx
x H
t
Hdx x
t BHdx
BdA A
d B
Ex.3 B = 4 t
2x
2W = 3.0m H = 2.0m t = 0.1s
increasing
counterclockwise
e
A commercial alternator
10-5 Induction and Energy Transfers
The work done by the applied force and the thermal energy produced in the wire.
R Blv
i
Blv dt
Bldx
dt Blx
d dt
d
Blx BA
Fv P
/ /
/ ) (
/ ,
E Ee
e
R v
L B R
R BLv
R i
P
R v
L B Fv
P
R v
L B iLB
F
th m
/ )
/ (
/ /
2 2 2 2
2
2 2 2
2 2
The Power
Eddy currents – induction stoves and
EM braking
Eddy currents –EM braking
Eddy currents – transformers
Eddy currents –metal detectors
The microphone
The tape recorder
Speaker
Voice-coil positioner of a HDD
10-6 Inductors and Inductance
Inductance:
: flux linkage
L N N
i
1 henry=1H= 1T‧m
2/A
Ex.4 Inductance of a solenoid:
A n
l L
lA n
i BA
nl i
N L
in B
BA nl
N
2 0
2 0
0
/
/ ) )(
( /
), )(
(
Ex.5 Inductance of a toroid
a b iNh
r dr iNh
r iN r
B iN
2 ln 2 =
=
2 hdr Bhdr
=
A d B
= 2 ,
b 0 a 0
b a b 0
a 0
a mH b h
N a
b iNh
i N i
L N ln 2.5
ln 2 2
2 0
0
10-7 Self-Induction
•An induced emf appears in any coil in which the current is changing.
dt L di dt
N Li d
N ( )
, E
10-8 RL Circuits
RC Circuits
C
C
t
t
e q q
RC e
C q
/ 0
C /
), 1
(
E
RL Circuits
/
(1
t L),
LL
i e
R R
E
Initially, an inductor acts to oppose changes in the current thru it.
0 iR L di
dt L di Ri
dt
E E
The inductive time constant
L L , unit: 1 H
R
1s
The decay of current
L
L t
t i e
R e i
dt iR L di
/
0 /
0
E
Ex.6 A multiloop RL circuit
• What is the current right after and long after the switch is closed?
R A i
R A i
eq
0 . 6
, 0 .
2
E E
s 10 .
0 2
ln
) 1
2 ( 1
0
0 /
L
t
t
R e R
L
E
E
Ex.7
the time the current takes to reach half its final equilibrium valueL = 53mH, R = 0.37Ω
10-9 Energy Stored in a Magnetic Field
2
2 2
0 0
= =
1 ( )
2 2
B
B
B
U i
B B E
di di
L iR i Li i R
dt dt
dU di
Li dU Lidi
dt dt
dU Lidi U Li U q
C
E E
10-10 Energy Density of a Magnetic Field
2 ) ( 1
2
2 1
) (
2
2
2 1
,
=
2 0
0 2 2
2 0
2 0
2 2
2
E B u
i n
A i n
L A
i l
L Al
u Li
Li Al U
u U
E B
B B
B
e
Ex.8 A long coaxial cable
a b dU i
U
dr r rl
i
dV u
dU
r u i
r B i
B
4 ln
) )(
2 8 (
2 ) 2 (
, 1 2
2 0 2
2 2 0
0 2 0
B 0
10-11 Mutual Induction
12 21
2 12
1 21
21 2
1 21
21 2
1 21 1
21 2
21
,
,
,
,
M dt M
M di dt
M di
dt N d
dt M di
N i
i M M N
E E
Ex.9 Two circular close-packed coils
mH 3
. 2
2 2
) )(
( 2
2 12 1
1
2 2 2 1 0 1
21 2
1
1 2 2 2 1 0
2 2 1
2 21
2
1 1 1 0 1
i M N
R
R N N i
M N
R
i R N N
R B
N N
R N B i
