chapter 7 electrodynamics
DESCRIPTION
Chapter 7 Electrodynamics. 7.0 Introduction 7.1 Electromotive Force 7.2 Electromagnetic Induction 7.3 Maxwell’s Equations. 0. 0. 7.0 Introduction. electrostatic. static. magnetostatic. =. conservation of charge. 7.0 (2). Maxwell’s equations:. 7.0 (3). =. Magnetic flux. - PowerPoint PPT PresentationTRANSCRIPT
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Chapter 7 Electrodynamics
7.0 Introduction
7.1 Electromotive Force
7.2 Electromagnetic Induction
7.3 Maxwell’s Equations
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?Et
? 0t
7.0 Introduction
electrostatic static
0
1E
magnetostatic
0B J
conservation of charge
? B
E
00?
0
0 ?B Jt
0
=
0
0E
0B
0Jt
?Et
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7.0 (2)
Maxwell’s equations:
0E
0B
BE
t
0 0 0B J Et
dJ displacement current
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7.0 (3)
dE
Magnetic flux
Induced electric field (force)
)(tB
induce
EB
E
•
=
BE
t
E da B dat t
B da
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7.0 (4)
E,B fields propagate in vacuum e.g. , BE
, ~ )( wtkxie
• E Bt
0 0B Et
aB
a aE induced by B
b aB induced by E
b bE induced by B
c bB induced by E
wave
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7.0 (5)
•
A.C. current can generate electromagnetic waveantennacyclotron massfree electron laser …..
E Bt
0 0 0B J Et
0( , )J x x t
aB
aE
bB
bE
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7.1 Electromotive Force
7.1.1 Ohm’s Law
7.1.2 Electromotive Force
7.1.3 Motional emf
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7.1.1 Ohm’s Law
Current density conductivity force per unit charge of the medium
resistivity
0 for perfect conductors
for vk
usually true
but not in plasma; especially, hot.
Ohm’s Law
•
•
( a formula based on experience)
J f
1
for f E v B
( )J E v B
J E
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7.1.1 (2)
Total current flowing from one electrode to the other
V=I R Ohm’s Law (based on experience)
Potential current resistance [ in ohm (Ω) ]
Note : for steady current and uniform conductivity
•
10E J
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7.1.1 (3)
Ex. 7.1
sol:
LV
AEAJAI
parallelin
seriesin
AL
R
I=?R=?
uniform
uniform
V
1 2 1 2,L L R R R
211 2
1 1 1,A A
R R R
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7.1.1 (4)
Ex. 7.3 Prove the field is uniform E
i.e.,
V=0 V=V0A=const =const
ˆ0 0 at the surfaces on the two endsJ J n
ˆ 0E n
0V
n
2 0 Laplace equationV
0( )V z
V zL
0 ˆV
E V zL
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7.1.1 (5)
L2
)ab(ln
R
Ex. 7.2 V ?Is
0ˆ
2E s
s
: line charge density
0ln ( )
2
a
b
bV E d
a
E V
10
0 02 [ ln ]
bI J da E da L V L
a
2
ln ( )
LV
ba
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7.1.1 (6)
The physics of Ohm’s Law and estimation of microscopic
the charge will be accelerated by before a collision
time interval of the acceleration is
E
a
2,
vmint mfp
thermal
mfp
mean free path
2
21
tamfp typical casefor very strong field and long mean free path
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2 thermalave
nfq FJ n f q v
v m
7.1.1 (7)The net drift velocity caused by the directional acceleration is
molecule density e charge
free electrons per molecule
Eq
=
mass of the molecule
RIVIP 2Power is dissipated by collision
Joule heating law
1
2 2 thermalave
av at
v
2
2 thermal
nf qJ E
mv
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b bab s sa aV E d f d f d
sf d f d
7.1.2 Electromotive Force
The current is the same all the way around the loop.
force electrostatic
electromotive force
0dE )0( E
outside the source
Produced by the charge accumulationdue to Iin > Iout
sourcef f E
E V
0f
sE f
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7.1.3 motional emf
,,
mag vmag v
Ff vB
q
B
,mag vF qvB
, causesmag vf d vBh u
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( ) ( )dx d d
vBh Bh Bhxdt dt dt
7.1.3 (2)
h
cossin
dd
=
sin)
cos)((
huBdf pull
cossin
uv
vBh
Work is done by the pull force, not . B
magnetic flux
pullf uB for equilibrium
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7.1.3 (3)
magnetic flux
for the loop
flux rule for motional emf
B da Bhx
d dxBh vBh
dt dt
d
dt
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( ) magw B d f d
7.1.3 (4)
a general proof
dtd
•
ribbon
( ) ( )d t dt t
ribbonB da
( )da v d dt
( ) ( )d
B v d B w ddt
magf
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7.1.3 (5)
Ex. 7.4
=?
0
a
magf ds
0
awsB ds
2
2
wBa
2
R 2
wBaI
R
ˆ( )magf v B ws w B
ˆwsB s
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7.2 Electromagnetic Induction
7.2.1 Faraday’s Law
7.2.2 The Induced Electric Field 7.2.3 Inductance
7.2.4 Energy in Magnetic Fields
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7.2.1 Faraday’s Law M. Faraday’s experiments
Induce induce induce
Faraday’s Law (integral form)
Faraday’s Law (differential form)
loop moves B moves B Area ,
[ ]I v B
[ ]I E
[ ]I E
( )emfd
E d E dadt
d
B da B dadt t
BE
t
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Lenz’s law : Nature abhors a change in flux ( the induced current will flow in such a direction that the flux it produces tends to cancel the change. )
7.2.1 (2)
A changing magnetic field induces an electric field.
(a) (b) & (c) induce that causesE
I
drive I
, notv B E
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7.2.1 (3)
sol:ˆ MnMKb
MB
0
at center , spread out near the ends
2
0max aM
Ex. 7.5
Induced ? )(t
ˆz r
loop
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7.2.1 (4) Ex. 7.6
Plug in, why ring jump?rI
Plug in, induces
B
B F
F
F
ring jump.
sI
I
B
induces rB I
v B
v B
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7.2.2 The Induced Electric Field
0 encB d I
dtd
dE
BE
t
0B J
0 ( 0)E
0B
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7.2.2 (2)
induced = ?E
sol:
dtBd
stBsdtd
dtd
dE
22 )]([ sE 2
=
2 dtBds
E
E
B
Ex. 7.7
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7.2.2 (3)
dtdB
adtd
dE 2
0BB
The charge ring is at rest
0B
What happens?sol:
torque on d ˆ( ) ( )dN r F b d E z b Ed
2 2ˆ ˆ [ ]dB dB
N dN zb E d zb a b adt dt
the angular momentum on the wheel
zbBaBdabdtNB ˆ0
202
0
Ex. 7.8. z
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7.2.2 (4)
sol:
Induced ?)( sE
quasistatic
z
B
( )I t
0 ˆ2
IB
s
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7.2.2 (5)
=
Constant K( s , t )
0 '2 '
Id dE d B da ds
dt dt s
0( ) ( )E s E s
0
0 1'
2 '
s
s
dIds
dt s
00 (ln ln )
2
dIs s
dt
0 ˆ( ) [ ln ]2
dIE s s K z
dt
s << c = I / (dI/dt)
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7.2.3 Inductance
121212 IMadB
21)( adA
mutual inductance
1 2A d
0 11 1 12
ˆ
4
d RB I I
R
0 1 124
I dd
R
0 1 11 4
I dA
R
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7.2.3 (2)
Neumann formula
The mutual inductance is a purely geometrical quantity
0 1 221 4
d dM
R
M21 = M12 = M 1 = M12 I2
1 = 2 if I1 = I2
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7.2.3 (3)
Ex. 7.10
sol:B1 is too complicated… 2 = ?
Instead, assume I running through solenoid 2
20 1 2M a n n
III 12
?
?2
M
n2 turns per unit length
n1 turns per unit length
2
1 I given
assume I too.
1 1 1, per turmn 21 2
20 1 2 2
20 1 2
2 2 1( )
n a B
a n n I
a n n I
I I I
2 0 2 2B n I
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7.2.3 (4)
• )(1 tI
dtdI
Mdtd 12
2
changing current in loop1, induces current in loop21I
• self inductance
)(tI
self-inductance (or inductance )
[ unit: henries (H) ]A
VoltH
sec11
• back emf
L I
will reduce it.dI
L Idt
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7.2.3 (5)
Ex. 7.11
sol: adBN
sNI
B
20
b
adss
hNI
N1
20
20 ln ( )2
N h bL
a
L(self-inductance)=?
b
a
N turns
20 ln ( )2
N Ih b
a
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7.2.3 (6)
Ex. 7.12
sol:
IRdtdI
L 0
0( )Rt
LI t keR
particular solution
)1()1()( 00 tt
LR
eR
eR
tI
R0
( ) ?I t
0if (0) 0 ,I kR
time constantL
R
general solution
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7.2.4 Energy in Magnetic Fields
From the work done, we find the energy
in , E
dEdVWe20
2)(
21
But, does no work.B
In back emf
In E.S.
test charge
q
21( )2B
d dI dW I L I LI
dt dt dt
21 1
2 2BW LI I 21
( )2kW mv
( )s s loopB da A da A d
1 1
( )2 2B loop loop
W I A d A I d
WB = ?
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7.2.4 (2)In volume
1( )
2B VW A J d
dBAV )(
21
0
dBAdBVV )(
21
21
0
2
0
)()()( BAABBA
B
2B
s
adBA )(
s0
dBWspaceallB 2
021
dEdVWelec20
2)(
21
dBdJAWmag2
021
)(21
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7.2.4 (3)
Ex. 7.13
sol:
bsasI
B ˆ2
0
< < 0B
20
0
1( ) (2 )
2 2B BI
W dW sdss
)length(
?BW
s as b
20 ln( )4
I b
a
21
2BW L I
0 ln ( )2
bL
a
20
4
b
a
I ds
s
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7.3 Maxwell’s Equations
7.3.1 Electrodynamics before Maxwell 7.3.2 How to fix Ampere’s Law 7.3.3 Maxwell’s Equations
7.3.4 Magnetic Charge
7.3.5 Maxwell’s Equation in Matter
7.3.6 Boundary Conditions
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7.3.1 Electrodynamics before Maxwell
0)()()(
B
ttB
E
but
?)()( 0 JB
=0
Ampere’s Law fails because 0 J
0E
0B
BE
t
0B J
(Gauss Law)
(no name)
(Faraday’s Law)
(Ampere’s Law)
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7.3.1
an other way to see that Ampere’s Law fails for nonsteady current
encIdB 0
they are not the same.
loop 1
2
For loop 1, Ienc = 0For loop 2, Ienc = I
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7.3.2 How to fix Ampere’s Law
)(][ 00 tE
Ett
J
continuity equations, charge conservation
such that, Ampere’s law shall be changed to
tE
JB
000
A changing electric field induces a magnetic field.
Jd displacement current
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7.3.2
adtE
JadB
)( 000
adtE
IdB enc
000
=
for the problem in 7.3.1
between capacitorsAQ
E00
11
IAdt
dQAt
E
00
11
IIdBloop 0
01 00
10
IIdBloop 02 0 0
loop 1
2
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7.3.3 Maxwell’s equations
0 B
Et
JB
000
tB
E
0 E
Gauss’s law
Faraday’s law
Ampere’s law with Maxwell’s correction
Force law
continuity equationt
J
( the continuity equation can be obtained from Maxwell’s equation )
( )F q E v B
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7.3.3
0 B
JEt
B
000
0tB
E
0 E
Since , produce , J
E
B
),( trJ
E
B
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7.3.4 Magnetic Charge
Maxwell equations in free space ( i.e., , )0e 0eJ
symmetric
BE
EB
00
With and , the symmetry is broken.If there were ,and .
e eJ
m mJ
mB 0
tB
JE m
0
tE
JB e
000 symmetric
tJ ee
t
J mm
and
So far, there is no experimental evidence of magnetic monopole.
0E
0B
Et
0B
0 0B Et
0
eE
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7.3.5 Maxwell’s Equation in Matter
bound charge bound current
Pb MJb
0 no correspondingbJ
tP
tb
polarization currentPJ
0
Pb Jt
da
tda
tdI b )(
daJadtP
P
Pb
Q
surface charge
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7.3.5 (2)
Pfbf
Pt
MJJJJJ fPbf
0
1Gauss's law ( )fE P
fDor
PED
0
Et
Pt
MJB f
000 )(
Ampere’s law ( with Maxwell’s term )
)()( 0000 PEt
JMB f
Dt
JH f
MBH
0
1
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7.3.5 (3)
In terms of free charges and currents, Maxwell’s equationsbecome
fD
Dt
JH f
0 B
tB
E
displacement current, and , are mixed.D H E B
one needs constitutive relations: ( , ) and ( , )D E B H E B
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for linear dielectric.
7.3.5 (4)
orExP e
0
ED
HxM m
BH
1
)1(0 ex
)1(0 mx
0 B
fE
tB
E
tE
JB f
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7.3.6 Boundary Condition
Maxwell’s equations in integral form
Over any closed surface S
for any surface bounded by the S closed loop L
L s
dE d B da
dt
,f encsD da Q
0
sB da
fencL s
dH d I D da
dt
1 1,D B
2 2,D B
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7.3.6
aaDaD f 21
0S 021 adB
dtd
EE
fDD 21
021 BB
021 EE
= =
)nK()n(KHH ff
21
nKHH f ˆ21 = =
nKBB f ˆ11
22
11
= =
fEE 2211