Chapter 7 Electrodynamics
7.0 Introduction
7.1 Electromotive Force
7.2 Electromagnetic Induction
7.3 Maxwell’s Equations
?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
7.0 (2)
Maxwell’s equations:
0E
0B
BE
t
0 0 0B J Et
dJ displacement current
7.0 (3)
dE
Magnetic flux
Induced electric field (force)
)(tB
induce
EB
E
•
=
BE
t
E da B dat t
B da
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
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
7.1 Electromotive Force
7.1.1 Ohm’s Law
7.1.2 Electromotive Force
7.1.3 Motional emf
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
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
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
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
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
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
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
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
7.1.3 motional emf
,,
mag vmag v
Ff vB
q
B
,mag vF qvB
, causesmag vf d vBh u
( ) ( )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
7.1.3 (3)
magnetic flux
for the loop
flux rule for motional emf
B da Bhx
d dxBh vBh
dt dt
d
dt
( ) 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
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
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
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
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
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
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
7.2.2 The Induced Electric Field
0 encB d I
dtd
dE
BE
t
0B J
0 ( 0)E
0B
7.2.2 (2)
induced = ?E
sol:
dtBd
stBsdtd
dtd
dE
22 )]([ sE 2
=
2 dtBds
E
E
B
Ex. 7.7
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
7.2.2 (4)
sol:
Induced ?)( sE
quasistatic
z
B
( )I t
0 ˆ2
IB
s
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)
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
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
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
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
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
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
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 = ?
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
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
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
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)
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
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
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
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
7.3.3
0 B
JEt
B
000
0tB
E
0 E
Since , produce , J
E
B
),( trJ
E
B
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
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
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
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
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
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
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