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Chapter 13 Maxwell’s Equations 麦克斯韦方程组
Maxwell summarized the experimental laws of electricity and magnetism—the laws of Coulomb, Gauss, Biot-Savart, Ampere, and Faraday.
He found that all the experimental laws hold in general except for Ampere’s law.
Dones not apply to discontinuous current
Invent displacement current to generalize Ampere’s law
§ 13-2 Maxwell’s Equations
麦克斯韦方程组
§ 13-1 Displacement Current
位移电流
Electric field E eE
Electrostatic field
--set up by static charges
Induced electric field nE
--set up by varying M-field
Is there another M-field Magnetic fieldB
B
magnetic field --set up by steady current
that it is set up by varying E-field ? ?
1.Question
§13-1 Displacement Current
Displacement Current 位移电流
2. Maxwell’s hypothesis
A varying electric field will set up a magnetic field in exactly the same way as ordinary conduction current.
E
Varying
InducingE
B
R
cIcI
Two different surfaces Two different surfaces SS11, ,
SS22 bounded by the same bounded by the same
circle circle LL
3. 3. Displacement current2S
1SL
The capacitor is electrified
ForFor SS11
L
IldB 0
For For SS22 L
ldB 0
Ampere’s law cannot be used in this problem.Ampere’s law cannot be used in this problem.
Conductive current
R
cIcI
2S1S
LThough there is any current going through the surface S2, there is a changing -flux going through it.
E
E
When Ic0, 0dt
dq
cI
dt
dq
0dt
Ed
Assume Assume SS----the area of plate, the area of plate,
0
E
DSD S q
D
--the area charge density --the area charge density on on SS at time at time tt..
R
cIcI
2S1S
L
E
thenthen
dt
dq
dt
d D
cI
dt
dI D
d
Sd
t
DS
t
Djd
--the density of displacement current--the density of displacement current
-- -- Displacement current
Definition Definition
l dc IIldB 0
--Generalized form of Ampere’s law--Generalized form of Ampere’s law
II ==IIcc++IIdd
generalized current( 全电流 )
Let
dt
dI D
c0
dt
dI E
c
000 E
-flux
NoteNotes s
The differences between The differences between IIdd and and IIc c ::
IIcc is formed by the motion of charges in cois formed by the motion of charges in co
nductor along one direction. nductor along one direction.
IIcc produces Joule thermal energy in conductor.produces Joule thermal energy in conductor.
IIdd set up a M-field in exactly the same way as IIc c .
IId d is formed by the varying of electric field. is formed by the varying of electric field.
IIdd never has thermal effect. never has thermal effect.
IId d can be found in the area that exists varying can be found in the area that exists varying
electric field( vacuum, dielectric, conductor).electric field( vacuum, dielectric, conductor).
[[ExampleExample]Two circle plates with radius ]Two circle plates with radius RR=0.1m =0.1m consist of a parallel plate capacitor. When consist of a parallel plate capacitor. When the capacitor is electrified, the E-field between thethe E-field between the p plates increases with lates increases with dE/dtdE/dt=10=101212VmVm-1-1ss-1-1. .
E
R
Find Find The displacement current The displacement current IIdd between between
two plates. two plates.
The magnitude and The magnitude and direction of the M-field direction of the M-field in the area of in the area of rr<<R R and and r >Rr >R
SolutionSolution
dt
dI D
d
dt
dDS
dt
dER 0
2
A28.0The distribution of E-field has axial symmetry. The distribution of E-field has axial symmetry.
E
R rL
So the induced M-field produced by the varying So the induced M-field produced by the varying E-field has the same form as E-field has the same form as aa cylindrical currentcylindrical current..
Drawing a circle Drawing a circle LL with with rr as shown in Fig.as shown in Fig.
Using generalized form of Ampere’s law, we haveUsing generalized form of Ampere’s law, we have
dt
dEr 2
00
Bdt
dEr
200
rB 2
L
ldB
rB 2dt
d D 0 dt
dEr 2
00 When When r<Rr<R,,
((r<Rr<R))
20
2 R
rIB d
dt
dI D
d
dt
d e 0
dt
dEr 2
0
i.e.
When When r>Rr>R,,
dt
dER2
00
Bdt
dE
r
R200
2
rB 2
L
ldB
rB 2dt
d D 0
((r>Rr>R))
r
IB d
2
0
dt
dI D
d
dt
dER2
0i.e.
When , 0dt
dE L
R
dI
’s direction:B L
B
When , 0dt
dE
L
R
dI
L
B
§13-2 Maxwell’s Equations
E-FieldE-Field
M-FieldM-Field
M-FieldM-Field
E-FieldE-Field
Electromagnetic wave
Maxwell’s Equations
ChargeCurrentChargeCurrent
)(tq
)(tI
Electrostatic field set up by static charges ：
qSdDS
)1(
Steady magnetic field set up by steady current ：
0)1( SdBS
L
)( IldH
1
0)1( ldEL
1. Maxwell’s equations
dt
dldE m
L
)2(
Induced electric field set up by varying M-field:
0)2( SdDS
Induced M-field set up by varying E-field (displacement current):
0)2( SdBS
dt
dldH D)(
2
In general,)2()1( EEE
)()( 21 DDD
)()( 21 BBB
)()( 21 HHH
We get
qSdDS
dt
dldE m
L
0S
SdB
dt
dIldH D
L
Maxwell’s Equations in integral form.
As
Maxwell’s Equations in differential form.
V
S
dVDSdD )(
dVBSdBV
S
)(
And S
L
SdEldE
)(
S
L
SdHldH
)(
D
0 B
t
BE
t
DjH
2. Electromagnetic waveSpecial example ：In the free space( 自由空间 )
No any charge and conductive current.
Maxwell’s Equations ：
0 D
0 B
t
BE
t
DH
For vacuum,ED
0HB
0
Making a Rotation( 旋度 ) for ,
t
BE
)(
Bt
Use H
0
Use t
D
0Use
t
E
00
2
2
00 t
E
Using vector formula,
EEE
2)()(
=0
We can get the equation from above,
02
2
002
t
EE
--differential form of E-field
Similarly,
02
2
002
t
BB
--differential form of M-field
The wave equations of electromagnetic field in vacuum.
The speed of E-M-wave is00
1
c
E-M-field spreads in the space to form the E-M-wave.
x
y
z
o
E
B
c Direction of propagation
