Download 7.3.3 Maxwell`s equations

Chapter 7 Electrodynamics
7.0 Introduction
7.1 Electromotive Force
7.2 Electromagnetic Induction
7.3 Maxwell’s Equations
7.0 Introduction
electrostatic
E 
1
0

 E  0
magnetostatic
B  0
  B  0 J

 J  0
t

?0
t

 E  ?
t
0

   E    ?
t
?  B
0

   B  0  J    ?
t
0 

?  0 0 E
=
conservation of charge
static
7.0 (2)
Maxwell’s equations:

E 
0
B  0
B
Jd
 E  
t

  B  0 J  0 0 E
t
displacement current
7.0 (3)
B
t


   E  da   t  B  da   t 
•  E  
   B  da
=
 
 E  d
Magnetic flux
Induced electric field (force)

induce E

B (t )
 
BE
7.0 (4)
•  E    B
t

  B  0 0 E
t
Ba
Ea induced by Ba
Bb induced by Ea
Eb induced by Bb
Bc induced by Eb
 
e.g. , E , B ~
e i ( kxwt )
wave
E,B fields propagate in vacuum
7.0 (5)
•

 E   B
t

  B  0 J  0 0 E
t
J ( x  x0 , t ) Ba
Ea
Bb
Eb
A.C. current can generate electromagnetic wave
antenna
cyclotron mass
free electron laser
…..
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
•


J
Current density
of the medium

for
1

conductivity force per unit charge
resistivity
 
 0
for perfect conductors
f  E v B
J   ( E  v  B)
•
f
J E
Ohm’s Law
for

k
 v
( a formula based on experience)
usually true
but not in plasma; especially, hot.
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
 E 
1

 J  0
7.1.1 (3)
Ex. 7.1
uniform

I=?
R=?
uniform
V
sol:
I  JA  EA  A
V
L
in series
L1, L2
in parallel
A1, A2
L
R
A
R  R1  R2
1 1
1
 
R R1 R2
7.1.1 (4)

Ex. 7.3 Prove the field E is uniform
 J  0
V=0
A=const V=V
0
 =const
J  nˆ  0 at the surfaces on the two ends
 E  nˆ  0
i.e.,
V
0
n
V0 z
 V ( z) 
L
2V  0
V0
E  V   zˆ
L
Laplace equation
7.1.1 (5)

Ex. 7.2
V
I ?
ŝ

E
sˆ
2 0 s
V  
a
b
 : line charge density

b
E d 
ln ( )
2 0
a
E  V


b 1  2 L V
I   J  da    E  da   L  V  2 0[ ln ] L
ln ( b )
0
0
a
a
ln ( b )
a
R
2L
7.1.1 (6)
The physics of Ohm’s Law and estimation of microscopic 

the charge will be accelerated by E before a collision
time interval of the acceleration is
mean free path
2 mfp 
  mfp
t  min 
,

v
a
 thermal

1
typical case mfp  at 2
2
for very strong field and
long mean free path
7.1.1 (7)
The net drift velocity caused by the directional acceleration is
1
a
vave  at 
2
2vthermal

nfq F = qE
J  n f q vave 
2vthermal m
mass of the molecule
molecule density e charge
free electrons per molecule
nf  q 2
J
E
2mvthermal

Joule heating law
P  VI  I 2 R
Power is dissipated by collision
7.1.2 Electromotive Force
Produced by
the charge accumulation
due to Iin > Iout
The current is the same all the way around
the loop.
force f  f source  E
electromotive force

 f d
 
 E  d  0
electrostatic
  fs  d
b

(  E  0)
b
Vab    E  d   f s  d 
a
a
E  V
 fs  d

f 0
E   f s outside the source
7.1.3 motional emf

B
Fmag ,v  qvB
Fmag ,v
f mag ,v 
 vB
q

 fmag ,v  d
 vBh causes u
7.1.3 (2)
f pull  uB for equilibrium



h
f pull  d   (uB)(
) sin 
cos
v sin 

u cos 
=
 vBh  
d
sin 

d
cos
h

Work is done by the pull force, not B .
dx
d
d
  vBh  Bh( )   ( Bhx)   
dt
dt
dt
 magnetic flux
7.1.3 (3)
magnetic flux
   B  da  Bhx
for the loop
d
dx
  Bh  vBh  
dt
dt
d
  
dt
flux rule for motional emf
7.1.3 (4)
• a general proof
d   (t  dt )  (t )
  ribbon
da  (v  d )dt
d
  B  (v  d )   B  ( w  d )
dt
   ( w  B)  d    f mag  d  
f mag
d
 
dt

ribbon
B  da
7.1.3 (5)
Ex. 7.4
=?
a
   f mag  ds
0
a
  wsB ds
0
wBa 2

2
 wBa 2
I 
R
2R
f mag  v  B  ws ( wˆ  B )
 wsB sˆ
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

loop moves Area ,  
Induce I [v  B ]

B moves   
induce I [ E ]

B  
induce I [ E ]
d

  emf   E  d   (  E )  da
dt
d

Faraday’s Law (integral form)
   B  da    B  da
dt
t
B
Faraday’s Law (differential form)
  E  
t
7.2.1 (2)
A changing magnetic field induces an electric field.
(a)
v  B, not E


(b) & (c) induce E that causes I
drive I
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 (3)
Ex. 7.5
r̂
ẑ  
loop
Induced  (t ) ?
sol:


K b  M  nˆ  Mˆ


B  0 M
at center , spread
out near the ends
 max   0 Ma 2
ˆ
7.2.1 (4)
Ex. 7.6
Plug in, I
induces B
B induces I r

Is

Ir
Plug in, why ring jump?

B
vB vB

F

B

F

F

ring jump.
7.2.2 The Induced Electric Field
B
 E  
t
  B  0 J
  E  0 (   0)
B  0
 
d
 E  d    dt
 Bd
 0 I enc
7.2.2 (2)
Ex. 7.7

induced E = ?
=
sol:

 
d
d
2
2 dB
 E  d    dt   dt [s B(t )]  s dt
E  2s




s dB ˆ
 E

B
E
2 dt
7.2.2 (3)
Ex. 7.8.
ẑ

 
B  B0
The charge ring  is at rest

B0
sol:
 
d
2 dB
 E  d    dt  a dt
What happens?
dN  r  F  b  ( d ) E  zˆ  (b Ed )
torque on d
dB
2 dB
ˆ   E  d  zb
ˆ  [ a
N   dN  zb
]  b a
dt
dt
2
the angular momentum on the wheel

2 0
2
N
dt


b

a
d
B


a
bB0 zˆ

B
0
7.2.2 (4)

Induced E ( s)  ?
̂
ẑ
I (t )
sol:
0 I
B 
ˆ
2 s
quasistatic

B
7.2.2 (5)
=
d
d 0 I
 E  d   dt  B  da   dt  2 s '   ds '
E ( s0 )  E ( s )
0 dI s 1

ds '

2 dt s0 s '
0 dI

(ln s  ln s0 )
2 dt
0 dI
 E ( s)  [
ln s  K ] zˆ
2 dt

Constant K( s , t )
s << c t
t = I / (dI/dt)
7.2.3 Inductance
0
d 1  Rˆ
B1 
I1 
 I1
2
4
R
mutual inductance
 
 2   B1  da2  M 21I1

0 I1 d 1

  (  A1 )  da2
A1 

 A1  d
0 I1

4

4
2
d 1
d
R
2

R
7.2.3 (2)
0
 M 21 
4

d
1d 2
R
Neumann formula
The mutual inductance is a purely geometrical quantity
M21 = M12 = M
1 = M12 I2
1 = 2 if I1 = I2
7.2.3 (3)
n2 turns per unit length
Ex. 7.10
1
2
sol:
n1 turns per
unit length
I given
assume I too.
B1 is too complicated… 2 = ?
Instead, assume I running through solenoid 2
I 2  I1  I
B2  0n2 I 2
1  n1  1, per turm  n1   a 2  B2
 0 a 2 n1 n2 I 2
 0 a 2 n1 n2 I
M  0  a 2n1 n2
 2
( I 2  I1  I )
2  ?
M ?
7.2.3 (4)
•
I1 ( t )
2  
d 2
dI
 M 1
dt
dt
changing current I1 in loop1, induces current in loop2
• self inductance
I (t )
  LI
self-inductance
(or inductance )
Volt  sec
[ unit: henries (H) ]
1H  1
A
• back emf
dI
  L
dt
I
 will reduce it.
7.2.3 (5)
Ex. 7.11
N turns
b
a
L(self-inductance)=?
sol:
 
  N  B  da
0 NI b1
N
h a ds
2
s
0 N 2 Ih
b

ln ( )
2
a
0 N 2 h b
L
ln ( )
2
a
0 NI
B
2s
7.2.3 (6)
Ex. 7.12
sol:
I (t )  ?
dI
 0  L  IR
dt
I (t ) 
0
R
 ke
0

R
t
L
R
general solution
particular solution
if
I (0)  0 , k  
I (t ) 
0
R
0
R
R
 t
(1  e L )

0
R
(1  e

t
t
)
L
t
R
time constant
7.2.4 Energy in Magnetic Fields
In E.S.

From the work done, we find the energy
test charge
q

in E ,

But, B does no work.
1
0 2
We   (V )dt   E dt
2
2
WB = ?
In back emf
d
dI d 1 2
WB   I  L I
 ( LI )
dt
dt dt 2
1 2
1 2 1
(Wk  mv )
 WB  LI  I 
2
2
2
    B  da   (  A)  da 
s
 WB 
1
I
2
1
 loop A  d  2
s
 loop ( A  I )d
loop A  d
7.2.4 (2)
In volume
1
WB   ( A  J )dt
2 V

1


V A  (  B)dt

B
 
 


  ( A  B)  B  (  A)  A  (  B)
2 0
2


B
1
1
2

B
dt 
  ( A  B )dt


V
V
2 0
2 0
 

s ( A  B )  da s   0
1
2
 WB 
B
dt

all
space
2 0
1
0 2
Welec   (V )dt   E dt
2
2
1  
1
2
Wmag   ( A  J )dt 
B
dt

2
2 0
7.2.4 (3)
Ex. 7.13
WB  ?
(length )
sol:
 0 I
B
ˆ a＜ s＜b
2s

B0
sa
sb
0 I 2
WB   dWB  
(
) (2 sds)
20 2 s
0 I 2 b ds

a s
1 2
4
WB  L I
2
2
0 I
b
0
b

ln( )
L 
ln ( )
4
a
2
a
1
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

E 
0
(Gauss Law)
B  0
(no name)
B
 E  
t
 B  0 J
but
(Faraday’s Law)
(Ampere’s Law)



B

  (  E )    (  )   (  B )  0
t
t
=


  (  B )   0 (  J ) ?

 J  0
Ampere’s
Law
fails
because
0
7.3.1
an other way to see that Ampere’s Law fails for nonsteady
current
loop 1
2

 B  d  0 I enc
For loop 1, Ienc = 0
For loop 2, Ienc = I
they are not the same.
7.3.2 How to fix Ampere’s Law
continuity equations, charge conservation





E
 J  
  [ 0  E ]    ( 0 )
t
t
t
such that, Ampere’s law shall be changed to



E
  B  0 J  0 0
t
Jd displacement current
A changing electric field induces a magnetic field.
7.3.2
=

 
E



B

d
a

(

J



)

d
a
0 0

 0
t

 
E 
B

d



I



 da
0 enc
0 0

t
loop 1
2
for the problem in 7.3.1
1
1 Q
E



between capacitors
0
0 A

E
1 dQ
1


I
t  0 A dt  0 A
 
1
 loop1 B  d   0  0 0 I  0 I
0
 
loop 2 B  d   0 I  0  0 I
7.3.3 Maxwell’s equations
 
 E 
0

 B  0


B
 E  
t


 
  B  0 J  0 0 E
t
Gauss’s law
Faraday’s law
Ampere’s law with Maxwell’s correction
Force law F  q( E  v  B)


continuity equation   J  
t
( the continuity equation can be obtained from
Maxwell’s equation )
7.3.3


Since
, J
produce
 
 E 
0

 B  0

 B
 E 
0
t


 
  B  0 0 E  0 J
t
 
J (r , t )


B
E
 
E, B
7.3.4 Magnetic Charge

Maxwell equations in free space ( i.e.,  e  0 , J e  0 )
E  0
B  0
B
 E 
0
t

  B  0 0 E
t

With  e and J e , the symmetry
is broken.

If there were  m ,and J m .



e
B


E



J

E 
0 m
t
0




E
  B  0  m
  B  0 J e  0 0
t


 m
 e
 Jm  


J


and
e
t
t
symmetric


EB


B   0 0 E
symmetric
So far, there is no experimental evidence of magnetic monopole.
7.3.5 Maxwell’s Equation in Matter
bound charge
b    P
b
P
  
t
t
JP
bound current


Jb    M
  J b  0  no corresponding 
polarization current

b
  JP  0
t
Q


( b  da )    da
t
t

P 

 da  J P  da
t
dI 
surface charge
b  P
7.3.5 (2)

   f  b   f    P
 
 

  
J  J f  Jb  J P  J f    M  P
t
Gauss's law   E 
1
0
(  f    P)

or   D   f

 
D  0E  P
Ampere’s law ( with Maxwell’s term )


  
 
  B  0 ( J f    M  P)  0 0 E
t
t



 

  ( B  0 M )  0 J f  0 ( 0 E  P)
t
 1  
 
 
H
BM
 H  J f  D
0
t
7.3.5 (3)
In terms of free charges and currents, Maxwell’s equations
become


B

 E  
 D   f
t

 
 
 B  0
 H  J f  D
t
D, H and E , B are mixed.
displacement current
one needs constitutive relations: D ( E , B ) and H ( E , B )
7.3.5 (4)
for linear dielectric.
or


P   0 xe E


D  E
f
E 


 B  0


M  xm H
   0 (1  xe )
 1 
  0 (1  xm )
H B
 

B
 E  
t



E
  B   J f  
t
7.3.6 Boundary Condition
Maxwell’s equations in integral form
 s D  da  Q f ,enc
 s B  da  0
Over any closed surface S
d
 L E  d   dt s B  da
d
 L H  d  I fenc  dt s D  da
D1, B1
D2 , B2
for any surface
bounded by the S
closed loop L
7.3.6
   
D1  a  D2  a   f a
   
d  
E1    E2      B  da
0
dt
S 0

D1  D2   f
=
=
E1  E2  0
1E1   2 E2   f
B1

B2
0
    
  
H1    H 2    K f  ( n̂   )    ( K f  n̂ )
B1 
1
2

B2  K f  nˆ
=
1
=
1
=
=

H1  H 2  K f  nˆ