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Chapter 7 Electrodynamics
7.0 Introduction
7.1 Ohm’s Law
7.2 Faraday’s Law
7.3 Maxwell’s Equations
7.0 Introduction
electrostatic
 1
 E  
0

 E  0
magnetostatic

 B  0


  B  0 J
=
conservation of charge


  J  0
t
static

?0
t
0
 
 
 E  ?
 E  ?
t
t

?  B
0

 
    B  0  J    ?
t

?  0 0 E

0
7.0 (2)
Maxwell’s equations.
 
 E 
0

 B  0



Jd
B
 E  
t


 
  B  0 J  0 0 E
t
displaceme nt current
7.0 (3)
=


B
•
 E  
t
 
  

 


E

d
a


B

d
a



   B  da


t
t
 
Magnetic flux
E

d


Induced electric field (force)

B (t )
induce

E
 
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 )
E,B field propagate in vacuum
7.0 (5)
•

 
 E   B
t


 
  B  0 J  0 0 E
t

J ( x  x0,t )

Ba

Eb 
Ba

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



f
Current density conductivity force per unit charge
of the medium
 
1
for perfect conductors
  resistivity
 0

•
   
for f  E  v  B

  
J  (E  v  B)
( a formula based on experience)
w


for >> v usually true
J  E
k
but not in plasma
Ohm’s Law
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
 1

 E   J  0

7.1.1 (3)
Ex7.1
uniform

I=?
R=?
uniform
V
sol:
V
I  JA  EA  A
L
in series
L1 , L 2
in parallel
A1, A 2
L
R
A
R  R1  R 2
1 1
1


R R1 R 2
7.1.1 (4)

Ex.7.3 Prove the field E is uniform

  J  0
V=0 A=const V=V0
 =const

J  nˆ  0 at the surfaces on the two ends

 E  nˆ  0
i.e.,
V
0
n
V0 z
V ( z ) 
L

V0
E  V   zˆ
L
2V  0
Laplace equation
7.1.1 (5)
Ex. 7.2

V
I ?
ŝ

ŝ
 : line ch arg e density
20s


b
a
V   b E  d  
ln ( )
20
a

E  V
 
  

b 1
I   J  da   E  da  L  V  20 [ ln ] L
0
0
a
2L
ln ( b )
a

V
R

ln ( b )
2L
a

E
7.1.1 (6)
The physics of Ohm’s Law and estimation of microscropic 

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
2v thermal



nfq  F = qE
J  n f q v ave 
2v thermal m
mass of the molecule
molecule density e charge
free electrons per molecule

nfq 2 
J
E
2mv thermal

Joule heating law
P  VI  I 2 R
Power is dissipated by collision
7.1.2 Electromotive Force
The current is the same all the way around
the loop.
force
 

f  f source  E
electrostatic
electromotive force
 
   f  d    f s  d
 

E

d


0
(  E  0)


b 
b 


Vab    E  d    f s  d    f s  d  
a
a



fs  0
E  V f  0 
E   f s outside the source
Produced by
the charge accumulation
due to Iin > Iout
7.1.3 motional emf

B

Fmag ,v  qvB


Fmag ,v
f mag ,v 
 vB
q



   f mag ,v  d   vBh cause 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 prove
d  (t  dt )  (t )
  ribbon
 
  B  da
ribbon
  
da  ( v  d )  dt
  
  
d
  B  ( v  d )   B  ( w  d )
dt


  
   ( w  B)  d     f mag  d   

f mag
 
d
dt
7.1.3 (5)
Ex.7.4
=?
a
  0 f mag  ds
a
 0 wsB ds
wBa 2

2
 WBa 2
I 
R
2R


 
f mag  v  B  ws( ŵ  B)
 wsB ŝ
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

Area 
loop moves

  
Induce I [ v  B]


B moves

 
induce I [ E ]
B

 
induce I [ E ]
 

d


 emf   E  d    (  E)  da
dt
d  
 
  B  da    B  da
Faraday’s Law (integral form)
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

drive I

(b) & (c) induce E

that causes 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
zˆ  
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)


Plug in, I induces B
Ex. 7.6


B induces Ir

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:
What happens?
 
d
2 dB
 E  d    dt  a dt

   

torque on d 
dN  r  F  b  (d) E  zˆ  (bEd)




 
2 dB
2 dB
N   dN  zˆb  E  d   zˆb[a
]  ba
dt
dt
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)  ?
ˆ
ẑ
I( t )
sol:

B 
0 I ˆ

2s
guasistatic

B
7.2.2 (5)
=
 
d  
d 0 I
 E  d    dt  B  da   dt  2s'    ds'
E(s 0 )  E(s)
 0 dI s 1

ds '

s0
2 dt s'
 0 dI

(ln s  ln s 0 )
2 dt

0 dI
 E(s)  [
ln s  K] ẑ
2 dt

Constant K( s , t )
s << c

I
dI
dt
7.2.3 Inductance

 0
d 1  Rˆ
B1 
I1 
 I1
2
4
R
 
 2   B1  da2  M 21I1


mutual inductance
  (  A1 )  da2


  A1  d  2

 0 I1 d 1 

 d 2


4
R

  0 I1 d 1
A1 

4 R
7.2.3 (2)
 

d   d 2
 M 21  0   1
4
R
Neumann formula
The mutual inductance is a purely geometrical quantity
M 21  M12  M
1   2 if
I1  I 2
1  M12 I 2
7.2.3 (3)
n2 turns per unit length
Ex. 7.10
1
sol:
2
I given
n1 turns per
unit length
assume I too.
B1 is too complicated………..
2  ?
Instead, assume I running through solenoid 2
I 2  I1  I
1  n1  1,per turm  n1   a 2  B2
B2   0 n 2 I 2
  0  a 2 n1 n 2  I 2
  0  a 2 n1 n 2  I
M  0  a 2 n1 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
I
 will reduce it.
dt
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 2h b
L
ln ( )
2
a
0 NI
B
2s
7.2.3 (6)
Ex. 7.12
I( t )  ?
sol:
dI
 0  L  IR
dt
I (t ) 
0
R
0
R
R
 t
 ke L
particular sol. general sol.
if
I(0)  0 , k  
I (t ) 
0
R
R

t
0
L
(1  e )
R

0
R
(1  e

t

)
L

R
time cons tant
7.2.4 Energy in Magnetic Fields
In E.S.

test charge
q
From the work done, we find the energy

1
0 2
E
W

(
V

)
d


E d
in ,
e


2
2

But , B does no work.
WB  ?
In back emf
d
dI d 1
WB   I  L I  ( LI 2 )
dt
dt dt 2
1 2 1
1 2
 WB  LI  I
( Wk  mv )
2  2
2 




  sB  da  s(  A)  da  loop A  d 
 
  
 WB  1 I  A  d   1  (A  I )d 
2 loop
2 loop
7.2.4 (2)
In volume
 
1
WB  V ( A  J )  d

2
B
 
 




1
  ( A  B)  B  (  A)  A  (  B)

A  (  B )d

V
2 0
2


B
1
1
2

B
d 
  ( A  B )d


V
V
2 0
2 0
 

(
A

B
)

d
a
0
s
s
1
2
 WB 
B d

all
space
2 0
1
0 2
Welec   (V )d   E d
2
2
1  
1
2
Wmag   ( A  J )d 
B
d

2
2 0
7.2.4 (3)
Ex. 7.13
WB  ?
(length )
sol:
 0 I
B
ˆ a＜ s＜b
2s
WB    dWB   
0 I 2 b ds

 a
4
s
0 I 2 b

ln( )
4
a

B0
s ＜a
s ＞b
0 I 2
) (2sds )
2 0 2s
1
(
1 2
 WB  L I
2

b
 L  0 ln ( )
2
a
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
B  0


B
E  
t


  B  0 J
but
(Gauss Law )
(no name )
(Faraday' s Law )
(Ampere ' s Law )



B

  (  E )    (  )   (  B )  0
t
t
=


  (  B )   0 (  J ) ?

Ampere’s Law fails because   J  0
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 , I enc  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
J d 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
for the problem in 7.3.1
1
1 Q
E



between capacitors
0 A
 0
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
Gauss’s law


B
 E  
Faraday’s law
t


 
  B  0 J  0 0 E Ampere’s law with Maxwell’s correction
t

  
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 , B
 
 E 
0

 B  0

 B
 E 
0
t


 
  B  0 0 E  0 J
t
 
J (r , t )


B
E
7.3.4 Magnetic Charge

Maxwell equations in free space ( i.e.,  e  0 , J e  0 )

E  0

B  0

Je
With  e and
If there were  m
 e
E 
0

  B  0  m

 m
 Jm  
t

 B
E 
0
symmetric
t


EB

 


  B  00 E
B   0 0 E
t
, the symmetry
is broken .

,and J m .



B
  E   0 J m 
t
symmetric



E
  B  0 J e  0 0
t

 e
 Je  
and
t
So far, there is no experimental evidence of magnetic monopole.
7.3.5 Maxwell’s Equation in Matter
bound charge
bound current


Jb    M
b    P
b
P
  
t
t

JP

  J b  0  no correspond ing 
polarization current

b
  JP  0
t
Q


( b  da )    da
t
t

P 

 da  J P  da
t
dI 
surface ch arg e
b  P
7.3.5 (2)

   f  b   f    P
 
 

  
J  J f  Jb  J P  J f    M  P
t
 1

Gauss ' s law   E  (  f    P)
0

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.


P   0 xe E
or


D  E
 f
 E 
0

 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 fenc
Over any closed surface S
 
s B  da  0
 
d  
L E  d    dt sB  da
 
d  
L H  d   Ifenc  dt sD  da
 
D1 , B1
 
D 2 , B2
for any surface
bounded by the S
closed loop L
7.3.6
   
D1  a  D2  a   f a

H1  H 2  K f  nˆ
=
0
=

B2
0
S 0
E1  E2  0
=
B1
1E1   2 E2   f
=

D1  D2   f
   
d  
E1    E2      B  da
dt
    
  
H1    H 2    K f  ( n̂   )    ( K f  n̂ )
B1 
1
2

B2  K f  nˆ
=
1
=
1