Chapter 1ocw.nctu.edu.tw/course/classical_electrodynamics/electrodynamics... · Classical...
Transcript of Chapter 1ocw.nctu.edu.tw/course/classical_electrodynamics/electrodynamics... · Classical...
Classical Electrodynamics
Chapter 1Introduction and Survey
A First Look at Quantum Physics
2011 Classical Electrodynamics Prof. Y. F. Chen
A First Look at Quantum Physics
2011 Classical Electrodynamics Prof. Y. F. Chen
Contents
§1.1 Coulomb’s law and electric field§1.2 Electric field and electric potential§1.3 Discontinuity of electric field and potential§1.4 Poisson and Laplace equation§1.5 Green’s theorem and Green function§1.6 Electrostatic potential energy and energy density
(1) The force between two point charges is given by:
31
11
041
xxxxqqF
(2) The electric field of a point charge can be defined via force:
31
1
0
1
21 4 xx
xxqqFE
2011 Classical Electrodynamics Prof. Y. F. Chen
§1.1 Coulomb’s law and electric field
Note 1: If there are many point charges:
i
i xxxxqxE 31
1
041)(
Note 2: If the source is a distribution:
''
')'(4
1)( 33
0
xdxx
xxxxE
Compare the two equations above: i
ii xxqx )(
Note 3: The locality of the charge density could not be precisely determined.
2011 Classical Electrodynamics Prof. Y. F. Chen
§1.1 Coulomb’s law and electric field
§1.2 Electric field and electric potential
(1) For 'xx
0
'3
'3
''
1''
1
''
'1'
'1
33
33
32
xxxx
xxxx
xxxx
xxxx
xxxx
2011 Classical Electrodynamics Prof. Y. F. Chen
(2) For 'xx
4
ˆˆ1
ˆ''
'1'
'1
22
3
332
dRaaR
danxxxx
xdxx
xdxx
RR
Consequently: '4'
12 xxxx
'x
xRaRxx ˆ
2011 Classical Electrodynamics Prof. Y. F. Chen
§1.2 Electric field and electric potential
0
3
0
32
0
33
0
33
0
)(
')'(4)'(4
1''
1)'(4
1
''')'(
41'
'')'(
41
x
xdxxxxdxx
x
xdxxxxxxd
xxxxxE
(3)
Note : The locality of the charge density could be precisely determined.
2011 Classical Electrodynamics Prof. Y. F. Chen
§1.2 Electric field and electric potential
(4) Gauss’s law:
Differential form:0
E
Integral form:00
33 ˆ
QxddanExdE
(5) (i)
''')'(
41 3
30
xdxxxxxE
0'
'''
'3''
353
xx
xxxxxx
xxxxxx
0 E
2011 Classical Electrodynamics Prof. Y. F. Chen
§1.2 Electric field and electric potential
(5) (ii) The electric field can be expressed as the negative gradient of the electric
potential:
'
'1)'(
41'
'')'(
41)( 3
0
33
0
xdxx
xxdxx
xxxxE
Note the curl of the gradient of any well-behaved scalar function of positionvanishes. As a result, we can obtain:
0
E
2011 Classical Electrodynamics Prof. Y. F. Chen
§1.2 Electric field and electric potential
(5) (iii) With Stokes’s theorem:
0ˆ dldldEdanE
0 E
(5) (iv) depends on the central nature of the force between charges, and
on the fact that the force is a function of relative distances only, but does
not depend on the inverse square nature.
0 E
2011 Classical Electrodynamics Prof. Y. F. Chen
§1.2 Electric field and electric potential
§1.3 Discontinuity of electric field and potential(1) The tangential component of the electric field is continuous:
0ˆ ldEdanE
tttt EElEE 2121 0
(2) The discontinuity of the normal component of the electric field means the
existence of the charges at the boundary:
0
3 ˆ
inQdanExdE
0
12 )ˆ(ˆQAnEnE
0012
1
AQEE in
nn
2011 Classical Electrodynamics Prof. Y. F. Chen
E1
E2( ) x
Side 1
Side 2 n
(3) The discontinuity of the electric potential is due to the existence of the dipolelayer, and it can be analogous to the situation of the capacitance.
For the situation of the single layer, the electric potential is continuous, but theelectric field is not, as shown in the left figure.
For the situation of the dipole layer, if the distance d is limited to zero, then theelectric field can be view to be continuous, but the electric potential is not, asshown in the right figure.
single layer
Position (m)-3 -2 -1 0 1 2 3
Elec
tric
field
(V/m
)
-2
0
2
4
6
8
Elec
tric
pote
ntia
l (V
)
-2
0
2
4
6
8
electric fieldelectric potential
dipole layer
Position (m)-2 -1 0 1 2 3
Elec
tric
field
(V/m
)
-4
-2
0
2
4
6
Elec
tric
pote
ntia
l (V
)
0
2
4
6
8
10
electric fieldelectric potential
d
2011 Classical Electrodynamics Prof. Y. F. Chen
§1.3 Discontinuity of electric field and potential
§1. 4 Poisson and Laplace equation
(1)
'
'')'(
41)( 3
30
xdxx
xxxxE
This expression is convenient to be used in the situation of free space or the chargedistribution being point charge.
(2)0
E
EE 00
2
: Poisson equation
If = 0: 02
: Laplace equation
When we deal with the problems involving the boundary condition or finite region,using Poisson equation of Laplace equation together with special mathematicaltechniques (for example, Green function) is a convenient way to solve the problem.
2011 Classical Electrodynamics Prof. Y. F. Chen
§1. 5 Green’s theorem and Green function
(1) Green’s first identity:
2
,let
vfvfvfvf
Integrate the above equation and use the divergence theorem:
xd
dan
danxd
32
3
ˆ
dan
xd 32
2011 Classical Electrodynamics Prof. Y. F. Chen
(2) Green’s second identity (also known as Green’s theorem):
)1......(2
Interchange and :
)2......(2
(1)-(2) and integrate:
dann
xd 322
2011 Classical Electrodynamics Prof. Y. F. Chen
§1. 5 Green’s theorem and Green function
(3) To solve the Poisson equation with the boundary condition, firstly we can solvethe impulse response with the same boundary condition:
)'-(4)' ,(2 xxxxG
With the replacement of: G ,
(i)
)(4')'-()'(4
')' ,(')'(''3
3232
xxdxxx
xdxxGxxd
(ii) ')' ,()'(1')'(')' ,('' 3
0
3232 xdxxGxxdxxxGxd
(iii) ''
)' ,()'(''
dan
xxGxdan
(iv) ''
)'()' ,(''
danxxxGda
n
2011 Classical Electrodynamics Prof. Y. F. Chen
§1. 5 Green’s theorem and Green function
''
)' ,()'(41'
')'()' ,(
41
')' ,()'(4
1)( 3
0
dan
xxGxdanxxxG
xdxxGxx
(4) Advanced discussions:
(i) Free space means no boundary condition:'-
1)' ,(xx
xxG
'
'1)'(
41')' ,()'(
41)( 3
0
3
0
xdxx
xxdxxGxx
2011 Classical Electrodynamics Prof. Y. F. Chen
§1. 5 Green’s theorem and Green function
(ii) For Neumann boundary condition: the simplest case isSn
GN 4'
SNN danxxxGxdxxGxx
''
)'()' ,(41')' ,()'(
41)( 3
0
whereS
daxS
')'(
Note that the electric potential for a point charge is:'4
1)(0 xx
Qx
If the total charge is expressed as the surface charge: ')'( daxQ
'
')'(
41)(
0
daxx
xx
Compare with the term of the red box with the Green function in thefree space:
'-1)' ,(
xxxxG
2011 Classical Electrodynamics Prof. Y. F. Chen
§1. 5 Green’s theorem and Green function
As a consequence, the interpretation of the surface integral of the redbox is the potential due to the surface charge density given above.
' )'(
' 0
0 nx
n
The discontinuities in the electric field across the surface then lead tozero field outside the volume V:
'
'
' ,0
0
0
12
012
n
n
nEE
EE
nn
nn
0
'1
n
E n
02 nE
'n̂Surface
2011 Classical Electrodynamics Prof. Y. F. Chen
§1. 5 Green’s theorem and Green function
(iii) For Dirichlet boundary condition: 0DG
''
)' ,()'(41')' ,()'(
41)( 3
0
dan
xxGxxdxxGxx DD
Note that the electric potential for a dipole is:
dxxQ
xxQx
''4
1)(0
With Taylor expansion: ......'
1'
1'
1
d
xxxxdxx
'1'
4'1
4)(
00 xxdQ
xxdQx
If the total charge is expressed as the surface charge: ')'( daxQ
And define dipole moment: danDddaP 'ˆ
2011 Classical Electrodynamics Prof. Y. F. Chen
§1. 5 Green’s theorem and Green function
'
'1''ˆ
41)(
0
daxx
nDx
Compare with the term of the blue box with the Green function in thefree space:
'-1)' ,(
xxxxG
00
DD
As a consequence, the interpretation of the surface integral of the bluebox is the potential due to the dipole layer D given above.
2011 Classical Electrodynamics Prof. Y. F. Chen
§1. 5 Green’s theorem and Green function
The discontinuities in the electric potential across the surface thenlead to zero potential outside the volume V:
01
02
D
'n̂Surface
0
0
12
012
,0
D
D
D
2011 Classical Electrodynamics Prof. Y. F. Chen
§1. 5 Green’s theorem and Green function
§1. 6 Electrostatic potential energy and energy density
(1) For a point charge, the work done on the charge is given by: )( iii xqW
(2) If the potential is produced by other charges, then the potential is given by:
1
1041)(
N
j ji
ji xx
qx
So that the potential energy of the charge qi is:
1
104
N
j ji
jii xx
qqW
(3) The total potential energy of all the charges due to all the forces acting betweenthem is:
i j ji
jiN
j ij ji
ji
xxqq
xxqq
W 0
1
10 81
41
It is understood that i = j terms (infinite “self-energy” terms) are omittedin the double sum.
2011 Classical Electrodynamics Prof. Y. F. Chen
(4) For a continuous charge distribution:
xdxxxdxdxxxxW 333
0
)()(21'
')'()(
81
(5)
xdEW
EEE
E
320
0
21
&
The potential energy expressed in (5) is definitely nonnegative. Thisseems to contradict our impression from (3) that the potential energy oftwo charges of opposite sign is negative. The reason for this apparentcontradictions is that (5) contains “self-energy” contributions to theenergy density, whereas the double sum in (3) is not.
2011 Classical Electrodynamics Prof. Y. F. Chen
§1. 6 Electrostatic potential energy and energy density